Abstract group 1296.3279: $(C_3^2\times C_6).S_4$

• Label: 1296.3279
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{3} \)
• Perm deg.: $25$
• Trans deg.: $144$
• Rank: $4$

Group information

Description:	$(C_3^2\times C_6).S_4$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_3^4.Q_8.C_2^2.D_6^2$, of order \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$4$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	1	242	114	242	108	372	216	1296
Conjugacy classes	1	1	41	2	41	2	16	4	108
Divisions	1	1	27	2	27	1	10	1	70
Autjugacy classes	1	1	5	2	5	1	4	1	20

Dimension	1	2	3	4	6	8	12
Irr. complex chars.	6	45	6	39	12	0	0	108
Irr. rational chars.	2	15	2	27	6	14	4	70

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$144$
Rank:	$4$
Inequivalent generating quadruples:	$3931200$

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:	${\langle a, b, c, d \mid b^{3}=d^{12}=1, a^{6}=d^{6}, c^{6}=d^{6}, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $25$ $\langle(2,3)(5,6)(10,11,14,17)(12,18,20,25)(13,21,23,16)(15,22,24,19), (7,8,9) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 37 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 21 \\ 21 & 16 \end{array}\right), \left(\begin{array}{rr} 31 & 24 \\ 6 & 25 \end{array}\right), \left(\begin{array}{rr} 37 & 6 \\ 12 & 19 \end{array}\right), \left(\begin{array}{rr} 5 & 27 \\ 12 & 37 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/42\Z)$
Direct product:	$C_3$ $\, \times\, $ $((C_3\times C_6).S_4)$
Semidirect product:	$C_3^3$ $\,\rtimes\,$ $(C_2.S_4)$	$C_3^2$ $\,\rtimes\,$ $(C_6.S_4)$	$C_3^2$ $\,\rtimes\,$ $(C_6.S_4)$	$C_3$ $\,\rtimes\,$ $(C_3\times C_6.S_4)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(Q_8\times C_3^3)$ . $S_3$	$(C_3^2\times C_6)$ . $S_4$	$C_6$ . $(C_3^2:S_4)$	$C_6$ . $(C_3^2:S_4)$	all 18

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

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_{3}^{3}$
Commutator length:	$1$

Subgroups

There are 3664 subgroups in 558 conjugacy classes, 82 normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_3^2:S_4$
Commutator:	$G' \simeq$ $C_3^2\times \SL(2,3)$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_3^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_3^3$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $(C_3^2\times C_6).S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2\times C_6$	$G/\operatorname{soc} \simeq$ $S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

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 1296: $(C_3^2\times C_6).S_4$

Order 648: $C_3^3\times \SL(2,3)$

Order 432: $C_3\times C_6.S_4$ x 12, $(C_3\times C_6).S_4$, $C_3^3:Q_{16}$

Order 324: $C_3^3:C_{12}$

Order 216: $C_3^2\times \SL(2,3)$ x 26, $C_3^2:C_{24}$, $Q_8\times C_3^3$, $C_3^3:Q_8$

Order 162: $C_3^3\times C_6$

Order 144: $C_6.S_4$ x 12, $C_6.S_4$ x 9, $C_3^2:Q_{16}$ x 4, $C_3^2:Q_{16}$

Order 108: $C_3^2:C_{12}$ x 13, $C_3^2\times C_{12}$, $C_3^3:C_4$

Order 81: $C_3^4$

Order 72: $C_3\times \SL(2,3)$ x 69, $Q_8\times C_3^2$ x 9, $C_3^2:Q_8$ x 4, $C_3:C_{24}$ x 4, $C_3^2:Q_8$, $C_3^2:C_8$

Order 54: $C_3^2\times C_6$ x 27

Order 48: $C_2.S_4$ x 9, $C_3:Q_{16}$ x 4, $C_3\times Q_{16}$

Order 36: $C_3^2:C_4$ x 13, $C_3:C_{12}$ x 13, $C_3\times C_{12}$ x 9

Order 27: $C_3^3$ x 27

Order 24: $\SL(2,3)$ x 18, $C_3\times Q_8$ x 10, $C_3:Q_8$ x 4, $C_3:C_8$ x 4, $C_{24}$

Order 18: $C_3\times C_6$ x 78

Order 16: $Q_{16}$

Order 12: $C_3:C_4$ x 13, $C_{12}$ x 10

Order 9: $C_3^2$ x 78

Order 8: $Q_8$ x 2, $C_8$

Order 6: $C_6$ x 27

Order 4: $C_4$ x 2

Order 3: $C_3$ x 27

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $(C_3^2\times C_6).S_4$

Order 648: $C_3^3\times \SL(2,3)$

Order 432: $(C_3\times C_6).S_4$, $C_3\times C_6.S_4$, $C_3^3:Q_{16}$

Order 324: $C_3^3:C_{12}$

Order 216: $C_3^2\times \SL(2,3)$ x 4, $C_3^2:C_{24}$, $Q_8\times C_3^3$, $C_3^3:Q_8$

