Abstract group 1728.34562: $(C_2\times C_4).S_3^3$

• Label: 1728.34562
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $25$
• Trans deg.: $96$
• Rank: $4$

Group information

Description:	$(C_2\times C_4).S_3^3$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	Group of order \(55296\)\(\medspace = 2^{11} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6	12
Elements	1	195	26	316	318	872	1728
Conjugacy classes	1	9	7	15	29	44	105
Divisions	1	9	7	11	29	28	85
Autjugacy classes	1	7	7	11	26	26	78

Dimension	1	2	4	8	16
Irr. complex chars.	16	32	43	14	0	105
Irr. rational chars.	16	24	21	21	3	85

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$96$
Rank:	$4$
Inequivalent generating quadruples:	$44291520$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid b^{6}=c^{12}=d^{12}=[a,c]=1, a^{2}=d^{6}, b^{a}=b^{5}c^{6}, d^{a}=c^{6}d^{7}, c^{b}=c^{5}, d^{b}=c^{6}d, d^{c}=d^{11} \rangle$
Permutation group:	Degree $25$ $\langle(15,16)(17,20)(18,22)(19,21)(24,25), (1,2)(3,6)(4,5)(9,13)(11,14)(15,17,19,22) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(D_6.D_{12})$ $\,\rtimes\,$ $S_3$	$(C_6.D_{12})$ $\,\rtimes\,$ $D_6$	$(C_6.D_{12})$ $\,\rtimes\,$ $D_6$	$(C_6.D_{12})$ $\,\rtimes\,$ $D_6$ (3)	all 47
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_6.D_6)$ . $D_6$ (2)	$D_6$ . $(D_{12}:S_3)$	$(C_6:D_{12})$ . $D_6$	$(D_6:C_{12})$ . $D_6$ (2)	all 66

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

Homology

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

Subgroups

There are 12932 subgroups in 1332 conjugacy classes, 174 normal (164 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2\times S_3^3$
Commutator:	$G' \simeq$ $C_3\times C_6^2$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2\times S_3^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6\times C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2^3$
Radical:	$R \simeq$ $(C_2\times C_4).S_3^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

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 1728: $(C_2\times C_4).S_3^3$

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

Order 576: $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$

Order 432: $C_6^2.D_6$ x 3, $C_6^2.D_6$ x 3, $C_6^2.D_6$ x 3, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_3:D_6^2$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_3^3:C_4^2$

Order 288: $D_6^2.C_2$ x 3, $C_6^2.C_2^3$ x 3, $C_6^2.C_2^3$ x 3, $C_6^2.C_2^3$ x 2, $D_6.D_{12}$ x 2, $C_6^2.C_2^3$ x 2, $D_6.D_{12}$ x 2, $C_6^2.C_2^3$ x 2, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_{12}.D_{12}$, $C_6^2.C_2^3$, $C_6^2.D_4$, $C_6^2:Q_8$, $C_6^2:D_4$, $C_2^3.S_3^2$, $C_6^2.D_4$, $C_2^3.S_3^2$, $D_6:D_{12}$, $D_6:D_{12}$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $D_6.D_{12}$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_{12}.D_{12}$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$

Order 216: $C_6:S_3^2$ x 5, $C_6.C_6^2$ x 5, $C_6^2.C_6$ x 4, $C_6.S_3^2$ x 3, $C_6^2:C_6$ x 2, $C_4\times C_3^2:S_3$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$ x 2, $C_6^2:S_3$, $S_3\times C_6^2$, $C_3\times C_6\times C_{12}$, $C_6^2.S_3$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_6.S_3^2$, $C_3^3:Q_8$

