Abstract group 864.4384: $D_6^2:S_3$

• Label: 864.4384
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $15$
• Trans deg.: $48$
• Rank: $4$

Group information

Description:	$D_6^2:S_3$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_6^3.C_2^6.C_2$, of order \(27648\)\(\medspace = 2^{10} \cdot 3^{3} \)
Composition factors:	$C_2$ x 5, $C_3$ 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	207	26	48	414	168	864
Conjugacy classes	1	15	7	4	49	14	90
Divisions	1	15	7	4	43	8	78
Autjugacy classes	1	7	7	2	22	4	43

Dimension	1	2	4	8
Irr. complex chars.	16	44	26	4	90
Irr. rational chars.	16	28	26	8	78

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$48$
Rank:	$4$
Inequivalent generating quadruples:	$5536440$

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:	$\langle a, b, c, d \mid a^{2}=b^{6}=c^{12}=d^{6}=[a,c]=[a,d]=[b,d]=1, b^{a}=b^{5}, c^{b}=c^{11}, d^{c}=d^{5} \rangle$
Permutation group:	Degree $15$ $\langle(4,5)(6,7)(8,9)(10,11)(12,14)(13,15), (2,3)(8,9), (6,8,7,9)(10,11)(12,15) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 35 & 0 \\ 9 & 17 \end{array}\right), \left(\begin{array}{rr} 17 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 26 & 9 \\ 9 & 26 \end{array}\right), \left(\begin{array}{rr} 1 & 27 \\ 27 & 10 \end{array}\right), \left(\begin{array}{rr} 25 & 8 \\ 32 & 29 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 24 & 25 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Direct product:	$C_2$ $\, \times\, $ $S_3$ $\, \times\, $ $(C_3:D_{12})$
Semidirect product:	$D_6^2$ $\,\rtimes\,$ $S_3$	$(D_6.D_6)$ $\,\rtimes\,$ $S_3$	$(S_3\times D_6)$ $\,\rtimes\,$ $D_6$	$D_6$ $\,\rtimes\,$ $(S_3\times D_6)$	all 59
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $D_6^2$ (3)	$C_2^2$ . $S_3^3$	$D_6$ . $(S_3\times D_6)$	$(C_6\times D_6)$ . $D_6$ (2)	all 15

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 9440 subgroups in 1218 conjugacy classes, 188 normal (54 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $S_3^3$
Commutator:	$G' \simeq$ $C_3^2\times C_6$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times S_3^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2^3$
Radical:	$R \simeq$ $D_6^2:S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

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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Subgroup information

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Classes of subgroups up to conjugation

Order 864: $D_6^2:S_3$

Order 432: $D_6:S_3^2$ x 8, $C_3:D_6^2$, $C_3\times D_6^2$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$

Order 288: $D_6^2:C_2$, $C_6^2:D_4$, $D_6\times D_{12}$

Order 216: $C_6:S_3^2$ x 8, $C_6\times S_3^2$ x 8, $C_3^2:D_{12}$ x 4, $C_3^2:D_{12}$ x 4, $C_3^2:D_{12}$ x 4, $C_3^2:D_{12}$ x 4, $C_6.S_3^2$ x 4, $C_6^2:C_6$ x 2, $S_3\times C_6^2$ x 2, $C_6^2:S_3$, $C_6^2.C_6$, $C_6.C_6^2$

Order 144: $C_6:D_{12}$ x 17, $D_6:D_6$ x 8, $S_3\times D_{12}$ x 8, $D_6^2$ x 4, $C_6^2:C_2^2$ x 2, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_6:D_{12}$, $C_6^2:C_2^2$, $C_6^2:C_4$, $C_6\times D_{12}$, $C_{12}\times D_6$, $D_6:D_6$, $D_6.D_6$

Order 108: $C_3^2\times D_6$ x 10, $C_3^2:D_6$ x 8, $C_3:S_3^2$ x 8, $C_3\times S_3^2$ x 8, $C_3^2:D_6$ x 4, $C_3^2:C_{12}$ x 2, $C_3^2:C_{12}$ x 2, $C_3\times C_6^2$

