Abstract group 17496.ln: $(C_3^3\times \He_3):D_{12}$

• Label: 17496.ln
• Order: \( 2^{3} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{2} \)
• Perm deg.: $24$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$(C_3^3\times \He_3):D_{12}$
Order:	\(17496\)\(\medspace = 2^{3} \cdot 3^{7} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^6.C_3.D_6^2.C_2^2$, of order \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \)
Composition factors:	$C_2$ x 3, $C_3$ x 7
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	657	2186	486	10278	3888	17496
Conjugacy classes	1	3	74	1	39	8	126
Divisions	1	3	57	1	28	4	94
Autjugacy classes	1	3	32	1	18	2	57

Dimension	1	2	4	6	8	12	16	24	48
Irr. complex chars.	4	13	22	24	24	18	0	21	0	126
Irr. rational chars.	4	7	13	12	14	12	8	21	3	94

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid b^{12}=c^{3}=d^{3}=e^{3}=f^{3}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,3)(2,4,7,12,16,9)(5,6,10,15,13,14)(8,11)(17,18)(19,20,21,22)(23,24), (1,2)(3,5,8,13,17,10)(4,6,9,14,12,15)(7,11)(16,18)(19,20,21,22,23,24)\rangle$
Transitive group:	36T10579	36T10676			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^5:D_6)$ $\,\rtimes\,$ $S_3$	$(C_3^6:D_6)$ $\,\rtimes\,$ $C_2$	$C_3^6$ $\,\rtimes\,$ $(C_3:D_4)$	$C_3^5$ $\,\rtimes\,$ $(C_3:D_{12})$	all 19
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^5$ . $(D_6:S_3)$	$(C_3^5:C_6)$ . $D_6$	$(C_3^5:C_6)$ . $D_6$ (2)	$C_3^5$ . $(C_3:D_{12})$	all 17

Elements of the group are displayed as permutations of degree 24.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{6}$
Commutator length:	$2$

Subgroups

There are 483456 subgroups in 8638 conjugacy classes, 58 normal (40 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $(C_3^3\times \He_3):D_{12}$
Commutator:	$G' \simeq$ $C_3^6.C_6$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_3^4:D_{12}$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2\times C_3^4:C_3$	$G/\operatorname{Fit} \simeq$ $D_4$
Radical:	$R \simeq$ $(C_3^3\times \He_3):D_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $C_3^2:D_{12}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\times C_3^4:C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 17496: $(C_3^3\times \He_3):D_{12}$

Order 8748: $C_3^6.C_6.C_2$, $(C_3^3\times \He_3):C_{12}$, $C_3^6:D_6$

Order 5832: $C_3^6:D_4$, $(C_3^3\times \He_3):D_4$, $C_3^5:D_{12}$

Order 4374: $C_3^3.\He_3.C_6$, $C_3^6.C_6$, $C_3^6.C_6$

Order 2916: $C_3^5.C_6.C_2$ x 3, $C_3^4:C_6^2$ x 2, $C_3^5:C_{12}$ x 2, $C_3^5.D_6$, $C_3^5.C_6.C_2$, $C_3^5:C_{12}$, $C_3^4:C_6^2$, $(C_3^3\times \He_3):C_4$, $C_3^5:D_6$, $C_3^5:D_6$

Order 2187: $C_3^2\times C_3^4:C_3$

Order 1944: $C_3^4:D_{12}$ x 4, $C_3^4:D_{12}$ x 4, $C_3^4:D_{12}$ x 3, $\He_3:\SOPlus(4,2)$ x 3, $C_3^4:D_{12}$, $(C_3\times \He_3):D_{12}$

Order 1458: $C_3^5:C_6$ x 6, $C_3^5:S_3$ x 5, $C_3^5:C_6$ x 5, $C_3^5:C_6$ x 4, $C_3^5:C_6$ x 3, $C_3^5:S_3$ x 2, $C_3^5:C_6$ x 2, $C_3^5:C_6$ x 2, $C_3^3\wr C_2$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:S_3$

Order 972: $C_3^3:S_3^2$ x 15, $C_3^3:S_3^2$ x 12, $C_3^3:S_3^2$ x 7, $C_3^4:C_{12}$ x 6, $C_3^4:C_{12}$ x 6, $C_3^3:C_6^2$ x 4, $C_3^4:C_{12}$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:C_6^2$ x 3, $(C_3^2\times \He_3):C_4$ x 3, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^3:S_3^2$, $C_2\times C_3^4:C_6$, $C_3^4:D_6$, $(C_3\times \He_3):C_{12}$