Order 192: $C_4^2:D_6$, $(C_2\times D_4).D_6$, $(C_2\times Q_8):D_6$

Order 144: $D_6:C_{12}$ x 16, $C_6.D_{12}$ x 13, $C_6.D_{12}$ x 12, $C_{12}:C_{12}$ x 9, $C_6.D_{12}$ x 8, $C_{12}:D_6$ x 8, $C_6.D_{12}$ x 7, $C_3^2:C_4^2$ x 7, $C_6^2.C_2^2$ x 6, $C_6:D_{12}$ x 6, $C_2^2.S_3^2$ x 4, $D_6.D_6$ x 4, $D_6^2$ x 3, $C_3^2:C_4^2$ x 2, $C_{12}:C_{12}$ x 2, $C_6^2:C_4$ x 2, $C_{12}\times D_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}:C_{12}$ x 2, $C_6.D_{12}$, $C_6^2.C_2^2$, $C_6^2:C_4$, $C_2^2.S_3^2$, $C_6^2:C_2^2$, $C_6:D_{12}$, $C_6^2:C_2^2$, $C_6\times D_{12}$, $C_2^2.S_3^2$, $D_6:D_6$, $C_6^2:C_4$, $C_6^2:C_4$

Order 108: $C_3^2:D_6$ x 6, $C_3^2:C_{12}$ x 5, $C_3^2:D_6$ x 4, $C_3^2\times D_6$ x 4, $C_3:S_3^2$ x 4, $C_3^2:C_{12}$ x 4, $C_3\times C_6^2$, $C_3^2\times C_{12}$, $C_3^3:C_4$

Order 96: $D_6.D_4$ x 5, $D_6.D_4$ x 5, $(C_4\times S_3):C_4$ x 4, $(C_2\times C_4).D_6$ x 4, $C_2^2.D_{12}$ x 2, $C_2^3.D_6$ x 2, $D_6.D_4$ x 2, $C_2^2:D_{12}$ x 2, $C_2^2.D_{12}$ x 2, $D_6:Q_8$ x 2, $D_{12}:C_4$ x 2, $C_2^3.D_6$, $C_2^3.D_6$, $C_2^3.D_6$, $C_2^3.D_6$, $C_4\times D_{12}$, $C_4^2:S_3$, $C_4^2:C_6$, $(C_2^2\times C_{12}):C_2$, $C_{12}.D_4$, $C_{12}.D_4$, $D_6:Q_8$, $D_6:D_4$, $C_2^3.D_6$, $C_{12}:D_4$, $C_2^3.D_6$, $D_6:Q_8$

Order 72: $C_6:C_{12}$ x 30, $C_{12}:S_3$ x 19, $S_3\times D_6$ x 18, $C_6:D_6$ x 14, $C_6\times C_{12}$ x 12, $C_6.D_6$ x 11, $C_3:D_{12}$ x 10, $C_6\times D_6$ x 9, $C_6.D_6$ x 5, $C_6.D_6$ x 5, $S_3\times C_{12}$ x 3, $C_3:D_{12}$ x 2, $C_6\wr C_2$ x 2, $C_3\times D_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_2\times C_6^2$, $C_3^2:Q_8$, $D_6:S_3$

Order 64: $C_4^2:C_2^2$

Order 54: $C_3^2\times C_6$ x 3, $C_3^2:S_3$ x 2, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2

Order 48: $D_6:C_4$ x 35, $C_4\times D_6$ x 15, $C_4:C_{12}$ x 10, $C_{12}:C_4$ x 10, $C_2^2:C_{12}$ x 9, $C_{12}:C_4$ x 9, $C_6.D_4$ x 9, $C_6.D_4$ x 7, $C_2\times D_{12}$ x 5, $C_2^2\times D_6$ x 4, $C_6:D_4$ x 4, $C_6.C_2^3$ x 3, $C_2^2\times C_{12}$ x 2, $C_6:Q_8$ x 2, $C_6\times Q_8$, $C_6\times D_4$, $C_4\times C_{12}$

Order 36: $C_6:S_3$ x 42, $C_6\times S_3$ x 31, $C_3:C_{12}$ x 27, $S_3^2$ x 16, $C_3\times C_{12}$ x 14, $C_3^2:C_4$ x 13, $C_6^2$ x 11

Order 32: $C_2^2.D_4$ x 3, $C_4^2:C_2$ x 2, $C_2^2:Q_8$ x 2, $C_4\times D_4$ x 2, $C_4^2:C_2$ x 2, $C_2^3:C_4$ x 2, $C_4^2:C_2$, $C_2^2\wr C_2$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 37, $C_2\times C_{12}$ x 29, $C_6:C_4$ x 28, $C_4\times S_3$ x 28, $D_{12}$ x 9, $C_3:D_4$ x 6, $C_2^2\times C_6$ x 5, $C_3:Q_8$ x 2, $C_3\times D_4$ x 2, $C_3\times Q_8$