Order 96: $C_2^3:D_6$, $D_4\times D_6$, $C_2^2\times D_{12}$

Order 72: $S_3\times D_6$ x 36, $C_3:D_{12}$ x 36, $C_6\times D_6$ x 28, $C_6:D_6$ x 17, $C_6:C_{12}$ x 10, $C_6^2:C_2$ x 4, $C_3:D_{12}$ x 4, $C_6\wr C_2$ x 4, $C_3\times D_{12}$ x 4, $S_3\times C_{12}$ x 4, $D_6:S_3$ x 4, $C_6.D_6$ x 4, $C_2\times C_6^2$ x 2, $C_6\times C_{12}$, $C_6.D_6$

Order 54: $S_3\times C_3^2$ x 6, $C_3^2:C_6$ x 4, $C_3^2\times C_6$ x 3, $C_3^2:S_3$ x 2

Order 48: $C_6:D_4$ x 15, $C_2\times D_{12}$ x 14, $S_3\times D_4$ x 8, $C_2^2\times D_6$ x 6, $C_2^3\times C_6$, $C_6\times D_4$, $C_2^2\times C_{12}$, $C_6.C_2^3$, $C_4\times D_6$

Order 36: $C_6\times S_3$ x 72, $C_6:S_3$ x 48, $S_3^2$ x 40, $C_6^2$ x 17, $C_3:C_{12}$ x 12, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2

Order 32: $C_2^2\times D_4$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 67, $C_3:D_4$ x 28, $D_{12}$ x 24, $C_2^2\times C_6$ x 16, $C_2\times C_{12}$ x 8, $C_6:C_4$ x 8, $C_4\times S_3$ x 4, $C_3\times D_4$ x 4

Order 18: $C_3\times S_3$ x 32, $C_3\times C_6$ x 27, $C_3:S_3$ x 22

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$

Order 12: $D_6$ x 124, $C_2\times C_6$ x 53, $C_{12}$ x 8, $C_3:C_4$ x 8

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 21, $D_4$ x 16, $C_2\times C_4$ x 6

Order 6: $C_6$ x 43, $S_3$ x 38

Order 4: $C_2^2$ x 39, $C_4$ x 4

Order 3: $C_3$ x 7

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 864: $D_6^2:S_3$

Order 432: $C_3:D_6^2$, $C_3\times D_6^2$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $D_6:S_3^2$

Order 288: $D_6^2:C_2$, $C_6^2:D_4$, $D_6\times D_{12}$

Order 216: $C_6^2:C_6$ x 2, $S_3\times C_6^2$ x 2, $C_6:S_3^2$ x 2, $C_6\times S_3^2$ x 2, $C_6^2:S_3$, $C_6^2.C_6$, $C_6.C_6^2$, $C_3^2:D_{12}$, $C_3^2:D_{12}$, $C_3^2:D_{12}$, $C_3^2:D_{12}$, $C_6.S_3^2$

Order 144: $C_6:D_{12}$ x 10, $D_6^2$ x 4, $C_6^2:C_2^2$ x 2, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_6:D_{12}$, $C_6^2:C_2^2$, $C_6^2:C_4$, $C_6\times D_{12}$, $C_{12}\times D_6$, $D_6:D_6$, $D_6:D_6$, $D_6.D_6$, $S_3\times D_{12}$

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

Order 96: $C_2^3:D_6$, $D_4\times D_6$, $C_2^2\times D_{12}$

Order 72: $C_6\times D_6$ x 16, $C_6:D_6$ x 11, $C_3:D_{12}$ x 9, $S_3\times D_6$ x 8, $C_6:C_{12}$ x 7, $C_2\times C_6^2$ x 2, $C_6\times C_{12}$, $C_6^2:C_2$, $C_6.D_6$, $C_3:D_{12}$, $C_6\wr C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $D_6:S_3$, $C_6.D_6$

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

Order 48: $C_6:D_4$ x 8, $C_2\times D_{12}$ x 7, $C_2^2\times D_6$ x 6, $C_2^3\times C_6$, $C_6\times D_4$, $C_2^2\times C_{12}$, $C_6.C_2^3$, $S_3\times D_4$, $C_4\times D_6$

Order 36: $C_6\times S_3$ x 26, $C_6:S_3$ x 19, $C_6^2$ x 11, $C_3:C_{12}$ x 6, $S_3^2$ x 4, $C_3\times C_{12}$, $C_3^2:C_4$

