Abstract group 1728.3085: $C_{12}^2.D_6$

• Label: 1728.3085
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $28$
• Trans deg.: $576$
• Rank: $3$

Group information

Description:	$C_{12}^2.D_6$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$C_2\times C_9.(C_2\times C_6^2).C_2^5$, of order \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \)
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	8	9	12	18	24	36
Elements	1	3	8	60	24	576	18	192	54	288	504	1728
Conjugacy classes	1	3	3	12	9	12	6	36	18	8	84	192
Divisions	1	3	3	9	9	5	2	17	6	2	13	70
Autjugacy classes	1	3	3	8	9	4	2	16	6	2	12	66

Dimension	1	2	4	6	8	12	16	24	48
Irr. complex chars.	16	92	84	0	0	0	0	0	0	192
Irr. rational chars.	8	18	13	8	7	6	3	5	2	70

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$576$
Rank:	$3$
Inequivalent generating triples:	$32256$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	8	16

Constructions

Presentation:	$\langle a, b, c \mid b^{24}=c^{36}=1, a^{2}=b^{12}c^{18}, b^{a}=b^{17}, c^{a}=b^{12}c^{19}, c^{b}=c^{35} \rangle$
Permutation group:	Degree $28$ $\langle(11,12)(13,14)(15,17)(16,18)(19,20)(21,22,25,23)(24,27,28,26), (2,4)(3,7) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{12}.D_9)$ $\,\rtimes\,$ $Q_8$ (2)	$C_9$ $\,\rtimes\,$ $(C_4^2.D_6)$	$C_3$ $\,\rtimes\,$ $(C_4^2.D_{18})$	$(C_3\times C_9)$ $\,\rtimes\,$ $(C_8:Q_8)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{12}^2$ . $D_6$	$C_{36}$ . $(S_3\times Q_8)$	$C_{36}$ . $(D_4:S_3)$	$C_4$ . $(D_{36}:S_3)$	all 86

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

Homology

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

Subgroups

There are 1400 subgroups in 322 conjugacy classes, 121 normal (113 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $S_3\times D_{18}$
Commutator:	$G' \simeq$ $C_6\times C_{18}$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_{12}$	$G/\Phi \simeq$ $S_3\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}\times C_{36}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{12}^2.D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_6.C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8:Q_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$

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

Order 1728: $C_{12}^2.D_6$

Order 864: $C_{12}.D_{36}$ x 2, $C_{12}^2.S_3$, $C_{36}:C_{24}$, $(C_2\times C_{12}).D_{18}$, $C_{18}.(S_3\times C_8)$, $C_{12}^2.C_6$

Order 576: $C_4^2.D_{18}$, $C_4^2.S_3^2$

Order 432: $C_{12}.D_{18}$ x 2, $C_6.D_4\times C_9$ x 2, $C_{12}:C_{36}$ x 2, $C_{18}:C_{24}$ x 2, $C_{36}.D_6$, $C_{12}\times C_{36}$, $C_{12}:C_{36}$

Order 288: $C_{36}:C_8$ x 3, $C_{18}.C_4^2$ x 3, $C_{12}.D_{12}$ x 2, $C_{36}:C_8$, $C_{12}^2.C_2$, $C_{12}^2.C_2$, $C_{12}:C_{24}$, $C_{12}.(C_4\times S_3)$, $C_3^2:(C_4\times C_8)$, $Q_8\times C_{36}$

Order 216: $C_6:C_{36}$ x 4, $C_6\times C_{36}$ x 3, $C_{12}.D_9$ x 3, $C_{12}.C_{18}$ x 2, $C_9:C_{24}$ x 2

Order 192: $C_4^2.D_6$, $C_4^2.D_6$

Order 144: $C_{18}:C_8$ x 6, $C_4\times C_{36}$ x 4, $C_4:C_{36}$ x 3, $C_{12}.D_6$ x 2, $C_6^2.C_2^2$ x 2, $C_{12}:C_{12}$ x 2, $C_6:C_{24}$ x 2, $C_{12}:C_{12}$, $Q_8\times C_{18}$, $C_{12}.D_6$, $C_{12}^2$

Order 108: $C_3:C_{36}$ x 5, $C_3\times C_{36}$ x 4, $C_6\times C_{18}$

Order 96: $C_6.C_4^2$ x 4, $C_{12}:C_8$ x 3, $C_{12}.D_4$ x 2, $C_{12}:C_8$, $C_{12}:Q_8$, $C_4:C_{24}$, $C_{24}:C_4$, $C_{24}:C_4$, $Q_8\times C_{12}$