Order 18: $C_3\times C_6$ x 23, $C_3:S_3$ x 20, $C_3\times S_3$ x 13

Order 16: $C_2^2:C_4$ x 14, $C_4:C_4$ x 8, $C_2^2\times C_4$ x 5, $C_4^2$ x 3, $C_2\times D_4$ x 3, $C_2^4$, $C_2\times Q_8$

Order 12: $D_6$ x 75, $C_{12}$ x 28, $C_3:C_4$ x 26, $C_2\times C_6$ x 25

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 18, $C_2^3$ x 9, $D_4$ x 5, $Q_8$

Order 6: $C_6$ x 29, $S_3$ x 27

Order 4: $C_2^2$ x 19, $C_4$ x 11

Order 3: $C_3$ x 7

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $(C_2\times C_4).S_3^3$

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

Order 576: $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$

Order 432: $C_6^2.D_6$ x 3, $C_6^2.D_6$ x 3, $C_6^2.D_6$ x 3, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_3:D_6^2$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_3^3:C_4^2$

Order 288: $D_6^2.C_2$ x 3, $C_6^2.C_2^3$ x 3, $C_6^2.C_2^3$ x 3, $C_6^2.C_2^3$ x 2, $D_6.D_{12}$ x 2, $C_6^2.C_2^3$ x 2, $D_6.D_{12}$ x 2, $C_6^2.C_2^3$ x 2, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_{12}.D_{12}$, $C_6^2.C_2^3$, $C_6^2.D_4$, $C_6^2:Q_8$, $C_6^2:D_4$, $C_2^3.S_3^2$, $C_6^2.D_4$, $C_2^3.S_3^2$, $D_6:D_{12}$, $D_6:D_{12}$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $D_6.D_{12}$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_{12}.D_{12}$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$

Order 216: $C_6.C_6^2$ x 5, $C_6^2.C_6$ x 4, $C_6:S_3^2$ x 3, $C_6^2:C_6$ x 2, $C_6.S_3^2$ x 2, $C_6^2:S_3$, $S_3\times C_6^2$, $C_3\times C_6\times C_{12}$, $C_6^2.S_3$, $C_4\times C_3^2:S_3$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_3^3:Q_8$, $C_3^2:D_{12}$

Order 192: $C_4^2:D_6$, $(C_2\times D_4).D_6$, $(C_2\times Q_8):D_6$

Order 144: $D_6:C_{12}$ x 16, $C_6.D_{12}$ x 13, $C_6.D_{12}$ x 12, $C_{12}:C_{12}$ x 9, $C_6.D_{12}$ x 8, $C_{12}:D_6$ x 8, $C_6.D_{12}$ x 7, $C_3^2:C_4^2$ x 7, $C_6^2.C_2^2$ x 6, $C_6:D_{12}$ x 6, $C_2^2.S_3^2$ x 4, $D_6.D_6$ x 4, $D_6^2$ x 3, $C_3^2:C_4^2$ x 2, $C_{12}:C_{12}$ x 2, $C_6^2:C_4$ x 2, $C_{12}\times D_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}:C_{12}$ x 2, $C_6.D_{12}$, $C_6^2.C_2^2$, $C_6^2:C_4$, $C_2^2.S_3^2$, $C_6^2:C_2^2$, $C_6:D_{12}$, $C_6^2:C_2^2$, $C_6\times D_{12}$, $C_2^2.S_3^2$, $D_6:D_6$, $C_6^2:C_4$, $C_6^2:C_4$

Order 108: $C_3^2:D_6$ x 6, $C_3^2:C_{12}$ x 5, $C_3^2:C_{12}$ x 4, $C_3^2:D_6$ x 3, $C_3^2\times D_6$ x 3, $C_3\times C_6^2$, $C_3:S_3^2$, $C_3^2\times C_{12}$, $C_3^3:C_4$

