Abstract group 10368.df: $C_6^3:(S_3\times D_4)$

• Label: 10368.df
• Order: \( 2^{7} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $20$
• Trans deg.: $24$
• Rank: $4$

Group information

Description:	$C_6^3:(S_3\times D_4)$
Order:	\(10368\)\(\medspace = 2^{7} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2\times C_3^4.C_2^6.C_2^5$, of order \(331776\)\(\medspace = 2^{12} \cdot 3^{4} \)
Composition factors:	$C_2$ x 7, $C_3$ x 4
Derived length:	$3$

This group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	12
Elements	1	1111	80	1320	3344	4512	10368
Conjugacy classes	1	19	12	12	97	36	177
Divisions	1	19	12	12	97	36	177
Autjugacy classes	1	11	8	6	41	14	81

Dimension	1	2	4	8	16
Irr. complex chars.	16	28	48	64	21	177
Irr. rational chars.	16	28	48	64	21	177

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$24$
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	8
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{6}=c^{4}=d^{6}=e^{6}=f^{6}=[b,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(2,4)(3,5)(8,9)(10,12)(11,15)(13,16)(14,17)(19,20), (1,2)(3,5)(10,12)(14,17) \!\cdots\! \rangle$
Transitive group:	24T10008				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(D_6^2:S_3)$ $\,\rtimes\,$ $D_6$	$D_6^2$ $\,\rtimes\,$ $(S_3\times D_6)$	$(C_6^3:D_4)$ $\,\rtimes\,$ $S_3$	$(C_6^3:S_3)$ $\,\rtimes\,$ $D_4$	all 63
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6:D_6)$ . $D_6^2$	$C_6$ . $(D_6^2:D_6)$	$C_6^2$ . $(C_6^2:D_4)$	$C_6^2$ . $(D_6^2:C_2)$	all 34

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 453152 subgroups in 12305 conjugacy classes, 195 normal (55 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^4.C_2^4.C_2^2$
Commutator:	$G' \simeq$ $C_3^4:C_2^3$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2\times S_3^2:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6^3$	$G/\operatorname{Fit} \simeq$ $C_2\times D_4$
Radical:	$R \simeq$ $C_6^3:(S_3\times D_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^3:D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

All subgroups of index up to 48 (order at least 216) are shown, as well as all normal subgroups of any index.

Order 10368: $C_6^3:(S_3\times D_4)$

Order 5184: $C_3^4:C_2\wr C_2^2$ x 2, $C_6.S_3^3:C_4$ x 2, $D_6^2:S_3^2$ x 2, $D_6^2:S_3^2$ x 2, $C_6^3:D_{12}$ x 2, $C_3\times C_6^3:D_4$ x 2, $C_3^4.C_2^4.C_2^2$, $C_6^3:(C_4\times S_3)$, $C_6^2.D_6^2$

Order 3456: $C_6^3:(C_2\times D_4)$ x 2

Order 2592: $(C_2\times C_6).S_3^3$ x 7, $(C_3\times S_3^2):D_{12}$ x 4, $C_6^2.\SOPlus(4,2)$ x 4, $C_6^2:\SOPlus(4,2)$ x 4, $(C_2\times C_6):S_3^3$ x 4, $C_6^2.\SOPlus(4,2)$ x 4, $(C_6\times S_3)\wr C_2$ x 4, $(C_2\times C_6).S_3^3$ x 3, $C_6^3:D_6$ x 3, $(C_6\times S_3^2).D_6$ x 2, $C_2\times S_3^2:S_3^2$ x 2, $C_6:D_6.S_3^2$ x 2, $C_6:D_6.S_3^2$ x 2, $C_6^3:C_{12}$ x 2, $C_3^4.C_2^4.C_2$, $C_3^4.C_2^3.C_2^2$, $C_3^4.C_2^4.C_2$, $C_3^3.C_2^3:D_6$, $C_2\times C_3^4.C_2^3.C_2$, $C_2^2\times C_3^4:C_2^3$, $C_3\times C_3.D_6^2:C_2$, $C_3^2\times D_6^2:C_2$, $C_3^3.C_2^4:S_3$, $C_3^4.C_2^2\wr C_2$, $C_2\times C_3^4.C_2^3.C_2$, $(C_3^3\times C_6).C_2^4$

Order 1728: $D_6^2:D_6$ x 4, $(S_3\times D_6):D_{12}$ x 4, $D_6^2:D_6$ x 4, $(C_2\times S_3^3):C_4$ x 4, $C_2^3:S_3^3$ x 3, $D_6^2:D_6$ x 3, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6^3:(C_2\times C_4)$ x 2, $C_6^3:C_2^3$

Order 1296: $C_6^2:S_3^2$ x 14, $C_6.S_3^3$ x 14, $C_6^2:S_3^2$ x 12, $C_6^2.S_3^2$ x 12, $C_6.S_3^3$ x 12, $C_6:S_3^3$ x 8, $S_3^2:S_3^2$ x 8, $C_2\times C_3^3:D_{12}$ x 4, $C_2\times C_3\wr D_4$ x 4, $C_6^3:C_6$ x 4, $C_6^2.S_3^2$ x 4, $C_6^2:S_3^2$ x 4, $(C_3\times C_6^2):C_{12}$ x 4, $C_6.S_3^3$ x 4, $(C_3^2\times C_6).D_{12}$ x 4, $(C_3\times S_3^2):C_{12}$ x 4, $C_6.S_3^3$ x 3, $C_6^2:S_3^2$ x 2, $C_6:S_3^3$ x 2, $C_3^3:(C_4\times D_6)$ x 2, $C_6^2.S_3^2$ x 2, $C_3^2\wr C_2.D_4$ x 2, $C_6^3:S_3$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2\times S_3^2$, $C_6^3:C_6$, $C_6^3:C_6$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6.S_3^3$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.C_6^2$, $C_6^2.C_6^2$

