Abstract group 864.4704: $S_3\times D_6^2$

• Label: 864.4704
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3^{2} \)
• Perm deg.: $13$
• Trans deg.: $48$
• Rank: $5$

Group information

Description:	$S_3\times D_6^2$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$C_6^3:C_2^2.S_4^2$, of order \(497664\)\(\medspace = 2^{11} \cdot 3^{5} \)
Composition factors:	$C_2$ x 5, $C_3$ x 3
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.

Group statistics

Order	1	2	3	6
Elements	1	255	26	582	864
Conjugacy classes	1	31	7	69	108
Divisions	1	31	7	69	108
Autjugacy classes	1	4	3	6	14

Dimension	1	2	4	8
Irr. complex chars.	32	48	24	4	108
Irr. rational chars.	32	48	24	4	108

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$48$
Rank:	$5$
Inequivalent generating 5-tuples:	$277760000$

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:	$\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{6}=d^{6}=e^{6}=[a,b]=[a,c]=[a,e]=[b,d]=[b,e]=[c,d]=[c,e]=1, d^{a}=d^{5}, c^{b}=c^{5}, e^{d}=e^{5} \rangle$
Permutation group:	Degree $13$ $\langle(2,3)(4,5), (4,5)(10,11)(12,13), (7,8)(12,13), (12,13), (4,5), (6,7,8), (9,10,11), (1,2,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 25 \\ 9 & 14 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 6 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $S_3$ ${}^3$
Semidirect product:	$C_6$ $\,\rtimes\,$ $D_6^2$	$C_3^3$ $\,\rtimes\,$ $C_2^5$	$(C_6\times D_6)$ $\,\rtimes\,$ $D_6$	$(C_6:D_6)$ $\,\rtimes\,$ $D_6$	all 33
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{30}\Z)$.

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{10}$
Commutator length:	$1$

Subgroups

There are 22400 subgroups in 3426 conjugacy classes, 628 normal (7 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:	$G' \simeq$ $C_3^3$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_3\times D_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2^3$
Radical:	$R \simeq$ $S_3\times D_6^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5$
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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Classes of subgroups up to conjugation

Order 864: $S_3\times D_6^2$

Order 432: $C_2\times S_3^3$ x 24, $C_3:D_6^2$ x 3, $C_3\times D_6^2$ x 3, $C_3:D_6^2$

Order 288: $C_2\times D_6^2$ x 3

Order 216: $S_3^3$ x 64, $C_6:S_3^2$ x 36, $C_6\times S_3^2$ x 36, $C_6:S_3^2$ x 12, $C_6^2:C_6$ x 3, $S_3\times C_6^2$ x 3, $C_6^2:S_3$

Order 144: $D_6^2$ x 87, $C_6^2:C_2^2$ x 6, $C_6^2:C_2^2$ x 3

Order 108: $C_3:S_3^2$ x 48, $C_3\times S_3^2$ x 48, $C_3^2:D_6$ x 18, $C_3^2\times D_6$ x 18, $C_3:S_3^2$ x 16, $C_3^2:D_6$ x 6, $C_3\times C_6^2$

Order 96: $C_2^3\times D_6$ x 3

Order 72: $S_3\times D_6$ x 372, $C_6\times D_6$ x 90, $C_6:D_6$ x 46, $C_2\times C_6^2$ x 3

Order 54: $C_3^2:C_6$ x 12, $S_3\times C_3^2$ x 12, $C_3^2:S_3$ x 4, $C_3^2\times C_6$ x 3

Order 48: $C_2^2\times D_6$ x 93, $C_2^3\times C_6$ x 3

Order 36: $S_3^2$ x 240, $C_6\times S_3$ x 204, $C_6:S_3$ x 108, $C_6^2$ x 25

Order 32: $C_2^5$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 463, $C_2^2\times C_6$ x 48

Order 18: $C_3\times S_3$ x 72, $C_3:S_3$ x 40, $C_3\times C_6$ x 33

Order 16: $C_2^4$ x 31

Order 12: $D_6$ x 450, $C_2\times C_6$ x 127

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 155

Order 6: $S_3$ x 76, $C_6$ x 69

Order 4: $C_2^2$ x 155

Order 3: $C_3$ x 7

Order 2: $C_2$ x 31

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 864: $S_3\times D_6^2$

Order 432: $C_3:D_6^2$, $C_3:D_6^2$, $C_3\times D_6^2$, $C_2\times S_3^3$

Order 288: $C_2\times D_6^2$

Order 216: $C_6^2:S_3$, $C_6^2:C_6$, $S_3\times C_6^2$, $C_6:S_3^2$, $C_6:S_3^2$, $C_6\times S_3^2$, $S_3^3$

Order 144: $D_6^2$ x 5, $C_6^2:C_2^2$, $C_6^2:C_2^2$

Order 108: $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3:S_3^2$, $C_3:S_3^2$, $C_3\times S_3^2$

Order 96: $C_2^3\times D_6$

Order 72: $C_6:D_6$ x 5, $C_6\times D_6$ x 5, $S_3\times D_6$ x 5, $C_2\times C_6^2$