Order 96: $D_6.D_4$ x 5, $D_6.D_4$ x 5, $(C_4\times S_3):C_4$ x 4, $(C_2\times C_4).D_6$ x 4, $C_2^2.D_{12}$ x 2, $C_2^3.D_6$ x 2, $D_6.D_4$ x 2, $C_2^2:D_{12}$ x 2, $C_2^2.D_{12}$ x 2, $D_6:Q_8$ x 2, $D_{12}:C_4$ x 2, $C_2^3.D_6$, $C_2^3.D_6$, $C_2^3.D_6$, $C_2^3.D_6$, $C_4\times D_{12}$, $C_4^2:S_3$, $C_4^2:C_6$, $(C_2^2\times C_{12}):C_2$, $C_{12}.D_4$, $C_{12}.D_4$, $D_6:Q_8$, $D_6:D_4$, $C_2^3.D_6$, $C_{12}:D_4$, $C_2^3.D_6$, $D_6:Q_8$

Order 72: $C_6:C_{12}$ x 29, $C_6:D_6$ x 12, $C_6\times C_{12}$ x 12, $C_6.D_6$ x 11, $C_6\times D_6$ x 9, $S_3\times D_6$ x 9, $C_{12}:S_3$ x 8, $C_3:D_{12}$ x 6, $C_6.D_6$ x 4, $C_6.D_6$ x 4, $S_3\times C_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_2\times C_6^2$, $C_3:D_{12}$, $C_6\wr C_2$, $C_3\times D_{12}$, $C_3^2:Q_8$, $D_6:S_3$

Order 64: $C_4^2:C_2^2$

Order 54: $C_3^2\times C_6$ x 3, $C_3^2:C_6$ x 2, $C_3^2:S_3$, $S_3\times C_3^2$

Order 48: $D_6:C_4$ x 35, $C_4\times D_6$ x 15, $C_4:C_{12}$ x 10, $C_{12}:C_4$ x 10, $C_2^2:C_{12}$ x 9, $C_{12}:C_4$ x 9, $C_6.D_4$ x 9, $C_6.D_4$ x 7, $C_2\times D_{12}$ x 5, $C_2^2\times D_6$ x 4, $C_6:D_4$ x 4, $C_6.C_2^3$ x 3, $C_2^2\times C_{12}$ x 2, $C_6:Q_8$ x 2, $C_6\times Q_8$, $C_6\times D_4$, $C_4\times C_{12}$

Order 36: $C_6:S_3$ x 28, $C_3:C_{12}$ x 27, $C_6\times S_3$ x 27, $C_3\times C_{12}$ x 12, $C_3^2:C_4$ x 11, $C_6^2$ x 10, $S_3^2$ x 3

Order 32: $C_2^2.D_4$ x 3, $C_4^2:C_2$ x 2, $C_2^2:Q_8$ x 2, $C_4\times D_4$ x 2, $C_4^2:C_2$ x 2, $C_2^3:C_4$ x 2, $C_4^2:C_2$, $C_2^2\wr C_2$

Order 27: $C_3^3$

Order 24: $C_2\times C_{12}$ x 28, $C_6:C_4$ x 27, $C_2\times D_6$ x 26, $C_4\times S_3$ x 15, $D_{12}$ x 5, $C_2^2\times C_6$ x 5, $C_3:D_4$ x 4, $C_3:Q_8$ x 2, $C_3\times Q_8$, $C_3\times D_4$

Order 18: $C_3\times C_6$ x 22, $C_3:S_3$ x 9, $C_3\times S_3$ x 9

Order 16: $C_2^2:C_4$ x 14, $C_4:C_4$ x 8, $C_2^2\times C_4$ x 5, $C_4^2$ x 3, $C_2\times D_4$ x 3, $C_2^4$, $C_2\times Q_8$

Order 12: $D_6$ x 46, $C_{12}$ x 26, $C_3:C_4$ x 24, $C_2\times C_6$ x 22

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 16, $C_2^3$ x 7, $D_4$ x 3, $Q_8$

Order 6: $C_6$ x 26, $S_3$ x 14

Order 4: $C_2^2$ x 14, $C_4$ x 11

Order 3: $C_3$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $(C_2\times C_4).S_3^3$ ($C_1$)

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

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

Order 288: $C_6^2.C_2^3$ ($S_3$), $D_6.D_{12}$ ($S_3$), $C_6^2.C_2^3$ ($S_3$)