Order 72: $C_2\times C_{36}$ x 10, $C_9:C_8$ x 9, $C_6:C_{12}$ x 4, $C_6\times C_{12}$ x 3, $C_3^2:C_8$ x 3, $C_3^2:Q_8$ x 2, $C_3:C_{24}$ x 2, $Q_8\times C_9$ x 2

Order 64: $C_8:Q_8$

Order 54: $C_3\times C_{18}$ x 3

Order 48: $C_6:C_8$ x 8, $C_4\times C_{12}$ x 5, $C_4:C_{12}$ x 3, $C_2\times C_{24}$ x 2, $C_6.D_4$ x 2, $C_{12}:C_4$ x 2, $C_6\times Q_8$, $C_6:Q_8$, $C_{12}:C_4$

Order 36: $C_{36}$ x 13, $C_3:C_{12}$ x 5, $C_3\times C_{12}$ x 4, $C_2\times C_{18}$ x 2, $C_6^2$

Order 32: $C_4:C_8$ x 3, $C_8:C_4$ x 2, $C_4\times C_8$, $C_4\times Q_8$

Order 27: $C_3\times C_9$

Order 24: $C_2\times C_{12}$ x 13, $C_3:C_8$ x 12, $C_6:C_4$ x 4, $C_3:Q_8$ x 2, $C_{24}$ x 2, $C_3\times Q_8$ x 2

Order 18: $C_{18}$ x 6, $C_3\times C_6$ x 3

Order 16: $C_2\times C_8$ x 4, $C_4:C_4$ x 3, $C_4^2$ x 3, $C_2\times Q_8$

Order 12: $C_{12}$ x 17, $C_3:C_4$ x 5, $C_2\times C_6$ x 3

Order 9: $C_9$ x 2, $C_3^2$

Order 8: $C_2\times C_4$ x 7, $C_8$ x 5, $Q_8$ x 2

Order 6: $C_6$ x 9

Order 4: $C_4$ x 9, $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $C_{12}^2.D_6$

Order 864: $C_{12}.D_{36}$ x 2, $C_{12}^2.S_3$, $C_{36}:C_{24}$, $(C_2\times C_{12}).D_{18}$, $C_{18}.(S_3\times C_8)$, $C_{12}^2.C_6$

Order 576: $C_4^2.D_{18}$, $C_4^2.S_3^2$

Order 432: $C_{12}.D_{18}$ x 2, $C_6.D_4\times C_9$ x 2, $C_{12}:C_{36}$ x 2, $C_{18}:C_{24}$ x 2, $C_{36}.D_6$, $C_{12}\times C_{36}$, $C_{12}:C_{36}$

Order 288: $C_{36}:C_8$ x 3, $C_{18}.C_4^2$ x 3, $C_{12}.D_{12}$ x 2, $C_{36}:C_8$, $C_{12}^2.C_2$, $C_{12}^2.C_2$, $C_{12}:C_{24}$, $C_{12}.(C_4\times S_3)$, $C_3^2:(C_4\times C_8)$, $Q_8\times C_{36}$

Order 216: $C_6:C_{36}$ x 4, $C_6\times C_{36}$ x 3, $C_{12}.D_9$ x 2, $C_9:C_{24}$ x 2, $C_{12}.C_{18}$

Order 192: $C_4^2.D_6$, $C_4^2.D_6$

Order 144: $C_{18}:C_8$ x 6, $C_4\times C_{36}$ x 4, $C_4:C_{36}$ x 3, $C_{12}.D_6$ x 2, $C_6^2.C_2^2$ x 2, $C_{12}:C_{12}$ x 2, $C_6:C_{24}$ x 2, $C_{12}:C_{12}$, $Q_8\times C_{18}$, $C_{12}.D_6$, $C_{12}^2$

Order 108: $C_3:C_{36}$ x 4, $C_3\times C_{36}$ x 4, $C_6\times C_{18}$

Order 96: $C_6.C_4^2$ x 4, $C_{12}:C_8$ x 3, $C_{12}.D_4$ x 2, $C_{12}:C_8$, $C_{12}:Q_8$, $C_4:C_{24}$, $C_{24}:C_4$, $C_{24}:C_4$, $Q_8\times C_{12}$

Order 72: $C_2\times C_{36}$ x 10, $C_9:C_8$ x 6, $C_6:C_{12}$ x 4, $C_6\times C_{12}$ x 3, $C_3^2:C_8$ x 2, $C_3:C_{24}$ x 2, $C_3^2:Q_8$, $Q_8\times C_9$