Order 729: $C_3^5:C_3$ x 20, $C_3^3\times \He_3$ x 6, $C_3^6$

Order 648: $C_3^3:D_{12}$ x 17, $C_3^3:D_{12}$, $C_3^4:D_4$, $C_3^4:D_4$, $C_3^3:D_{12}$, $C_3^3:D_{12}$, $(C_3\times \He_3):D_4$

Order 486: $C_3^4:C_6$ x 56, $C_3^4:C_6$ x 33, $C_3^4:C_6$ x 25, $C_3^4:C_6$ x 18, $C_3^4:C_6$ x 16, $C_3^4:C_6$ x 14, $C_3^4:S_3$ x 14, $C_3^4:C_6$ x 13, $C_3^4:S_3$ x 11, $C_3^4:C_6$ x 8, $C_3^4:C_6$ x 6, $S_3\times C_3^4$ x 2, $C_3^4:S_3$, $C_3^4:S_3$, $C_3^4:C_6$

Order 324: $C_3^2:S_3^2$ x 38, $C_3\wr C_2^2$ x 21, $C_3^2:S_3^2$ x 21, $C_3\wr C_4$ x 18, $C_3^2:S_3^2$ x 16, $C_3^2:S_3^2$ x 8, $C_3^2\times S_3^2$ x 5, $C_3^4:C_4$ x 3, $C_3^3:D_6$ x 3, $C_3^2:S_3^2$ x 3, $C_3^3:C_{12}$ x 2, $C_3^2:C_6^2$ x 2, $C_3^3:D_6$, $C_3^2:C_6^2$, $C_3^3:C_{12}$, $C_3^3:C_{12}$, $C_3^3:D_6$, $(C_3\times \He_3):C_4$

Order 243: $C_3^2\times \He_3$ x 139, $C_3^4:C_3$ x 105, $C_3^5$ x 46

Order 216: $S_3^2:S_3$ x 5, $C_3^2:D_{12}$ x 4, $C_3^2:D_{12}$ x 4, $C_3^2:D_{12}$ x 4, $C_3^2:D_{12}$ x 3, $\He_3:D_4$ x 3, $C_3^2:D_{12}$

Order 162: $C_3^3:C_6$ x 272, $C_3^2\wr C_2$ x 166, $C_3^3:C_6$ x 98, $C_3^3:C_6$ x 84, $S_3\times C_3^3$ x 48, $S_3\times \He_3$ x 48, $C_3^3:S_3$ x 33, $C_3^3:S_3$ x 15, $C_3^3:S_3$ x 11, $C_6\times \He_3$ x 6, $C_3^3\times C_6$

Order 108: $C_3:S_3^2$ x 56, $C_3\times S_3^2$ x 28, $C_3:S_3^2$ x 18, $C_3^2:D_6$ x 15, $C_3^2:D_6$ x 12, $C_3^2:D_6$ x 7, $C_3^2:C_{12}$ x 6, $C_3^3:C_4$ x 6, $C_3^2:C_{12}$ x 6, $C_3^2\times D_6$ x 5, $C_3^2:C_{12}$ x 4, $C_3^2:C_{12}$ x 4, $C_3^2:D_6$ x 3, $\He_3:C_4$ x 3

Order 81: $C_3^4$ x 744, $C_3\times \He_3$ x 718

Order 72: $C_3:D_{12}$ x 17, $\SOPlus(4,2)$, $C_6^2:C_2$, $C_3:D_{12}$, $D_6:S_3$

Order 54: $C_3^2:C_6$ x 429, $C_3^2:S_3$ x 256, $S_3\times C_3^2$ x 240, $C_3^2:C_6$ x 105, $C_2\times \He_3$ x 18, $C_3^2\times C_6$ x 17, $C_3^2:S_3$ x 15

Order 36: $C_6:S_3$ x 38, $C_6\times S_3$ x 26, $S_3^2$ x 25, $C_3:C_{12}$ x 18, $C_3^2:C_4$ x 3, $C_3^2:C_4$, $C_3\times C_{12}$, $C_6^2$

