Abstract group 1296.3576: $C_2\times C_3^4:D_4$

• Label: 1296.3576
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_2\times C_3^4:D_4$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^4.Q_8.C_6.C_2^4.C_2$, of order \(124416\)\(\medspace = 2^{9} \cdot 3^{5} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6
Elements	1	235	80	324	656	1296
Conjugacy classes	1	7	14	2	30	54
Divisions	1	7	14	2	30	54
Autjugacy classes	1	4	2	1	3	11

Dimension	1	2	4	8
Irr. complex chars.	8	2	32	12	54
Irr. rational chars.	8	2	32	12	54

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$5460$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	8
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{4}=c^{3}=d^{3}=e^{3}=f^{6}=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(1,2)(3,5)(4,6)(12,13), (3,6)(7,8)(9,10)(11,12)(13,14), (7,8), (3,6)(4,5) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 2 & 0 & 1 & 0 \\ 2 & 2 & 2 & 0 \\ 2 & 2 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 2 & 2 & 2 & 0 \\ 2 & 0 & 1 & 0 \\ 1 & 1 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 2 & 0 & 1 & 0 \\ 2 & 2 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 1 & 0 & 2 & 0 \\ 1 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 2 & 1 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 2 & 2 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 2 & 2 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 2 & 2 & 0 \\ 2 & 2 & 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	24T2862	24T2864	36T2116	36T2119	all 7
Direct product:	$C_2$ $\, \times\, $ $(C_3^4:D_4)$
Semidirect product:	$(C_3^3:S_3)$ $\,\rtimes\,$ $D_4$	$(C_3^3\times C_6)$ $\,\rtimes\,$ $D_4$	$C_3^4$ $\,\rtimes\,$ $(C_2\times D_4)$	$(C_3^4:C_2^3)$ $\,\rtimes\,$ $C_2$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^3:S_3)$ . $C_2^3$	$(C_3^3:D_6)$ . $C_2^2$			more information

Elements of the group are displayed as matrices in $\GL_{4}(\F_{3})$.

Homology

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

Subgroups

There are 15310 subgroups in 841 conjugacy classes, 29 normal (9 characteristic).

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

Special subgroups

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

Order 648: $C_3^4:D_4$ x 4, $C_3^4:C_2^3$ x 2, $C_2\times C_3^4:C_4$

Order 324: $C_3^2:S_3^2$ x 6, $C_3^2:C_6^2$ x 2, $C_3^4:C_4$ x 2, $C_3^3:D_6$

Order 216: $C_6:S_3^2$ x 8

Order 162: $C_3^2\wr C_2$ x 4, $C_3^3:S_3$ x 2, $C_3^3\times C_6$

Order 144: $S_3^2:C_2^2$ x 4

Order 108: $C_3:S_3^2$ x 32, $C_3^2:D_6$ x 14, $C_3^2:D_6$ x 8, $C_3^2\times D_6$ x 8

Order 81: $C_3^4$

Order 72: $S_3\times D_6$ x 20, $\SOPlus(4,2)$ x 16, $C_2\times C_3^2:C_4$ x 7, $C_6:D_6$ x 2

Order 54: $C_3^2:S_3$ x 28, $C_3^2:C_6$ x 16, $S_3\times C_3^2$ x 16, $C_3^2\times C_6$ x 14

Order 36: $S_3^2$ x 72, $C_6:S_3$ x 54, $C_6\times S_3$ x 32, $C_3^2:C_4$ x 14, $C_6^2$ x 2

Order 27: $C_3^3$ x 14

Order 24: $C_2\times D_6$ x 8

Order 18: $C_3:S_3$ x 92, $C_3\times S_3$ x 64, $C_3\times C_6$ x 48

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 54, $C_2\times C_6$ x 8

Order 9: $C_3^2$ x 44

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 6: $S_3$ x 44, $C_6$ x 30

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

Order 3: $C_3$ x 14

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $C_2\times C_3^4:D_4$

Order 648: $C_3^4:C_2^3$, $C_2\times C_3^4:C_4$, $C_3^4:D_4$

Order 324: $C_3^2:S_3^2$ x 2, $C_3^3:D_6$, $C_3^2:C_6^2$, $C_3^4:C_4$

Order 216: $C_6:S_3^2$

Order 162: $C_3^3:S_3$ x 2, $C_3^3\times C_6$, $C_3^2\wr C_2$

Order 144: $S_3^2:C_2^2$

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

Order 81: $C_3^4$

Order 72: $S_3\times D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $C_6:D_6$, $\SOPlus(4,2)$

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

Order 36: $C_6:S_3$ x 10, $S_3^2$ x 4, $C_3^2:C_4$ x 2, $C_6\times S_3$ x 2, $C_6^2$

Order 27: $C_3^3$ x 2

Order 24: $C_2\times D_6$

Order 18: $C_3:S_3$ x 15, $C_3\times C_6$ x 8, $C_3\times S_3$ x 2

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 5, $C_2\times C_6$

Order 9: $C_3^2$ x 7

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

Order 6: $S_3$ x 5, $C_6$ x 3

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $C_2\times C_3^4:D_4$ ($C_1$)

Order 648: $C_3^4:D_4$ ($C_2$) x 4, $C_3^4:C_2^3$ ($C_2$) x 2, $C_2\times C_3^4:C_4$ ($C_2$)

Order 324: $C_3^2:S_3^2$ ($C_2^2$) x 4, $C_3^4:C_4$ ($C_2^2$) x 2, $C_3^3:D_6$ ($C_2^2$)

Order 162: $C_3^3\times C_6$ ($D_4$), $C_3^3:S_3$ ($C_2^3$), $C_3^3:S_3$ ($D_4$)

Order 81: $C_3^4$ ($C_2\times D_4$)

Order 18: $C_3\times C_6$ ($\SOPlus(4,2)$) x 4

Order 9: $C_3^2$ ($S_3^2:C_2^2$) x 4

Order 2: $C_2$ ($C_3^4:D_4$)

Order 1: $C_1$ ($C_2\times C_3^4:D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $C_2\times C_3^4:D_4$ ($C_1$)

Order 648: $C_3^4:C_2^3$ ($C_2$), $C_2\times C_3^4:C_4$ ($C_2$), $C_3^4:D_4$ ($C_2$)

Order 324: $C_3^3:D_6$ ($C_2^2$), $C_3^2:S_3^2$ ($C_2^2$), $C_3^4:C_4$ ($C_2^2$)

Order 162: $C_3^3\times C_6$ ($D_4$), $C_3^3:S_3$ ($C_2^3$), $C_3^3:S_3$ ($D_4$)

Order 81: $C_3^4$ ($C_2\times D_4$)

Order 18: $C_3\times C_6$ ($\SOPlus(4,2)$)

Order 9: $C_3^2$ ($S_3^2:C_2^2$)

Order 2: $C_2$ ($C_3^4:D_4$)

Order 1: $C_1$ ($C_2\times C_3^4:D_4$)

Series

Derived series

$C_2\times C_3^4:D_4$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

Chief series

$C_2\times C_3^4:D_4$

$\rhd$

$C_3^4:D_4$

$\rhd$

$C_3^2:S_3^2$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_2\times C_3^4:D_4$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 26 larger groups in the database.

This group is a maximal quotient of 7 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 $54 \times 54$ rational character table. Alternatively, you may search for characters of this group with desired properties.