Order 64: $C_8:Q_8$

Order 54: $C_3\times C_{18}$ x 3

Order 48: $C_6:C_8$ x 8, $C_4\times C_{12}$ x 5, $C_4:C_{12}$ x 3, $C_2\times C_{24}$ x 2, $C_6.D_4$ x 2, $C_{12}:C_4$ x 2, $C_6\times Q_8$, $C_6:Q_8$, $C_{12}:C_4$

Order 36: $C_{36}$ x 12, $C_3\times C_{12}$ x 4, $C_3:C_{12}$ x 4, $C_2\times C_{18}$ x 2, $C_6^2$

Order 32: $C_4:C_8$ x 3, $C_8:C_4$ x 2, $C_4\times C_8$, $C_4\times Q_8$

Order 27: $C_3\times C_9$

Order 24: $C_2\times C_{12}$ x 13, $C_3:C_8$ x 8, $C_6:C_4$ x 4, $C_{24}$ x 2, $C_3:Q_8$, $C_3\times Q_8$

Order 18: $C_{18}$ x 6, $C_3\times C_6$ x 3

Order 16: $C_2\times C_8$ x 4, $C_4:C_4$ x 3, $C_4^2$ x 3, $C_2\times Q_8$

Order 12: $C_{12}$ x 16, $C_3:C_4$ x 4, $C_2\times C_6$ x 3

Order 9: $C_9$ x 2, $C_3^2$

Order 8: $C_2\times C_4$ x 7, $C_8$ x 4, $Q_8$

Order 6: $C_6$ x 9

Order 4: $C_4$ x 8, $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $C_{12}^2.D_6$ ($C_1$)

Order 864: $C_{12}.D_{36}$ ($C_2$) x 2, $C_{12}^2.S_3$ ($C_2$), $C_{36}:C_{24}$ ($C_2$), $(C_2\times C_{12}).D_{18}$ ($C_2$), $C_{18}.(S_3\times C_8)$ ($C_2$), $C_{12}^2.C_6$ ($C_2$)

Order 432: $C_{12}.D_{18}$ ($C_2^2$) x 2, $C_6.D_4\times C_9$ ($C_4$) x 2, $C_{12}:C_{36}$ ($C_2^2$) x 2, $C_{18}:C_{24}$ ($C_2^2$) x 2, $C_{36}.D_6$ ($C_4$), $C_{12}\times C_{36}$ ($C_2^2$), $C_{12}:C_{36}$ ($C_4$)

Order 288: $C_{12}^2.C_2$ ($S_3$), $C_{36}:C_8$ ($S_3$)

Order 216: $C_6:C_{36}$ ($C_2\times C_4$) x 4, $C_6\times C_{36}$ ($C_2\times C_4$) x 2, $C_{12}.D_9$ ($Q_8$) x 2, $C_6\times C_{36}$ ($C_2^3$)

Order 144: $C_{18}:C_8$ ($D_6$) x 2, $C_6^2.C_2^2$ ($C_3:C_4$) x 2, $C_{12}:C_{12}$ ($D_6$) x 2, $C_{12}:C_{12}$ ($C_3:C_4$), $C_4\times C_{36}$ ($D_6$), $C_{12}.D_6$ ($C_3:C_4$), $C_{12}^2$ ($D_6$)

Order 108: $C_3:C_{36}$ ($\OD_{16}$) x 2, $C_6\times C_{18}$ ($C_2^2\times C_4$), $C_3\times C_{36}$ ($D_4:C_2$), $C_3\times C_{36}$ ($C_2\times Q_8$)

Order 96: $C_{12}:Q_8$ ($D_9$)

Order 72: $C_6:C_{12}$ ($C_6:C_4$) x 4, $C_2\times C_{36}$ ($C_4\times S_3$) x 2, $C_6\times C_{12}$ ($C_6:C_4$) x 2, $C_2\times C_{36}$ ($C_2\times D_6$), $C_6\times C_{12}$ ($C_2\times D_6$)

Order 54: $C_3\times C_{18}$ ($\OD_{16}:C_2$), $C_3\times C_{18}$ ($C_2\times \OD_{16}$), $C_3\times C_{18}$ ($C_4\times Q_8$)

Order 48: $C_6.D_4$ ($C_9:C_4$) x 2, $C_{12}:C_4$ ($D_{18}$) x 2, $C_6:Q_8$ ($C_9:C_4$), $C_4\times C_{12}$ ($D_{18}$), $C_4\times C_{12}$ ($S_3^2$), $C_{12}:C_4$ ($C_9:C_4$)