Order 27: $C_3^3$ x 2110, $\He_3$ x 426

Order 24: $C_3:D_4$ x 5, $D_{12}$ x 4

Order 18: $C_3:S_3$ x 274, $C_3\times S_3$ x 212, $C_3\times C_6$ x 58

Order 12: $D_6$ x 20, $C_3:C_4$ x 6, $C_2\times C_6$ x 5, $C_{12}$ x 4

Order 9: $C_3^2$ x 864

Order 8: $D_4$

Order 6: $S_3$ x 50, $C_6$ x 28

Order 4: $C_2^2$ x 2, $C_4$

Order 3: $C_3$ x 57

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 17496: $(C_3^3\times \He_3):D_{12}$

Order 8748: $C_3^6.C_6.C_2$, $(C_3^3\times \He_3):C_{12}$, $C_3^6:D_6$

Order 5832: $C_3^6:D_4$, $(C_3^3\times \He_3):D_4$, $C_3^5:D_{12}$

Order 4374: $C_3^3.\He_3.C_6$, $C_3^6.C_6$, $C_3^6.C_6$

Order 2916: $C_3^5.C_6.C_2$ x 3, $C_3^4:C_6^2$ x 2, $C_3^5:C_{12}$ x 2, $C_3^5.D_6$, $C_3^5.C_6.C_2$, $C_3^5:C_{12}$, $C_3^4:C_6^2$, $(C_3^3\times \He_3):C_4$, $C_3^5:D_6$, $C_3^5:D_6$

Order 2187: $C_3^2\times C_3^4:C_3$

Order 1944: $C_3^4:D_{12}$ x 2, $C_3^4:D_{12}$ x 2, $C_3^4:D_{12}$, $C_3^4:D_{12}$, $\He_3:\SOPlus(4,2)$, $(C_3\times \He_3):D_{12}$

Order 1458: $C_3^5:C_6$ x 6, $C_3^5:S_3$ x 4, $C_3^5:C_6$ x 4, $C_3^5:C_6$ x 3, $C_3^5:C_6$ x 2, $C_3^5:S_3$ x 2, $C_3^5:C_6$ x 2, $C_3^5:C_6$ x 2, $C_3^3\wr C_2$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:S_3$

Order 972: $C_3^3:S_3^2$ x 6, $C_3^3:S_3^2$ x 5, $C_3^3:S_3^2$ x 5, $C_3^4:C_{12}$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:C_6^2$ x 2, $C_3^4:C_{12}$ x 2, $C_3^3:C_6^2$ x 2, $C_3^4:C_{12}$ x 2, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $(C_3^2\times \He_3):C_4$, $C_2\times C_3^4:C_6$, $C_3^4:D_6$, $(C_3\times \He_3):C_{12}$

Order 729: $C_3^5:C_3$ x 16, $C_3^3\times \He_3$ x 6, $C_3^6$

Order 648: $C_3^3:D_{12}$ x 5, $C_3^3:D_{12}$, $C_3^4:D_4$, $C_3^4:D_4$, $C_3^3:D_{12}$, $C_3^3:D_{12}$, $(C_3\times \He_3):D_4$

Order 486: $C_3^4:C_6$ x 20, $C_3^4:C_6$ x 15, $C_3^4:C_6$ x 13, $C_3^4:C_6$ x 12, $C_3^4:C_6$ x 8, $C_3^4:C_6$ x 8, $C_3^4:S_3$ x 8, $C_3^4:S_3$ x 6, $C_3^4:C_6$ x 6, $C_3^4:C_6$ x 6, $C_3^4:C_6$ x 4, $S_3\times C_3^4$ x 2, $C_3^4:S_3$, $C_3^4:S_3$, $C_3^4:C_6$

Order 324: $C_3^2:S_3^2$ x 16, $C_3\wr C_2^2$ x 9, $C_3^2:S_3^2$ x 6, $C_3\wr C_4$ x 6, $C_3^2:S_3^2$ x 5, $C_3^2:S_3^2$ x 5, $C_3^2\times S_3^2$ x 3, $C_3^4:C_4$ x 3, $C_3^3:D_6$ x 3, $C_3^3:C_{12}$ x 2, $C_3^2:C_6^2$ x 2, $C_3^3:D_6$, $C_3^2:C_6^2$, $C_3^3:C_{12}$, $C_3^3:C_{12}$, $C_3^3:D_6$, $C_3^2:S_3^2$, $(C_3\times \He_3):C_4$

Order 243: $C_3^2\times \He_3$ x 68, $C_3^4:C_3$ x 45, $C_3^5$ x 24

Order 216: $S_3^2:S_3$ x 3, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$, $\He_3:D_4$, $C_3^2:D_{12}$