Order 1152: $D_6^2:D_4$, $D_6^2:D_4$

Order 864: $C_6^3:C_2^2$ x 20, $C_2^2:S_3^3$ x 14, $C_6.D_6^2$ x 14, $C_6.D_6^2$ x 12, $C_6.D_6^2$ x 12, $C_6^3:C_2^2$ x 7, $D_6^2:C_6$ x 7, $D_6^2:S_3$ x 7, $C_2^2.S_3^3$ x 7, $S_3^3:C_2^2$ x 4, $C_6^2:D_{12}$ x 4, $D_6^2:S_3$ x 4, $(S_3\times D_6):D_6$ x 4, $D_6^2:C_6$ x 4, $C_6^2:C_4\times S_3$ x 4, $S_3^2:D_{12}$ x 4, $D_6:\SOPlus(4,2)$ x 4, $S_3^3:C_4$ x 4, $C_6^2.D_{12}$ x 4, $(C_6\times S_3^2):C_4$ x 4, $(C_3\times C_6^2).D_4$ x 4, $(S_3\times D_6):C_{12}$ x 4, $C_6^2.(C_4\times S_3)$ x 4, $S_3\times D_6^2$ x 3, $C_6^3:C_2^2$ x 3, $C_6^3:C_2^2$ x 3, $C_2^2.S_3^3$ x 3, $C_6^3:C_2^2$ x 2, $(C_6\times C_{12}):D_6$ x 2, $(C_6\times C_{12}):D_6$ x 2, $C_6^3:C_4$ x 2, $C_6^3:C_4$ x 2, $C_6:D_6^2$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^2:D_{12}$, $C_6^2:D_{12}$, $C_6^3.C_2^2$

Order 648: $C_3:S_3^3$ x 20, $C_3\times C_6.S_3^2$ x 14, $C_3^4:C_2^3$ x 12, $C_3^4:D_4$ x 12, $C_3^3:D_{12}$ x 12, $C_3^3:D_{12}$ x 12, $C_3^4:C_2^3$ x 8, $C_3^3:D_{12}$ x 8, $C_3\wr D_4$ x 8, $C_3^4:C_2^3$ x 6, $C_3^4:C_2^3$ x 6, $C_3^4:C_2^3$ x 6, $C_3^3:(C_4\times S_3)$ x 6, $C_3^4:C_2^3$ x 4, $C_3^4:C_2^3$ x 4, $C_3^4:C_2^3$ x 4, $C_2\times C_3\wr C_4$ x 4, $C_3:S_3^3$ x 4, $C_3^3:(C_4\times S_3)$ x 4, $C_6\times C_3^2:C_{12}$ x 4, $C_3^4:Q_8$ x 4, $C_3^4:D_4$ x 2, $C_3^4:D_4$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3\times C_6^3$, $C_3^4:C_2^3$, $C_3:C_6^3$, $C_2\times C_3^3:C_{12}$, $C_6\times C_3^2:C_{12}$

Order 576: $D_6^2:C_2^2$ x 5, $D_6^2:C_2^2$ x 4, $D_6^2:C_2^2$ x 4, $D_6^2:C_2^2$ x 4, $D_6^2:C_2^2$ x 2, $C_2^4:S_3^2$ x 2, $D_6^2:C_4$ x 2, $C_3\times C_2^4:D_6$ x 2, $C_2^4:S_3^2$ x 2, $(C_2\times C_{12}):D_{12}$ x 2, $(C_2\times D_6):D_{12}$ x 2, $D_6^2:C_4$ x 2, $C_6^2.C_2^4$, $C_6^2.(C_2^2\times C_4)$, $(C_6\times D_4).D_6$

Order 432: $C_6^2:D_6$ x 124, $C_3:D_6^2$ x 34, $C_2\times S_3^3$ x 34, $C_6^2:D_6$ x 24, $C_6^2.D_6$ x 24, $C_2.S_3^3$ x 24, $D_6:S_3^2$ x 24, $C_6^3:C_2$ x 20, $D_6:C_6^2$ x 20, $C_6^2.D_6$ x 20, $C_6^2.D_6$ x 20, $C_6^2.D_6$ x 20, $S_3^3:C_2$ x 16, $C_6^2:D_6$ x 14, $C_6^2:D_6$ x 14, $D_6:S_3^2$ x 14, $D_6:S_3^2$ x 14, $C_2.S_3^3$ x 14, $C_6^3:C_2$ x 13, $C_6^2.D_6$ x 12, $C_6^2.D_6$ x 12, $C_2.S_3^3$ x 12, $C_2.S_3^3$ x 12, $D_6:S_3^2$ x 12, $C_6^2.D_6$ x 12, $S_3^2:D_6$ x 8, $C_3\times D_6^2$ x 7, $C_6^2.D_6$ x 7, $C_6^2.D_6$ x 7, $C_2\times C_3^2:D_{12}$ x 4, $C_6\times \SOPlus(4,2)$ x 4, $C_3^2:C_4\times D_6$ x 4, $C_6^2:D_6$ x 4, $(C_3\times C_6^2):C_4$ x 4, $C_6^2:C_{12}$ x 4, $C_2.S_3^3$ x 4, $C_6.\SOPlus(4,2)$ x 4, $(C_3\times S_3^2):C_4$ x 4, $C_6.\SOPlus(4,2)$ x 4, $S_3^2:C_{12}$ x 4, $(C_3^2\times D_6):C_4$ x 4, $C_3:D_6^2$ x 3, $C_6^3:C_2$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^3:C_2$, $D_6\times C_6^2$, $C_{12}:C_6^2$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2:C_{12}$, $C_6^2:C_{12}$