Order 32: $C_2^2\times D_4$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 27, $C_2^2\times C_6$ x 10, $C_3:D_4$ x 7, $D_{12}$ x 6, $C_2\times C_{12}$ x 5, $C_6:C_4$ x 5, $C_4\times S_3$, $C_3\times D_4$

Order 18: $C_3\times C_6$ x 16, $C_3\times S_3$ x 12, $C_3:S_3$ x 9

Order 16: $C_2\times D_4$ x 5, $C_2^4$ x 2, $C_2^2\times C_4$

Order 12: $D_6$ x 36, $C_2\times C_6$ x 24, $C_{12}$ x 4, $C_3:C_4$ x 4

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 9, $D_4$ x 4, $C_2\times C_4$ x 3

Order 6: $C_6$ x 22, $S_3$ x 15

Order 4: $C_2^2$ x 13, $C_4$ x 2

Order 3: $C_3$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 864: $D_6^2:S_3$ ($C_1$)

Order 432: $D_6:S_3^2$ ($C_2$) x 8, $C_3:D_6^2$ ($C_2$), $C_3\times D_6^2$ ($C_2$), $C_6^2.D_6$ ($C_2$), $C_6^2.D_6$ ($C_2$), $C_6^2.D_6$ ($C_2$), $C_6^2.D_6$ ($C_2$), $C_6^2.D_6$ ($C_2$)

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

Order 144: $D_6^2$ ($S_3$), $C_6:D_{12}$ ($S_3$), $D_6.D_6$ ($S_3$)

Order 108: $C_3^2:D_6$ ($C_2^3$) x 4, $C_3^2\times D_6$ ($C_2^3$) x 4, $C_3^2\times D_6$ ($D_4$) x 4, $C_3^2:D_6$ ($C_2^3$) x 2, $C_3^2:C_{12}$ ($C_2^3$) x 2, $C_3^2:C_{12}$ ($C_2^3$) x 2, $C_3\times C_6^2$ ($C_2^3$)

Order 72: $C_6\times D_6$ ($D_6$) x 4, $S_3\times D_6$ ($D_6$) x 4, $C_3:D_{12}$ ($D_6$) x 4, $C_6.D_6$ ($D_6$) x 4, $C_6:D_6$ ($D_6$) x 2, $C_6:C_{12}$ ($D_6$) x 2, $C_6.D_6$ ($D_6$)

Order 54: $S_3\times C_3^2$ ($C_2\times D_4$) x 4, $C_3^2\times C_6$ ($C_2\times D_4$) x 2, $C_3^2\times C_6$ ($C_2^4$)

Order 36: $C_6\times S_3$ ($C_2\times D_6$) x 8, $C_3:C_{12}$ ($C_2\times D_6$) x 4, $C_6:S_3$ ($C_2\times D_6$) x 4, $C_6\times S_3$ ($C_3:D_4$) x 4, $C_6\times S_3$ ($D_{12}$) x 4, $C_6^2$ ($C_2\times D_6$) x 3, $C_3^2:C_4$ ($C_2\times D_6$) x 2

Order 27: $C_3^3$ ($C_2^2\times D_4$)

Order 24: $C_2\times D_6$ ($S_3^2$) x 2, $C_6:C_4$ ($S_3^2$)

Order 18: $C_3\times S_3$ ($C_6:D_4$) x 4, $C_3\times S_3$ ($C_2\times D_{12}$) x 4, $C_3\times C_6$ ($C_2^2\times D_6$) x 3, $C_3\times C_6$ ($C_6:D_4$) x 2, $C_3\times C_6$ ($S_3\times D_4$) x 2, $C_3\times C_6$ ($C_2\times D_{12}$) x 2

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

Order 9: $C_3^2$ ($C_2^3:D_6$), $C_3^2$ ($D_4\times D_6$), $C_3^2$ ($C_2^2\times D_{12}$)

Order 6: $S_3$ ($C_6:D_{12}$) x 4, $C_6$ ($D_6^2$) x 3, $C_6$ ($D_6:D_6$) x 2, $C_6$ ($C_6:D_{12}$) x 2, $C_6$ ($S_3\times D_{12}$) x 2

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

Order 3: $C_3$ ($D_6^2:C_2$), $C_3$ ($C_6^2:D_4$), $C_3$ ($D_6\times D_{12}$)