Order 162: $C_3^2\wr C_2$ x 66, $C_3^3:C_6$ x 61, $C_3^3:C_6$ x 28, $S_3\times C_3^3$ x 25, $C_3^3:C_6$ x 22, $C_3^3:S_3$ x 13, $S_3\times \He_3$ x 12, $C_6\times \He_3$ x 6, $C_3^3:S_3$ x 6, $C_3^3:S_3$ x 4, $C_3^3\times C_6$

Order 108: $C_3:S_3^2$ x 16, $C_3\times S_3^2$ x 12, $C_3:S_3^2$ x 10, $C_3^2:D_6$ x 6, $C_3^2:D_6$ x 5, $C_3^2:D_6$ x 5, $C_3^3:C_4$ x 4, $C_3^2:C_{12}$ x 4, $C_3^2\times D_6$ x 3, $C_3^2:C_{12}$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$, $\He_3:C_4$

Order 81: $C_3^4$ x 183, $C_3\times \He_3$ x 170

Order 72: $C_3:D_{12}$ x 5, $\SOPlus(4,2)$, $C_6^2:C_2$, $C_3:D_{12}$, $D_6:S_3$

Order 54: $C_3^2:C_6$ x 121, $S_3\times C_3^2$ x 88, $C_3^2:S_3$ x 51, $C_3^2:C_6$ x 20, $C_3^2\times C_6$ x 11, $C_2\times \He_3$ x 6, $C_3^2:S_3$ x 4

Order 36: $C_6:S_3$ x 16, $C_6\times S_3$ x 12, $S_3^2$ x 12, $C_3:C_{12}$ x 6, $C_3^2:C_4$ x 3, $C_3^2:C_4$, $C_3\times C_{12}$, $C_6^2$

Order 27: $C_3^3$ x 406, $\He_3$ x 70

Order 24: $C_3:D_4$ x 3, $D_{12}$ x 2

Order 18: $C_3\times S_3$ x 84, $C_3:S_3$ x 62, $C_3\times C_6$ x 32

Order 12: $D_6$ x 12, $C_3:C_4$ x 4, $C_2\times C_6$ x 3, $C_{12}$ x 2

Order 9: $C_3^2$ x 235

Order 8: $D_4$

Order 6: $S_3$ x 24, $C_6$ x 18

Order 4: $C_2^2$ x 2, $C_4$

Order 3: $C_3$ x 32

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 17496: $(C_3^3\times \He_3):D_{12}$ ($C_1$)

Order 8748: $C_3^6.C_6.C_2$ ($C_2$), $(C_3^3\times \He_3):C_{12}$ ($C_2$), $C_3^6:D_6$ ($C_2$)

Order 4374: $C_3^6.C_6$ ($C_2^2$)

Order 2916: $C_3^5.D_6$ ($S_3$), $(C_3^3\times \He_3):C_4$ ($S_3$), $C_3^5:D_6$ ($S_3$)

Order 2187: $C_3^2\times C_3^4:C_3$ ($D_4$)

Order 1458: $C_3^5:C_6$ ($D_6$) x 2, $C_3^5:C_6$ ($D_6$)

Order 729: $C_3^6$ ($C_3:D_4$), $C_3^3\times \He_3$ ($C_3:D_4$), $C_3^3\times \He_3$ ($D_{12}$)

Order 486: $C_3^4:C_6$ ($S_3^2$) x 3

Order 243: $C_3^5$ ($C_3:D_{12}$) x 2, $C_3^5$ ($D_6:S_3$), $C_3^4:C_3$ ($\SOPlus(4,2)$)

Order 162: $C_3^2\wr C_2$ ($C_3^2:D_6$) x 6, $C_3^2\wr C_2$ ($C_3:S_3^2$)

Order 81: $C_3^4$ ($C_3^2:D_{12}$) x 3, $C_3^4$ ($\He_3:D_4$) x 3, $C_3^4$ ($S_3^2:S_3$), $C_3^4$ ($C_3^2:D_{12}$), $C_3\times \He_3$ ($C_3^2:D_{12}$), $C_3\times \He_3$ ($S_3^2:S_3$)

Order 54: $C_3^2:C_6$ ($C_3^2:S_3^2$) x 2

Order 27: $C_3^3$ ($C_3^3:D_{12}$) x 2, $C_3^3$ ($C_3^4:D_4$), $C_3^3$ ($C_3^3:D_{12}$), $C_3^3$ ($\He_3:D_{12}$)