Order 384: $C_2\wr C_2^2\times S_3$ x 2

Order 324: $C_3^2:C_6^2$ x 12, $C_3^2:S_3^2$ x 12, $C_3^3:D_6$ x 9, $C_3^2:S_3^2$ x 8, $C_3^2:S_3^2$ x 8, $C_3^2:S_3^2$ x 6, $C_3\wr C_2^2$ x 6, $C_3^2\times S_3^2$ x 6, $C_3^3:C_{12}$ x 6, $C_3^3:D_6$ x 4, $D_6\times C_3^3$ x 4, $C_3\wr C_4$ x 4, $C_3^2\times C_6^2$ x 3, $C_3^3:C_{12}$, $C_3^3:C_{12}$

Order 288: $D_6^2:C_2$ x 36, $D_{12}:D_6$ x 24, $C_2^3:S_3^2$ x 14, $C_6^2:C_2^3$ x 13, $D_4:S_3^2$ x 12, $D_4\times S_3^2$ x 12, $D_6\wr C_2$ x 12, $D_6^2:C_2$ x 10, $D_6:D_{12}$ x 10, $C_6^2:D_4$ x 9, $C_6^2:D_4$ x 9, $C_2^3.S_3^2$ x 9, $D_6^2.C_2$ x 9, $C_6^2:D_4$ x 8, $C_6^2.D_4$ x 8, $C_6^2.C_2^3$ x 7, $D_{12}:D_6$ x 7, $D_{12}:D_6$ x 6, $C_2\times D_6^2$ x 5, $C_6^2:D_4$ x 4, $C_6^2.D_4$ x 4, $C_6^2.D_4$ x 4, $C_2\times C_6^2:C_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2.C_2^3$ x 2, $(C_2\times C_6^2):C_4$ x 2, $(C_6\times C_{12}):C_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:C_2^3$

Order 216: $C_6:S_3^2$ x 190, $C_6^2:S_3$ x 119, $S_3^3$ x 84, $C_6^2:C_6$ x 62, $C_6^2:C_6$ x 62, $C_3^2:D_{12}$ x 62, $C_3^2:D_{12}$ x 62, $C_6.S_3^2$ x 62, $C_6\times S_3^2$ x 52, $C_6^2:C_6$ x 26, $S_3\times C_6^2$ x 26, $C_3^2:D_{12}$ x 24, $C_6.S_3^2$ x 24, $C_6:S_3^2$ x 20, $C_6^2.C_6$ x 20, $C_6.C_6^2$ x 20, $S_3^2:S_3$ x 16, $C_6.S_3^2$ x 14, $C_6.S_3^2$ x 14, $C_6^3$ x 13, $C_3^2:D_{12}$ x 12, $C_3^3:Q_8$ x 12, $C_3^3:D_4$ x 12, $C_3^2:D_{12}$ x 8, $S_3^2:C_6$ x 8, $C_3^2:C_4\times S_3$ x 8, $C_2\times C_3^3:C_4$ x 4, $C_2\times C_3^2:C_{12}$ x 4, $C_6^2:S_3$ x 4, $C_3^3:Q_8$ x 4, $D_4\times C_3^3$ x 2, $C_3^2:D_{12}$ x 2, $C_4\times C_3^2:S_3$ x 2, $C_3\times C_6\times C_{12}$, $C_6^2.S_3$

Order 162: $C_3^3:S_3$ x 2, $C_3^3\times C_6$

Order 144: $S_3^2:C_2^2$ x 8, $D_6:D_6$ x 8, $C_6^2:C_2^2$ x 2, $C_6.D_{12}$, $D_6^2$

Order 128: $C_2^4:D_4$

Order 108: $C_3^2:D_6$ x 4, $C_3\times C_6^2$ x 2

Order 81: $C_3^4$

Order 72: $S_3\times D_6$ x 18, $\SOPlus(4,2)$ x 16, $C_2\times C_6^2$ x 2, $C_6:D_6$ x 2, $C_6.D_6$ x 2

Order 64: $C_2\wr C_2^2$ x 8

Order 54: $C_3^2\times C_6$ x 2

Order 48: $S_3\times D_4$ x 64

Order 36: $S_3^2$ x 40, $C_6:S_3$ x 4, $C_6^2$ x 2

Order 27: $C_3^3$ x 2

Order 24: $C_2\times D_6$ x 12, $C_2^2\times C_6$ x 2

Order 18: $C_3\times C_6$ x 2

Order 16: $C_2\times D_4$ x 12

Order 12: $D_6$ x 212, $C_2\times C_6$ x 2

Order 9: $C_3^2$ x 2

Order 8: $D_4$ x 24, $C_2^3$ x 3

Order 6: $S_3$ x 87, $C_6$ x 2

Order 4: $C_2^2$ x 30

Order 3: $C_3$ x 2

Order 2: $C_2$ x 16

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 48 (order at least 216) are shown, as well as all normal subgroups of any index.