Order 36: $C_3:C_{12}$ ($C_3:\OD_{16}$) x 2, $C_3\times C_{12}$ ($D_{12}:C_2$), $C_3\times C_{12}$ ($S_3\times Q_8$), $C_2\times C_{18}$ ($C_4\times D_6$), $C_{36}$ ($S_3\times Q_8$), $C_{36}$ ($D_4:S_3$), $C_6^2$ ($C_6.C_2^3$)

Order 27: $C_3\times C_9$ ($C_8:Q_8$)

Order 24: $C_6:C_4$ ($C_{18}:C_4$) x 4, $C_2\times C_{12}$ ($C_{18}:C_4$) x 2, $C_2\times C_{12}$ ($C_6.D_6$) x 2, $C_2\times C_{12}$ ($S_3\times D_6$), $C_2\times C_{12}$ ($C_2\times D_{18}$)

Order 18: $C_3\times C_6$ ($C_{12}.C_2^3$), $C_3\times C_6$ ($C_3:C_4\times Q_8$), $C_3\times C_6$ ($C_6:\OD_{16}$), $C_{18}$ ($C_3:(C_4\times Q_8)$), $C_{18}$ ($S_3\times \OD_{16}$), $C_{18}$ ($C_8.D_6$)

Order 16: $C_4^2$ ($S_3\times D_9$)

Order 12: $C_3:C_4$ ($C_9:\OD_{16}$) x 2, $C_2\times C_6$ ($C_2^2.D_{18}$), $C_2\times C_6$ ($D_6.D_6$), $C_{12}$ ($D_{36}:C_2$), $C_{12}$ ($Q_8\times D_9$), $C_{12}$ ($C_{12}.D_6$), $C_{12}$ ($D_{12}:S_3$)

Order 9: $C_3^2$ ($C_4^2.D_6$), $C_9$ ($C_4^2.D_6$)

Order 8: $C_2\times C_4$ ($C_6.D_{18}$) x 2, $C_2\times C_4$ ($S_3\times D_{18}$)

Order 6: $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_{12}.(C_4\times S_3)$), $C_6$ ($S_3\times C_3:\OD_{16}$), $C_6$ ($C_{36}.C_2^3$), $C_6$ ($C_9:C_4\times Q_8$), $C_6$ ($C_{18}:\OD_{16}$)

Order 4: $C_2^2$ ($D_6.D_{18}$), $C_4$ ($C_{12}.D_{18}$), $C_4$ ($D_{36}:S_3$)

Order 3: $C_3$ ($C_4^2.D_{18}$), $C_3$ ($C_4^2.S_3^2$)

Order 2: $C_2$ ($(C_2\times C_{12}).D_{18}$), $C_2$ ($(C_2\times C_{12}).D_{18}$), $C_2$ ($S_3\times C_9:\OD_{16}$)

Order 1: $C_1$ ($C_{12}^2.D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $C_{12}^2.D_6$ ($C_1$)

Order 864: $C_{12}.D_{36}$ ($C_2$) x 2, $C_{12}^2.S_3$ ($C_2$), $C_{36}:C_{24}$ ($C_2$), $(C_2\times C_{12}).D_{18}$ ($C_2$), $C_{18}.(S_3\times C_8)$ ($C_2$), $C_{12}^2.C_6$ ($C_2$)

Order 432: $C_{12}.D_{18}$ ($C_2^2$) x 2, $C_6.D_4\times C_9$ ($C_4$) x 2, $C_{12}:C_{36}$ ($C_2^2$) x 2, $C_{18}:C_{24}$ ($C_2^2$) x 2, $C_{36}.D_6$ ($C_4$), $C_{12}\times C_{36}$ ($C_2^2$), $C_{12}:C_{36}$ ($C_4$)

Order 288: $C_{12}^2.C_2$ ($S_3$), $C_{36}:C_8$ ($S_3$)

Order 216: $C_6:C_{36}$ ($C_2\times C_4$) x 4, $C_6\times C_{36}$ ($C_2\times C_4$) x 2, $C_6\times C_{36}$ ($C_2^3$), $C_{12}.D_9$ ($Q_8$)

Order 144: $C_{18}:C_8$ ($D_6$) x 2, $C_6^2.C_2^2$ ($C_3:C_4$) x 2, $C_{12}:C_{12}$ ($D_6$) x 2, $C_{12}:C_{12}$ ($C_3:C_4$), $C_4\times C_{36}$ ($D_6$), $C_{12}.D_6$ ($C_3:C_4$), $C_{12}^2$ ($D_6$)

Order 108: $C_3:C_{36}$ ($\OD_{16}$), $C_6\times C_{18}$ ($C_2^2\times C_4$), $C_3\times C_{36}$ ($D_4:C_2$), $C_3\times C_{36}$ ($C_2\times Q_8$)