Order 2: $C_2$ ($D_6:S_3^2$) x 2, $C_2$ ($C_2\times S_3^3$)

Order 1: $C_1$ ($D_6^2:S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 864: $D_6^2:S_3$ ($C_1$)

Order 432: $C_3:D_6^2$ ($C_2$), $C_3\times D_6^2$ ($C_2$), $C_6^2.D_6$ ($C_2$), $C_6^2.D_6$ ($C_2$), $C_6^2.D_6$ ($C_2$), $C_6^2.D_6$ ($C_2$), $C_6^2.D_6$ ($C_2$), $D_6:S_3^2$ ($C_2$)

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

Order 144: $D_6^2$ ($S_3$), $C_6:D_{12}$ ($S_3$), $D_6.D_6$ ($S_3$)

Order 108: $C_3^2:D_6$ ($C_2^3$) x 2, $C_3^2\times D_6$ ($C_2^3$) x 2, $C_3\times C_6^2$ ($C_2^3$), $C_3^2:D_6$ ($C_2^3$), $C_3^2\times D_6$ ($D_4$), $C_3^2:C_{12}$ ($C_2^3$), $C_3^2:C_{12}$ ($C_2^3$)

Order 72: $C_6\times D_6$ ($D_6$) x 4, $C_6:D_6$ ($D_6$) x 2, $C_6:C_{12}$ ($D_6$) x 2, $S_3\times D_6$ ($D_6$), $C_6.D_6$ ($D_6$), $C_3:D_{12}$ ($D_6$), $C_6.D_6$ ($D_6$)

Order 54: $C_3^2\times C_6$ ($C_2^4$), $C_3^2\times C_6$ ($C_2\times D_4$), $S_3\times C_3^2$ ($C_2\times D_4$)

Order 36: $C_6\times S_3$ ($C_2\times D_6$) x 4, $C_6^2$ ($C_2\times D_6$) x 3, $C_3:C_{12}$ ($C_2\times D_6$) x 2, $C_6:S_3$ ($C_2\times D_6$) x 2, $C_3^2:C_4$ ($C_2\times D_6$), $C_6\times S_3$ ($C_3:D_4$), $C_6\times S_3$ ($D_{12}$)

Order 27: $C_3^3$ ($C_2^2\times D_4$)

Order 24: $C_2\times D_6$ ($S_3^2$) x 2, $C_6:C_4$ ($S_3^2$)

Order 18: $C_3\times C_6$ ($C_2^2\times D_6$) x 3, $C_3\times C_6$ ($C_6:D_4$), $C_3\times C_6$ ($S_3\times D_4$), $C_3\times C_6$ ($C_2\times D_{12}$), $C_3\times S_3$ ($C_6:D_4$), $C_3\times S_3$ ($C_2\times D_{12}$)

Order 12: $C_2\times C_6$ ($S_3\times D_6$) x 3, $D_6$ ($S_3\times D_6$) x 2, $D_6$ ($C_3:D_{12}$), $C_3:C_4$ ($S_3\times D_6$)

Order 9: $C_3^2$ ($C_2^3:D_6$), $C_3^2$ ($D_4\times D_6$), $C_3^2$ ($C_2^2\times D_{12}$)

Order 6: $C_6$ ($D_6^2$) x 3, $C_6$ ($D_6:D_6$), $C_6$ ($C_6:D_{12}$), $C_6$ ($S_3\times D_{12}$), $S_3$ ($C_6:D_{12}$)

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

Order 3: $C_3$ ($D_6^2:C_2$), $C_3$ ($C_6^2:D_4$), $C_3$ ($D_6\times D_{12}$)

Order 2: $C_2$ ($C_2\times S_3^3$), $C_2$ ($D_6:S_3^2$)

Order 1: $C_1$ ($D_6^2:S_3$)

Series

Derived series

$D_6^2:S_3$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_1$

Chief series

$D_6^2:S_3$

$\rhd$

$C_6^2.D_6$

$\rhd$

$C_6.C_6^2$

$\rhd$

$C_6:C_{12}$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_6^2:S_3$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 134 larger groups in the database.

This group is a maximal quotient of 177 larger groups in the database.

Character theory

Complex character table

See the $90 \times 90$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $78 \times 78$ rational character table.