Order 10368: $C_6^3:(S_3\times D_4)$

Order 5184: $C_3^4.C_2^4.C_2^2$, $C_3^4:C_2\wr C_2^2$, $C_6.S_3^3:C_4$, $D_6^2:S_3^2$, $D_6^2:S_3^2$, $C_6^3:(C_4\times S_3)$, $C_6^3:D_{12}$, $C_6^2.D_6^2$, $C_3\times C_6^3:D_4$

Order 3456: $C_6^3:(C_2\times D_4)$

Order 2592: $(C_2\times C_6).S_3^3$ x 3, $(C_2\times C_6).S_3^3$ x 2, $(C_2\times C_6):S_3^3$ x 2, $C_6^3:D_6$ x 2, $C_3^4.C_2^4.C_2$, $C_3^4.C_2^3.C_2^2$, $C_3^4.C_2^4.C_2$, $C_3^3.C_2^3:D_6$, $C_2\times C_3^4.C_2^3.C_2$, $C_2^2\times C_3^4:C_2^3$, $C_3\times C_3.D_6^2:C_2$, $C_3^2\times D_6^2:C_2$, $C_3^3.C_2^4:S_3$, $C_3^4.C_2^2\wr C_2$, $C_2\times C_3^4.C_2^3.C_2$, $(C_3^3\times C_6).C_2^4$, $(C_6\times S_3^2).D_6$, $(C_3\times S_3^2):D_{12}$, $C_2\times S_3^2:S_3^2$, $C_6:D_6.S_3^2$, $C_6:D_6.S_3^2$, $C_6^2.\SOPlus(4,2)$, $C_6^2:\SOPlus(4,2)$, $C_6^2.\SOPlus(4,2)$, $(C_6\times S_3)\wr C_2$, $C_6^3:C_{12}$

Order 1728: $C_2^3:S_3^3$ x 2, $D_6^2:D_6$ x 2, $D_6^2:D_6$, $C_6^3:C_2^3$, $C_6^3:D_4$, $C_6^3:D_4$, $C_6^3:D_4$, $C_6^3:D_4$, $C_6^3:(C_2\times C_4)$, $(S_3\times D_6):D_{12}$, $D_6^2:D_6$, $(C_2\times S_3^3):C_4$

Order 1296: $C_6^2:S_3^2$ x 6, $C_6^2:S_3^2$ x 4, $C_6^2.S_3^2$ x 4, $C_6.S_3^3$ x 4, $C_6.S_3^3$ x 4, $C_6:S_3^3$ x 3, $C_6^3:C_6$ x 3, $C_6^2.S_3^2$ x 3, $C_6^2:S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6^3:S_3$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2\times S_3^2$, $C_6:S_3^3$, $C_2\times C_3^3:D_{12}$, $C_2\times C_3\wr D_4$, $C_3^3:(C_4\times D_6)$, $S_3^2:S_3^2$, $C_6^3:C_6$, $C_6^3:C_6$, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $(C_3\times C_6^2):C_{12}$, $C_6.S_3^3$, $(C_3^2\times C_6).D_{12}$, $(C_3\times S_3^2):C_{12}$, $C_3^2\wr C_2.D_4$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.C_6^2$, $C_6^2.C_6^2$

Order 1152: $D_6^2:D_4$, $D_6^2:D_4$

Order 864: $C_6^3:C_2^2$ x 12, $C_6^3:C_2^2$ x 4, $C_2^2:S_3^3$ x 4, $C_6.D_6^2$ x 4, $C_6.D_6^2$ x 4, $C_6.D_6^2$ x 4, $D_6^2:C_6$ x 4, $D_6^2:S_3$ x 4, $C_2^2.S_3^3$ x 4, $S_3\times D_6^2$ x 2, $C_6^3:C_2^2$ x 2, $C_6^3:C_2^2$ x 2, $C_6^3:C_2^2$ x 2, $(C_6\times C_{12}):D_6$ x 2, $(C_6\times C_{12}):D_6$ x 2, $C_2^2.S_3^3$ x 2, $C_6:D_6^2$, $S_3^3:C_2^2$, $C_6^3:C_2^2$, $C_6^2:D_{12}$, $D_6^2:S_3$, $(S_3\times D_6):D_6$, $D_6^2:C_6$, $C_6^2:C_4\times S_3$, $S_3^2:D_{12}$, $D_6:\SOPlus(4,2)$, $S_3^3:C_4$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^2:D_{12}$, $C_6^2:D_{12}$, $C_6^3.C_2^2$, $C_6^3:C_4$, $C_6^3:C_4$, $C_6^2.D_{12}$, $(C_6\times S_3^2):C_4$, $(C_3\times C_6^2).D_4$, $(S_3\times D_6):C_{12}$, $C_6^2.(C_4\times S_3)$

Order 648: $C_3^4:C_2^3$ x 6, $C_3^4:C_2^3$ x 5, $C_3^4:C_2^3$ x 4, $C_3:S_3^3$ x 4, $C_3^4:D_4$ x 4, $C_3^3:D_{12}$ x 4, $C_3^3:(C_4\times S_3)$ x 4, $C_3^3:D_{12}$ x 4, $C_3\times C_6.S_3^2$ x 4, $C_3^4:C_2^3$ x 3, $C_3^4:C_2^3$ x 3, $C_6\times C_3^2:C_{12}$ x 3, $C_3^4:C_2^3$ x 2, $C_3^4:C_2^3$ x 2, $C_3^4:C_2^3$ x 2, $C_3^4:Q_8$ x 2, $C_3\times C_6^3$, $C_3^4:C_2^3$, $C_3:C_6^3$, $C_2\times C_3\wr C_4$, $C_3:S_3^3$, $C_3^3:D_{12}$, $C_3\wr D_4$, $C_3^3:(C_4\times S_3)$, $C_3^4:D_4$, $C_2\times C_3^3:C_{12}$, $C_3^4:D_4$, $C_6\times C_3^2:C_{12}$, $C_3^3:D_{12}$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$