Order 216: $C_6.C_6^2$ ($C_2^3$) x 5, $C_6^2.C_6$ ($C_2^3$) x 4, $C_6^2:C_6$ ($C_2^3$) x 2, $C_6^2:S_3$ ($C_2^3$), $S_3\times C_6^2$ ($C_2^3$), $C_3\times C_6\times C_{12}$ ($C_2^3$), $C_6^2.S_3$ ($C_2^3$)

Order 144: $C_6.D_{12}$ ($D_6$) x 3, $C_6.D_{12}$ ($D_6$) x 3, $C_6.D_{12}$ ($D_6$) x 2, $D_6:C_{12}$ ($D_6$) x 2, $C_{12}:C_{12}$ ($D_6$) x 2, $C_6^2.C_2^2$ ($D_6$) x 2, $D_6.D_6$ ($D_6$) x 2, $C_6.D_{12}$ ($D_6$), $C_6.D_{12}$ ($D_6$), $C_3^2:C_4^2$ ($D_6$), $C_2^2.S_3^2$ ($D_6$), $C_6:D_{12}$ ($D_6$)

Order 108: $C_3^2:D_6$ ($D_4:C_2$) x 2, $C_3^2\times D_6$ ($D_4:C_2$) x 2, $C_3\times C_6^2$ ($C_2^4$)

Order 72: $C_6:C_{12}$ ($C_2\times D_6$) x 10, $C_6.D_6$ ($C_2\times D_6$) x 4, $C_6\times C_{12}$ ($C_2\times D_6$) x 3, $C_6:D_6$ ($C_2\times D_6$) x 2, $C_6\times D_6$ ($C_2\times D_6$) x 2

Order 54: $C_3^2\times C_6$ ($D_4:C_2^2$) x 2, $C_3^2\times C_6$ ($D_4:C_2^2$)

Order 48: $D_6:C_4$ ($S_3^2$), $C_{12}:C_4$ ($S_3^2$), $C_6.D_4$ ($S_3^2$)

Order 36: $C_6^2$ ($C_2^2\times D_6$) x 3, $C_6\times S_3$ ($D_{12}:C_2$) x 2, $C_6\times S_3$ ($D_4:S_3$) x 2

Order 27: $C_3^3$ ($C_4^2:C_2^2$)

Order 24: $C_6:C_4$ ($S_3\times D_6$) x 5, $C_2\times C_{12}$ ($S_3\times D_6$) x 3, $C_2\times D_6$ ($S_3\times D_6$)

Order 18: $C_3\times C_6$ ($D_4:D_6$) x 4, $C_3\times C_6$ ($D_4:D_6$) x 2, $C_3\times C_6$ ($Q_8:D_6$), $C_3\times C_6$ ($D_4:D_6$), $C_3\times C_6$ ($D_4:D_6$)

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

Order 9: $C_3^2$ ($C_4^2:D_6$), $C_3^2$ ($(C_2\times D_4).D_6$), $C_3^2$ ($(C_2\times Q_8):D_6$)

Order 8: $C_2\times C_4$ ($S_3^3$)

Order 6: $C_6$ ($D_4:S_3^2$) x 2, $C_6$ ($D_{12}:D_6$), $C_6$ ($D_{12}:D_6$), $C_6$ ($D_{12}:D_6$), $C_6$ ($D_{12}:D_6$), $C_6$ ($D_{12}:D_6$), $C_6$ ($D_{12}:D_6$), $C_6$ ($D_{12}:D_6$)

Order 4: $C_2^2$ ($C_2\times S_3^3$)

Order 3: $C_3$ ($C_2^2.D_6^2$), $C_3$ ($C_2^2.D_6^2$), $C_3$ ($C_2^2.D_6^2$)

Order 2: $C_2$ ($C_6.D_6^2$), $C_2$ ($D_{12}:S_3^2$), $C_2$ ($D_{12}:S_3^2$)

Order 1: $C_1$ ($(C_2\times C_4).S_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $(C_2\times C_4).S_3^3$ ($C_1$)

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

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

Order 288: $C_6^2.C_2^3$ ($S_3$), $D_6.D_{12}$ ($S_3$), $C_6^2.C_2^3$ ($S_3$)