Order 162: $C_3^3\times C_6$

Order 144: $C_3^2:Q_{16}$, $C_3^2:Q_{16}$, $C_6.S_4$, $C_6.S_4$

Order 108: $C_3^2:C_{12}$ x 2, $C_3^2\times C_{12}$, $C_3^3:C_4$

Order 81: $C_3^4$

Order 72: $C_3\times \SL(2,3)$ x 4, $Q_8\times C_3^2$ x 3, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_3^2:C_8$, $C_3:C_{24}$

Order 54: $C_3^2\times C_6$ x 5

Order 48: $C_2.S_4$, $C_3\times Q_{16}$, $C_3:Q_{16}$

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

Order 27: $C_3^3$ x 5

Order 24: $C_3\times Q_8$ x 4, $\SL(2,3)$ x 2, $C_3:Q_8$, $C_{24}$, $C_3:C_8$

Order 18: $C_3\times C_6$ x 7

Order 16: $Q_{16}$

Order 12: $C_{12}$ x 4, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 7

Order 8: $Q_8$ x 2, $C_8$

Order 6: $C_6$ x 5

Order 4: $C_4$ x 2

Order 3: $C_3$ x 5

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $(C_3^2\times C_6).S_4$ ($C_1$)

Order 648: $C_3^3\times \SL(2,3)$ ($C_2$)

Order 432: $(C_3\times C_6).S_4$ ($C_3$)

Order 216: $C_3^2\times \SL(2,3)$ ($S_3$) x 12, $Q_8\times C_3^3$ ($S_3$), $C_3^2\times \SL(2,3)$ ($C_6$)

Order 72: $C_3\times \SL(2,3)$ ($C_3\times S_3$) x 12, $C_3\times \SL(2,3)$ ($C_3:S_3$) x 9, $Q_8\times C_3^2$ ($C_3:S_3$) x 4, $Q_8\times C_3^2$ ($C_3\times S_3$)

Order 54: $C_3^2\times C_6$ ($S_4$)

Order 27: $C_3^3$ ($C_2.S_4$)

Order 24: $\SL(2,3)$ ($C_3^2:C_6$) x 9, $C_3\times Q_8$ ($C_3^2:C_6$) x 4, $C_3\times Q_8$ ($C_3^2:S_3$)

Order 18: $C_3\times C_6$ ($C_3:S_4$) x 4, $C_3\times C_6$ ($C_3\times S_4$)

Order 9: $C_3^2$ ($C_6.S_4$) x 4, $C_3^2$ ($C_6.S_4$)

Order 8: $Q_8$ ($C_3^3:C_6$)

Order 6: $C_6$ ($C_3^2:S_4$) x 4, $C_6$ ($C_3^2:S_4$)

Order 3: $C_3$ ($C_3\times C_6.S_4$) x 4, $C_3$ ($(C_3\times C_6).S_4$)

Order 2: $C_2$ ($C_3^3:S_4$)

Order 1: $C_1$ ($(C_3^2\times C_6).S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $(C_3^2\times C_6).S_4$ ($C_1$)

Order 648: $C_3^3\times \SL(2,3)$ ($C_2$)

Order 432: $(C_3\times C_6).S_4$ ($C_3$)

Order 216: $Q_8\times C_3^3$ ($S_3$), $C_3^2\times \SL(2,3)$ ($C_6$), $C_3^2\times \SL(2,3)$ ($S_3$)

Order 72: $Q_8\times C_3^2$ ($C_3:S_3$), $Q_8\times C_3^2$ ($C_3\times S_3$), $C_3\times \SL(2,3)$ ($C_3:S_3$), $C_3\times \SL(2,3)$ ($C_3\times S_3$)

Order 54: $C_3^2\times C_6$ ($S_4$)

Order 27: $C_3^3$ ($C_2.S_4$)

Order 24: $\SL(2,3)$ ($C_3^2:C_6$), $C_3\times Q_8$ ($C_3^2:S_3$), $C_3\times Q_8$ ($C_3^2:C_6$)

Order 18: $C_3\times C_6$ ($C_3:S_4$), $C_3\times C_6$ ($C_3\times S_4$)

Order 9: $C_3^2$ ($C_6.S_4$), $C_3^2$ ($C_6.S_4$)

Order 8: $Q_8$ ($C_3^3:C_6$)

Order 6: $C_6$ ($C_3^2:S_4$), $C_6$ ($C_3^2:S_4$)

Order 3: $C_3$ ($(C_3\times C_6).S_4$), $C_3$ ($C_3\times C_6.S_4$)

Order 2: $C_2$ ($C_3^3:S_4$)

Order 1: $C_1$ ($(C_3^2\times C_6).S_4$)

Series

Derived series

$(C_3^2\times C_6).S_4$

$\rhd$

$C_3^2\times \SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$(C_3^2\times C_6).S_4$

$\rhd$

$C_3^3\times \SL(2,3)$

$\rhd$

$C_3^2\times \SL(2,3)$

$\rhd$

$C_3\times \SL(2,3)$

$\rhd$

$\SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_3^2\times C_6).S_4$

$\rhd$

$C_3^2\times \SL(2,3)$

Upper central series

$C_1$

$\lhd$

$C_6$

Character theory

Complex character table

See the $108 \times 108$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

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