Order 576: $D_6^2:C_2^2$ x 3, $D_6^2:C_2^2$ x 3, $D_6^2:C_2^2$ x 2, $D_6^2:C_2^2$ x 2, $D_6^2:C_2^2$, $C_6^2.C_2^4$, $C_2^4:S_3^2$, $C_6^2.(C_2^2\times C_4)$, $D_6^2:C_4$, $C_3\times C_2^4:D_6$, $C_2^4:S_3^2$, $(C_2\times C_{12}):D_{12}$, $(C_6\times D_4).D_6$, $(C_2\times D_6):D_{12}$, $D_6^2:C_4$

Order 432: $C_6^2:D_6$ x 25, $C_3:D_6^2$ x 18, $C_6^3:C_2$ x 12, $D_6:C_6^2$ x 12, $C_6^2.D_6$ x 12, $C_6^2.D_6$ x 12, $C_6^2.D_6$ x 12, $C_6^3:C_2$ x 9, $C_2\times S_3^3$ x 8, $C_6^2:D_6$ x 6, $C_6^2.D_6$ x 6, $C_2.S_3^3$ x 6, $D_6:S_3^2$ x 6, $C_6^2.D_6$ x 6, $C_3\times D_6^2$ x 4, $C_6^2:D_6$ x 4, $C_6^2.D_6$ x 4, $C_6^2:D_6$ x 4, $C_6^2.D_6$ x 4, $C_2.S_3^3$ x 4, $C_2.S_3^3$ x 4, $D_6:S_3^2$ x 4, $D_6:S_3^2$ x 4, $D_6:S_3^2$ x 4, $C_2.S_3^3$ x 4, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_3:D_6^2$ x 2, $S_3^2:D_6$ x 2, $C_6^3:C_2$ x 2, $C_2.S_3^3$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^3:C_2$, $D_6\times C_6^2$, $C_2\times C_3^2:D_{12}$, $C_6\times \SOPlus(4,2)$, $C_3^2:C_4\times D_6$, $S_3^3:C_2$, $C_{12}:C_6^2$, $C_6^2:D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $(C_3\times C_6^2):C_4$, $C_6^2:C_{12}$, $C_6.\SOPlus(4,2)$, $(C_3\times S_3^2):C_4$, $C_6.\SOPlus(4,2)$, $S_3^2:C_{12}$, $(C_3^2\times D_6):C_4$, $C_6^2:C_{12}$, $C_6^2:C_{12}$

Order 384: $C_2\wr C_2^2\times S_3$

Order 324: $C_3^3:D_6$ x 7, $C_3^2:C_6^2$ x 6, $C_3^2:S_3^2$ x 4, $C_3^2\times C_6^2$ x 3, $C_3^3:D_6$ x 3, $D_6\times C_3^3$ x 3, $C_3^3:C_{12}$ x 3, $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3\wr C_2^2$ x 2, $C_3^2\times S_3^2$ x 2, $C_3^2:S_3^2$, $C_3\wr C_4$, $C_3^3:C_{12}$, $C_3^3:C_{12}$

Order 288: $D_6^2:C_2$ x 19, $C_6^2:C_2^3$ x 8, $C_2^3:S_3^2$ x 7, $D_{12}:D_6$ x 6, $D_6^2:C_2$ x 6, $D_6:D_{12}$ x 6, $C_6^2:D_4$ x 5, $C_6^2:D_4$ x 5, $C_2^3.S_3^2$ x 5, $D_6^2.C_2$ x 5, $C_6^2.C_2^3$ x 4, $D_4:S_3^2$ x 4, $D_4\times S_3^2$ x 4, $D_{12}:D_6$ x 3, $D_{12}:D_6$ x 3, $D_6\wr C_2$ x 3, $C_2\times D_6^2$ x 3, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2.C_2^3$ x 2, $C_6^2:D_4$ x 2, $C_6^2.D_4$ x 2, $C_2\times C_6^2:C_4$, $C_6^2.D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $(C_2\times C_6^2):C_4$, $(C_6\times C_{12}):C_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2:C_2^3$, $C_6^2:D_4$

Order 216: $C_6:S_3^2$ x 52, $C_6^2:S_3$ x 50, $C_6.S_3^2$ x 19, $C_6^2:C_6$ x 15, $S_3\times C_6^2$ x 15, $C_6^2:C_6$ x 15, $C_6^2:C_6$ x 15, $C_3^2:D_{12}$ x 15, $C_3^2:D_{12}$ x 15, $C_6\times S_3^2$ x 14, $C_6^2.C_6$ x 12, $C_6.C_6^2$ x 12, $S_3^3$ x 10, $C_6^3$ x 9, $C_6:S_3^2$ x 6, $C_3^2:D_{12}$ x 6, $C_6.S_3^2$ x 6, $C_3^2:D_{12}$ x 4, $C_6.S_3^2$ x 4, $C_3^3:Q_8$ x 4, $C_3^3:D_4$ x 4, $C_6.S_3^2$ x 4, $S_3^2:S_3$ x 2, $C_6^2:S_3$ x 2, $C_3^3:Q_8$ x 2, $C_2\times C_3^3:C_4$, $C_2\times C_3^2:C_{12}$, $C_3^2:D_{12}$, $S_3^2:C_6$, $C_3^2:C_4\times S_3$, $D_4\times C_3^3$, $C_3\times C_6\times C_{12}$, $C_6^2.S_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$