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

Order 48: $C_2^2\times D_6$ x 6, $C_2^3\times C_6$

Order 36: $C_6:S_3$ x 5, $C_6\times S_3$ x 5, $C_6^2$ x 4, $S_3^2$ x 4

Order 32: $C_2^5$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 12, $C_2^2\times C_6$ x 4

Order 18: $C_3\times C_6$ x 4, $C_3:S_3$ x 4, $C_3\times S_3$ x 4

Order 16: $C_2^4$ x 4

Order 12: $D_6$ x 11, $C_2\times C_6$ x 7

Order 9: $C_3^2$ x 3

Order 8: $C_2^3$ x 7

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

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 432: $C_2\times S_3^3$ ($C_2$) x 24, $C_3:D_6^2$ ($C_2$) x 3, $C_3\times D_6^2$ ($C_2$) x 3, $C_3:D_6^2$ ($C_2$)

Order 216: $S_3^3$ ($C_2^2$) x 64, $C_6:S_3^2$ ($C_2^2$) x 36, $C_6\times S_3^2$ ($C_2^2$) x 36, $C_6:S_3^2$ ($C_2^2$) x 12, $C_6^2:C_6$ ($C_2^2$) x 3, $S_3\times C_6^2$ ($C_2^2$) x 3, $C_6^2:S_3$ ($C_2^2$)

Order 144: $D_6^2$ ($S_3$) x 3

Order 108: $C_3:S_3^2$ ($C_2^3$) x 48, $C_3\times S_3^2$ ($C_2^3$) x 48, $C_3^2:D_6$ ($C_2^3$) x 18, $C_3^2\times D_6$ ($C_2^3$) x 18, $C_3:S_3^2$ ($C_2^3$) x 16, $C_3^2:D_6$ ($C_2^3$) x 6, $C_3\times C_6^2$ ($C_2^3$)

Order 72: $S_3\times D_6$ ($D_6$) x 36, $C_6\times D_6$ ($D_6$) x 6, $C_6:D_6$ ($D_6$) x 3

Order 54: $C_3^2:C_6$ ($C_2^4$) x 12, $S_3\times C_3^2$ ($C_2^4$) x 12, $C_3^2:S_3$ ($C_2^4$) x 4, $C_3^2\times C_6$ ($C_2^4$) x 3

Order 36: $S_3^2$ ($C_2\times D_6$) x 48, $C_6\times S_3$ ($C_2\times D_6$) x 36, $C_6:S_3$ ($C_2\times D_6$) x 18, $C_6^2$ ($C_2\times D_6$) x 3

Order 27: $C_3^3$ ($C_2^5$)

Order 24: $C_2\times D_6$ ($S_3^2$) x 3

Order 18: $C_3\times S_3$ ($C_2^2\times D_6$) x 24, $C_3:S_3$ ($C_2^2\times D_6$) x 12, $C_3\times C_6$ ($C_2^2\times D_6$) x 9

Order 12: $D_6$ ($S_3\times D_6$) x 18, $C_2\times C_6$ ($S_3\times D_6$) x 3

Order 9: $C_3^2$ ($C_2^3\times D_6$) x 3

Order 6: $S_3$ ($D_6^2$) x 12, $C_6$ ($D_6^2$) x 9

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

Order 3: $C_3$ ($C_2\times D_6^2$) x 3

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

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

Normal subgroups up to automorphism (quotient in parentheses)

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

Order 432: $C_3:D_6^2$ ($C_2$), $C_3:D_6^2$ ($C_2$), $C_3\times D_6^2$ ($C_2$), $C_2\times S_3^3$ ($C_2$)

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

Order 144: $D_6^2$ ($S_3$)

Order 108: $C_3\times C_6^2$ ($C_2^3$), $C_3^2:D_6$ ($C_2^3$), $C_3^2:D_6$ ($C_2^3$), $C_3^2\times D_6$ ($C_2^3$), $C_3:S_3^2$ ($C_2^3$), $C_3:S_3^2$ ($C_2^3$), $C_3\times S_3^2$ ($C_2^3$)

Order 72: $C_6:D_6$ ($D_6$), $C_6\times D_6$ ($D_6$), $S_3\times D_6$ ($D_6$)

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

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

Order 27: $C_3^3$ ($C_2^5$)

Order 24: $C_2\times D_6$ ($S_3^2$)

Order 18: $C_3\times C_6$ ($C_2^2\times D_6$), $C_3:S_3$ ($C_2^2\times D_6$), $C_3\times S_3$ ($C_2^2\times D_6$)

Order 12: $C_2\times C_6$ ($S_3\times D_6$), $D_6$ ($S_3\times D_6$)

Order 9: $C_3^2$ ($C_2^3\times D_6$)

Order 6: $C_6$ ($D_6^2$), $S_3$ ($D_6^2$)

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

Order 3: $C_3$ ($C_2\times D_6^2$)

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

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

Series

Derived series

$S_3\times D_6^2$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$S_3\times D_6^2$

$\rhd$

$C_2\times S_3^3$

$\rhd$

$C_6\times S_3^2$

$\rhd$

$C_3^2\times D_6$

$\rhd$

$C_6\times S_3$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$S_3\times D_6^2$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 38 larger groups in the database.

This group is a maximal quotient of 149 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 $108 \times 108$ rational character table. Alternatively, you may search for characters of this group with desired properties.