Order 96: $C_{12}:Q_8$ ($D_9$)

Order 72: $C_6:C_{12}$ ($C_6:C_4$) x 4, $C_2\times C_{36}$ ($C_4\times S_3$) x 2, $C_6\times C_{12}$ ($C_6:C_4$) x 2, $C_2\times C_{36}$ ($C_2\times D_6$), $C_6\times C_{12}$ ($C_2\times D_6$)

Order 54: $C_3\times C_{18}$ ($\OD_{16}:C_2$), $C_3\times C_{18}$ ($C_2\times \OD_{16}$), $C_3\times C_{18}$ ($C_4\times Q_8$)

Order 48: $C_6.D_4$ ($C_9:C_4$) x 2, $C_{12}:C_4$ ($D_{18}$) x 2, $C_6:Q_8$ ($C_9:C_4$), $C_4\times C_{12}$ ($D_{18}$), $C_4\times C_{12}$ ($S_3^2$), $C_{12}:C_4$ ($C_9:C_4$)

Order 36: $C_3\times C_{12}$ ($D_{12}:C_2$), $C_3\times C_{12}$ ($S_3\times Q_8$), $C_3:C_{12}$ ($C_3:\OD_{16}$), $C_2\times C_{18}$ ($C_4\times D_6$), $C_{36}$ ($S_3\times Q_8$), $C_{36}$ ($D_4:S_3$), $C_6^2$ ($C_6.C_2^3$)

Order 27: $C_3\times C_9$ ($C_8:Q_8$)

Order 24: $C_6:C_4$ ($C_{18}:C_4$) x 4, $C_2\times C_{12}$ ($C_{18}:C_4$) x 2, $C_2\times C_{12}$ ($C_6.D_6$) x 2, $C_2\times C_{12}$ ($S_3\times D_6$), $C_2\times C_{12}$ ($C_2\times D_{18}$)

Order 18: $C_3\times C_6$ ($C_{12}.C_2^3$), $C_3\times C_6$ ($C_3:C_4\times Q_8$), $C_3\times C_6$ ($C_6:\OD_{16}$), $C_{18}$ ($C_3:(C_4\times Q_8)$), $C_{18}$ ($S_3\times \OD_{16}$), $C_{18}$ ($C_8.D_6$)

Order 16: $C_4^2$ ($S_3\times D_9$)

Order 12: $C_2\times C_6$ ($C_2^2.D_{18}$), $C_2\times C_6$ ($D_6.D_6$), $C_{12}$ ($D_{36}:C_2$), $C_{12}$ ($Q_8\times D_9$), $C_{12}$ ($C_{12}.D_6$), $C_{12}$ ($D_{12}:S_3$), $C_3:C_4$ ($C_9:\OD_{16}$)

Order 9: $C_3^2$ ($C_4^2.D_6$), $C_9$ ($C_4^2.D_6$)

Order 8: $C_2\times C_4$ ($C_6.D_{18}$) x 2, $C_2\times C_4$ ($S_3\times D_{18}$)

Order 6: $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_{12}.(C_4\times S_3)$), $C_6$ ($S_3\times C_3:\OD_{16}$), $C_6$ ($C_{36}.C_2^3$), $C_6$ ($C_9:C_4\times Q_8$), $C_6$ ($C_{18}:\OD_{16}$)

Order 4: $C_2^2$ ($D_6.D_{18}$), $C_4$ ($C_{12}.D_{18}$), $C_4$ ($D_{36}:S_3$)

Order 3: $C_3$ ($C_4^2.D_{18}$), $C_3$ ($C_4^2.S_3^2$)

Order 2: $C_2$ ($(C_2\times C_{12}).D_{18}$), $C_2$ ($(C_2\times C_{12}).D_{18}$), $C_2$ ($S_3\times C_9:\OD_{16}$)

Order 1: $C_1$ ($C_{12}^2.D_6$)

Series

Derived series

$C_{12}^2.D_6$

$\rhd$

$C_6\times C_{18}$

$\rhd$

$C_1$

Chief series

$C_{12}^2.D_6$

$\rhd$

$C_{18}.(S_3\times C_8)$

$\rhd$

$C_{12}:C_{36}$

$\rhd$

$C_6\times C_{36}$

$\rhd$

$C_3\times C_{36}$

$\rhd$

$C_3\times C_{18}$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_3^2$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{12}^2.D_6$

$\rhd$

$C_6\times C_{18}$

$\rhd$

$C_3\times C_9$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_4^2$

Character theory

Complex character table

See the $192 \times 192$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $70 \times 70$ rational character table.