Order 162: $C_3^3\times C_6$, $C_3^3:S_3$

Order 144: $C_6^2:C_2^2$ x 2, $C_6.D_{12}$, $D_6^2$, $S_3^2:C_2^2$, $D_6:D_6$

Order 128: $C_2^4:D_4$

Order 108: $C_3\times C_6^2$, $C_3^2:D_6$

Order 81: $C_3^4$

Order 72: $S_3\times D_6$ x 3, $C_2\times C_6^2$ x 2, $C_6:D_6$ x 2, $\SOPlus(4,2)$ x 2, $C_6.D_6$ x 2

Order 64: $C_2\wr C_2^2$ x 4

Order 54: $C_3^2\times C_6$

Order 48: $S_3\times D_4$ x 5

Order 36: $S_3^2$ x 7, $C_6^2$ x 2, $C_6:S_3$ x 2

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 2, $C_2^2\times C_6$

Order 18: $C_3\times C_6$ x 2

Order 16: $C_2\times D_4$ x 2

Order 12: $D_6$ x 31, $C_2\times C_6$

Order 9: $C_3^2$ x 2

Order 8: $D_4$ x 6, $C_2^3$ x 2

Order 6: $S_3$ x 27, $C_6$

Order 4: $C_2^2$ x 10

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 10368: $C_6^3:(S_3\times D_4)$ ($C_1$)

Order 5184: $C_3^4:C_2\wr C_2^2$ ($C_2$) x 2, $C_6.S_3^3:C_4$ ($C_2$) x 2, $D_6^2:S_3^2$ ($C_2$) x 2, $D_6^2:S_3^2$ ($C_2$) x 2, $C_6^3:D_{12}$ ($C_2$) x 2, $C_3\times C_6^3:D_4$ ($C_2$) x 2, $C_3^4.C_2^4.C_2^2$ ($C_2$), $C_6^3:(C_4\times S_3)$ ($C_2$), $C_6^2.D_6^2$ ($C_2$)

Order 2592: $C_6^2.\SOPlus(4,2)$ ($C_2^2$) x 4, $C_6^2:\SOPlus(4,2)$ ($C_2^2$) x 4, $C_6^2.\SOPlus(4,2)$ ($C_2^2$) x 4, $(C_6\times S_3)\wr C_2$ ($C_2^2$) x 4, $C_6:D_6.S_3^2$ ($C_2^2$) x 2, $C_6:D_6.S_3^2$ ($C_2^2$) x 2, $(C_2\times C_6).S_3^3$ ($C_2^2$) x 2, $(C_2\times C_6):S_3^3$ ($C_2^2$) x 2, $C_6^3:C_{12}$ ($C_2^2$) x 2, $C_6^3:D_6$ ($C_2^2$) x 2, $C_3^4.C_2^3.C_2^2$ ($C_2^2$), $C_3^3.C_2^3:D_6$ ($C_2^2$), $C_2^2\times C_3^4:C_2^3$ ($C_2^2$), $C_3^2\times D_6^2:C_2$ ($C_2^2$), $C_3^3.C_2^4:S_3$ ($C_2^2$), $C_3^4.C_2^2\wr C_2$ ($C_2^2$), $C_2\times C_3^4.C_2^3.C_2$ ($C_2^2$)

Order 1728: $C_6^3:D_4$ ($S_3$) x 2

Order 1296: $C_6^2:S_3^2$ ($C_2^3$) x 4, $(C_3\times C_6^2):C_{12}$ ($C_2^3$) x 4, $C_6.S_3^3$ ($D_4$) x 4, $C_6:S_3^3$ ($D_4$) x 2, $C_6.S_3^3$ ($D_4$) x 2, $C_6^3:S_3$ ($D_4$), $C_6^2:S_3^2$ ($C_2^3$), $C_6^2:S_3^2$ ($C_2^3$), $C_6^2\times S_3^2$ ($C_2^3$), $C_6^3:C_6$ ($C_2^3$), $C_6^3:C_6$ ($D_4$), $C_6^2.S_3^2$ ($C_2^3$), $C_6^2.S_3^2$ ($D_4$), $C_6^2.S_3^2$ ($D_4$), $C_6^2.S_3^2$ ($C_2^3$), $C_6^2.C_6^2$ ($C_2^3$)

Order 864: $D_6^2:S_3$ ($D_6$) x 4, $(C_3\times C_6^2).D_4$ ($D_6$) x 4, $C_6^3:C_2^2$ ($D_6$) x 2, $D_6^2:C_6$ ($D_6$) x 2, $C_6^3:C_4$ ($D_6$) x 2

Order 648: $C_3^4:C_2^3$ ($C_2\times D_4$) x 4, $C_3^4:C_2^3$ ($C_2\times D_4$) x 4, $C_3\times C_6.S_3^2$ ($C_2\times D_4$) x 4, $C_3^4:C_2^3$ ($C_2\times D_4$) x 2, $C_6\times C_3^2:C_{12}$ ($C_2\times D_4$) x 2, $C_3\times C_6^3$ ($C_2\times D_4$), $C_3^4:C_2^3$ ($C_2^4$), $C_3^4:C_2^3$ ($C_2\times D_4$)