Order 18: $C_3:S_3$ ($C_3^3:S_3^2$)

Order 9: $C_3^2$ ($C_3^4:D_{12}$) x 3, $C_3^2$ ($\He_3:\SOPlus(4,2)$) x 3, $C_3^2$ ($C_3^4:D_{12}$), $C_3^2$ ($(C_3\times \He_3):D_{12}$)

Order 3: $C_3$ ($C_3^5.C_3.D_4$), $C_3$ ($C_3.C_3^5.D_4$)

Order 1: $C_1$ ($(C_3^3\times \He_3):D_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 17496: $(C_3^3\times \He_3):D_{12}$ ($C_1$)

Order 8748: $C_3^6.C_6.C_2$ ($C_2$), $(C_3^3\times \He_3):C_{12}$ ($C_2$), $C_3^6:D_6$ ($C_2$)

Order 4374: $C_3^6.C_6$ ($C_2^2$)

Order 2916: $C_3^5.D_6$ ($S_3$), $(C_3^3\times \He_3):C_4$ ($S_3$), $C_3^5:D_6$ ($S_3$)

Order 2187: $C_3^2\times C_3^4:C_3$ ($D_4$)

Order 1458: $C_3^5:C_6$ ($D_6$) x 2, $C_3^5:C_6$ ($D_6$)

Order 729: $C_3^6$ ($C_3:D_4$), $C_3^3\times \He_3$ ($C_3:D_4$), $C_3^3\times \He_3$ ($D_{12}$)

Order 486: $C_3^4:C_6$ ($S_3^2$) x 3

Order 243: $C_3^5$ ($C_3:D_{12}$) x 2, $C_3^5$ ($D_6:S_3$), $C_3^4:C_3$ ($\SOPlus(4,2)$)

Order 162: $C_3^2\wr C_2$ ($C_3^2:D_6$) x 2, $C_3^2\wr C_2$ ($C_3:S_3^2$)

Order 81: $C_3^4$ ($C_3^2:D_{12}$), $C_3^4$ ($\He_3:D_4$), $C_3^4$ ($S_3^2:S_3$), $C_3^4$ ($C_3^2:D_{12}$), $C_3\times \He_3$ ($C_3^2:D_{12}$), $C_3\times \He_3$ ($S_3^2:S_3$)

Order 54: $C_3^2:C_6$ ($C_3^2:S_3^2$) x 2

Order 27: $C_3^3$ ($C_3^3:D_{12}$) x 2, $C_3^3$ ($C_3^4:D_4$), $C_3^3$ ($C_3^3:D_{12}$), $C_3^3$ ($\He_3:D_{12}$)

Order 18: $C_3:S_3$ ($C_3^3:S_3^2$)

Order 9: $C_3^2$ ($C_3^4:D_{12}$), $C_3^2$ ($C_3^4:D_{12}$), $C_3^2$ ($\He_3:\SOPlus(4,2)$), $C_3^2$ ($(C_3\times \He_3):D_{12}$)

Order 3: $C_3$ ($C_3^5.C_3.D_4$), $C_3$ ($C_3.C_3^5.D_4$)

Order 1: $C_1$ ($(C_3^3\times \He_3):D_{12}$)

Series

Derived series

$(C_3^3\times \He_3):D_{12}$

$\rhd$

$(C_3^3\times \He_3):D_{12}$

$\rhd$

$C_3^6.C_6$

$\rhd$

$C_3^6.C_6$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$(C_3^3\times \He_3):D_{12}$

$\rhd$

$(C_3^3\times \He_3):D_{12}$

$\rhd$

$C_3^6.C_6.C_2$

$\rhd$

$C_3^6.C_6.C_2$

$\rhd$

$C_3^6.C_6$

$\rhd$

$C_3^6.C_6$

$\rhd$

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

$\rhd$

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

$\rhd$

$C_3^6$

$\rhd$

$C_3^6$

$\rhd$

$C_3^3\times \He_3$

$\rhd$

$C_3^5$

$\rhd$

$C_3^5$

$\rhd$

$C_3^5$

$\rhd$

$C_3^5$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$(C_3^3\times \He_3):D_{12}$

$\rhd$

$(C_3^3\times \He_3):D_{12}$

$\rhd$

$C_3^6.C_6$

$\rhd$

$C_3^6.C_6$

$\rhd$

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

$\rhd$

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

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 9 larger groups in the database.

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

Character theory

Complex character table

See the $126 \times 126$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $94 \times 94$ rational character table.