Order 216: $C_6.C_6^2$ ($C_2^3$) x 5, $C_6^2.C_6$ ($C_2^3$) x 4, $C_6^2:C_6$ ($C_2^3$) x 2, $C_6^2:S_3$ ($C_2^3$), $S_3\times C_6^2$ ($C_2^3$), $C_3\times C_6\times C_{12}$ ($C_2^3$), $C_6^2.S_3$ ($C_2^3$)

Order 144: $C_6.D_{12}$ ($D_6$) x 3, $C_6.D_{12}$ ($D_6$) x 3, $C_6.D_{12}$ ($D_6$) x 2, $D_6:C_{12}$ ($D_6$) x 2, $C_{12}:C_{12}$ ($D_6$) x 2, $C_6^2.C_2^2$ ($D_6$) x 2, $D_6.D_6$ ($D_6$) x 2, $C_6.D_{12}$ ($D_6$), $C_6.D_{12}$ ($D_6$), $C_3^2:C_4^2$ ($D_6$), $C_2^2.S_3^2$ ($D_6$), $C_6:D_{12}$ ($D_6$)

Order 108: $C_3\times C_6^2$ ($C_2^4$), $C_3^2:D_6$ ($D_4:C_2$), $C_3^2\times D_6$ ($D_4:C_2$)

Order 72: $C_6:C_{12}$ ($C_2\times D_6$) x 10, $C_6.D_6$ ($C_2\times D_6$) x 4, $C_6\times C_{12}$ ($C_2\times D_6$) x 3, $C_6:D_6$ ($C_2\times D_6$) x 2, $C_6\times D_6$ ($C_2\times D_6$) x 2

Order 54: $C_3^2\times C_6$ ($D_4:C_2^2$) x 2, $C_3^2\times C_6$ ($D_4:C_2^2$)

Order 48: $D_6:C_4$ ($S_3^2$), $C_{12}:C_4$ ($S_3^2$), $C_6.D_4$ ($S_3^2$)

Order 36: $C_6^2$ ($C_2^2\times D_6$) x 3, $C_6\times S_3$ ($D_{12}:C_2$), $C_6\times S_3$ ($D_4:S_3$)

Order 27: $C_3^3$ ($C_4^2:C_2^2$)

Order 24: $C_6:C_4$ ($S_3\times D_6$) x 5, $C_2\times C_{12}$ ($S_3\times D_6$) x 3, $C_2\times D_6$ ($S_3\times D_6$)

Order 18: $C_3\times C_6$ ($D_4:D_6$) x 4, $C_3\times C_6$ ($D_4:D_6$) x 2, $C_3\times C_6$ ($Q_8:D_6$), $C_3\times C_6$ ($D_4:D_6$), $C_3\times C_6$ ($D_4:D_6$)

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

Order 9: $C_3^2$ ($C_4^2:D_6$), $C_3^2$ ($(C_2\times D_4).D_6$), $C_3^2$ ($(C_2\times Q_8):D_6$)

Order 8: $C_2\times C_4$ ($S_3^3$)

Order 6: $C_6$ ($D_4:S_3^2$) x 2, $C_6$ ($D_{12}:D_6$), $C_6$ ($D_{12}:D_6$), $C_6$ ($D_{12}:D_6$), $C_6$ ($D_{12}:D_6$), $C_6$ ($D_{12}:D_6$), $C_6$ ($D_{12}:D_6$), $C_6$ ($D_{12}:D_6$)

Order 4: $C_2^2$ ($C_2\times S_3^3$)

Order 3: $C_3$ ($C_2^2.D_6^2$), $C_3$ ($C_2^2.D_6^2$), $C_3$ ($C_2^2.D_6^2$)

Order 2: $C_2$ ($C_6.D_6^2$), $C_2$ ($D_{12}:S_3^2$), $C_2$ ($D_{12}:S_3^2$)

Order 1: $C_1$ ($(C_2\times C_4).S_3^3$)

Series

Derived series

$(C_2\times C_4).S_3^3$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_1$

Chief series

$(C_2\times C_4).S_3^3$

$\rhd$

$C_2^2.S_3^3$

$\rhd$

$C_6^2.D_6$

$\rhd$

$C_6.C_6^2$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_2\times C_4).S_3^3$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

Character theory

Complex character table

See the $105 \times 105$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $85 \times 85$ rational character table.