Order 432: $C_6^2:D_6$ ($C_2\times D_6$) x 4, $(C_3\times C_6^2):C_4$ ($C_2\times D_6$) x 4, $C_3\times D_6^2$ ($C_2\times D_6$) x 2, $C_6^3:C_2$ ($C_2\times D_6$) x 2, $C_6^2.D_6$ ($C_2\times D_6$) x 2

Order 324: $C_3^3:D_6$ ($C_2^2\wr C_2$) x 2, $C_3^2:C_6^2$ ($C_2^2\times D_4$) x 2, $C_3^2:C_6^2$ ($C_2^2\wr C_2$) x 2, $C_3^2\times C_6^2$ ($C_2^2\times D_4$)

Order 288: $D_6^2:C_2$ ($S_3^2$)

Order 216: $C_6:S_3^2$ ($S_3\times D_4$) x 4, $C_6.S_3^2$ ($S_3\times D_4$) x 4, $C_6^3$ ($S_3\times D_4$) x 2, $C_6^2:C_6$ ($C_2^2\times D_6$) x 2, $C_6^2.C_6$ ($S_3\times D_4$) x 2

Order 162: $C_3^3:S_3$ ($C_2\wr C_2^2$) x 2, $C_3^3\times C_6$ ($C_2^3:D_4$)

Order 144: $C_6.D_{12}$ ($S_3\times D_6$), $D_6^2$ ($S_3\times D_6$), $C_6^2:C_2^2$ ($S_3\times D_6$), $C_6^2:C_2^2$ ($\SOPlus(4,2)$)

Order 108: $C_3^2:D_6$ ($D_4\times D_6$) x 4, $C_3\times C_6^2$ ($D_4\times D_6$) x 2

Order 81: $C_3^4$ ($C_2^4:D_4$)

Order 72: $C_2\times C_6^2$ ($S_3^2:C_2^2$), $C_2\times C_6^2$ ($D_6:D_6$), $C_6:D_6$ ($D_6^2$), $C_6:D_6$ ($S_3^2:C_2^2$), $C_6.D_6$ ($S_3^2:C_2^2$), $C_6.D_6$ ($D_6:D_6$)

Order 54: $C_3^2\times C_6$ ($C_2^4:D_6$) x 2

Order 36: $C_6:S_3$ ($D_4\times S_3^2$) x 2, $C_6:S_3$ ($D_6\wr C_2$) x 2, $C_6^2$ ($D_6^2:C_2$), $C_6^2$ ($C_6^2:D_4$)

Order 27: $C_3^3$ ($C_2\wr C_2^2\times S_3$) x 2

Order 24: $C_2^2\times C_6$ ($S_3^3:C_2$) x 2

Order 18: $C_3\times C_6$ ($D_6^2:C_2^2$), $C_3\times C_6$ ($D_6^2:C_2^2$)

Order 12: $C_2\times C_6$ ($S_3^3:C_2^2$) x 2

Order 9: $C_3^2$ ($D_6^2:D_4$), $C_3^2$ ($D_6^2:D_4$)

Order 8: $C_2^3$ ($S_3^2:S_3^2$)

Order 6: $C_6$ ($D_6^2:D_6$) x 2

Order 4: $C_2^2$ ($C_2\times S_3^2:S_3^2$)

Order 3: $C_3$ ($C_6^3:(C_2\times D_4)$) x 2

Order 2: $C_2$ ($C_3^4.C_2^4.C_2^2$)

Order 1: $C_1$ ($C_6^3:(S_3\times D_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 10368: $C_6^3:(S_3\times D_4)$ ($C_1$)

Order 5184: $C_3^4.C_2^4.C_2^2$ ($C_2$), $C_3^4:C_2\wr C_2^2$ ($C_2$), $C_6.S_3^3:C_4$ ($C_2$), $D_6^2:S_3^2$ ($C_2$), $D_6^2:S_3^2$ ($C_2$), $C_6^3:(C_4\times S_3)$ ($C_2$), $C_6^3:D_{12}$ ($C_2$), $C_6^2.D_6^2$ ($C_2$), $C_3\times C_6^3:D_4$ ($C_2$)

Order 2592: $C_3^4.C_2^3.C_2^2$ ($C_2^2$), $C_3^3.C_2^3:D_6$ ($C_2^2$), $C_2^2\times C_3^4:C_2^3$ ($C_2^2$), $C_3^2\times D_6^2:C_2$ ($C_2^2$), $C_3^3.C_2^4:S_3$ ($C_2^2$), $C_3^4.C_2^2\wr C_2$ ($C_2^2$), $C_2\times C_3^4.C_2^3.C_2$ ($C_2^2$), $C_6:D_6.S_3^2$ ($C_2^2$), $C_6:D_6.S_3^2$ ($C_2^2$), $C_6^2.\SOPlus(4,2)$ ($C_2^2$), $C_6^2:\SOPlus(4,2)$ ($C_2^2$), $(C_2\times C_6).S_3^3$ ($C_2^2$), $(C_2\times C_6):S_3^3$ ($C_2^2$), $C_6^2.\SOPlus(4,2)$ ($C_2^2$), $(C_6\times S_3)\wr C_2$ ($C_2^2$), $C_6^3:C_{12}$ ($C_2^2$), $C_6^3:D_6$ ($C_2^2$)

Order 1728: $C_6^3:D_4$ ($S_3$)

Order 1296: $C_6^3:S_3$ ($D_4$), $C_6^2:S_3^2$ ($C_2^3$), $C_6^2:S_3^2$ ($C_2^3$), $C_6^2\times S_3^2$ ($C_2^3$), $C_6:S_3^3$ ($D_4$), $C_6^3:C_6$ ($C_2^3$), $C_6^3:C_6$ ($D_4$), $C_6^2.S_3^2$ ($C_2^3$), $C_6^2.S_3^2$ ($D_4$), $C_6^2.S_3^2$ ($D_4$), $C_6^2:S_3^2$ ($C_2^3$), $(C_3\times C_6^2):C_{12}$ ($C_2^3$), $C_6.S_3^3$ ($D_4$), $C_6.S_3^3$ ($D_4$), $C_6^2.S_3^2$ ($C_2^3$), $C_6^2.C_6^2$ ($C_2^3$)

Order 864: $C_6^3:C_2^2$ ($D_6$), $D_6^2:S_3$ ($D_6$), $D_6^2:C_6$ ($D_6$), $C_6^3:C_4$ ($D_6$), $(C_3\times C_6^2).D_4$ ($D_6$)

Order 648: $C_3^4:C_2^3$ ($C_2\times D_4$) x 2, $C_3^4:C_2^3$ ($C_2\times D_4$) x 2, $C_6\times C_3^2:C_{12}$ ($C_2\times D_4$) x 2, $C_3\times C_6^3$ ($C_2\times D_4$), $C_3^4:C_2^3$ ($C_2^4$), $C_3^4:C_2^3$ ($C_2\times D_4$), $C_3^4:C_2^3$ ($C_2\times D_4$), $C_3\times C_6.S_3^2$ ($C_2\times D_4$)

Order 432: $C_3\times D_6^2$ ($C_2\times D_6$), $C_6^3:C_2$ ($C_2\times D_6$), $C_6^2:D_6$ ($C_2\times D_6$), $(C_3\times C_6^2):C_4$ ($C_2\times D_6$), $C_6^2.D_6$ ($C_2\times D_6$)

Order 324: $C_3^3:D_6$ ($C_2^2\wr C_2$) x 2, $C_3^2\times C_6^2$ ($C_2^2\times D_4$), $C_3^2:C_6^2$ ($C_2^2\times D_4$), $C_3^2:C_6^2$ ($C_2^2\wr C_2$)

Order 288: $D_6^2:C_2$ ($S_3^2$)

Order 216: $C_6^3$ ($S_3\times D_4$), $C_6^2:C_6$ ($C_2^2\times D_6$), $C_6:S_3^2$ ($S_3\times D_4$), $C_6^2.C_6$ ($S_3\times D_4$), $C_6.S_3^2$ ($S_3\times D_4$)

Order 162: $C_3^3\times C_6$ ($C_2^3:D_4$), $C_3^3:S_3$ ($C_2\wr C_2^2$)

Order 144: $C_6.D_{12}$ ($S_3\times D_6$), $D_6^2$ ($S_3\times D_6$), $C_6^2:C_2^2$ ($S_3\times D_6$), $C_6^2:C_2^2$ ($\SOPlus(4,2)$)

Order 108: $C_3\times C_6^2$ ($D_4\times D_6$), $C_3^2:D_6$ ($D_4\times D_6$)

Order 81: $C_3^4$ ($C_2^4:D_4$)

Order 72: $C_2\times C_6^2$ ($S_3^2:C_2^2$), $C_2\times C_6^2$ ($D_6:D_6$), $C_6:D_6$ ($D_6^2$), $C_6:D_6$ ($S_3^2:C_2^2$), $C_6.D_6$ ($S_3^2:C_2^2$), $C_6.D_6$ ($D_6:D_6$)

Order 54: $C_3^2\times C_6$ ($C_2^4:D_6$)

Order 36: $C_6^2$ ($D_6^2:C_2$), $C_6^2$ ($C_6^2:D_4$), $C_6:S_3$ ($D_4\times S_3^2$), $C_6:S_3$ ($D_6\wr C_2$)

Order 27: $C_3^3$ ($C_2\wr C_2^2\times S_3$)

Order 24: $C_2^2\times C_6$ ($S_3^3:C_2$)

Order 18: $C_3\times C_6$ ($D_6^2:C_2^2$), $C_3\times C_6$ ($D_6^2:C_2^2$)

Order 12: $C_2\times C_6$ ($S_3^3:C_2^2$)

Order 9: $C_3^2$ ($D_6^2:D_4$), $C_3^2$ ($D_6^2:D_4$)

Order 8: $C_2^3$ ($S_3^2:S_3^2$)

Order 6: $C_6$ ($D_6^2:D_6$)

Order 4: $C_2^2$ ($C_2\times S_3^2:S_3^2$)

Order 3: $C_3$ ($C_6^3:(C_2\times D_4)$)

Order 2: $C_2$ ($C_3^4.C_2^4.C_2^2$)

Order 1: $C_1$ ($C_6^3:(S_3\times D_4)$)

Series

Derived series

$C_6^3:(S_3\times D_4)$

$\rhd$

$C_3^4:C_2^3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_6^3:(S_3\times D_4)$

$\rhd$

$D_6^2:S_3^2$

$\rhd$

$C_2^2\times C_3^4:C_2^3$

$\rhd$

$C_6^2:S_3^2$

$\rhd$

$C_3^4:C_2^3$

$\rhd$

$C_3^2:C_6^2$

$\rhd$

$C_3^3\times C_6$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^3:(S_3\times D_4)$

$\rhd$

$C_3^4:C_2^3$

$\rhd$

$C_3^3\times C_6$

$\rhd$

$C_3^4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $177 \times 177$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.