Abstract group 288.376: $C_4.\SOPlus(4,2)$

• Label: 288.376
• Order: \( 2^{5} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $22$
• Trans deg.: $48$
• Rank: $2$

Group information

Description:	$C_4.\SOPlus(4,2)$
Order:	\(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$D_6^2:C_2^3$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Composition factors:	$C_2$ x 5, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	19	8	44	8	96	64	48	288
Conjugacy classes	1	2	2	4	2	4	5	4	24
Divisions	1	2	2	4	2	2	4	1	18
Autjugacy classes	1	2	2	3	2	2	3	1	16

Dimension	1	2	4	8	16
Irr. complex chars.	8	2	13	1	0	24
Irr. rational chars.	4	4	7	2	1	18

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$48$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	16
Arbitrary	4	8	12

Constructions

Presentation:	$\langle a, b, c, d \mid b^{2}=c^{24}=d^{3}=[a,b]=1, a^{2}=c^{12}, c^{a}=bc, d^{a}=c^{16}, c^{b}=c^{5}d, d^{b}=d^{2}, d^{c}=d^{2} \rangle$
Permutation group:	Degree $22$ $\langle(1,2)(3,5)(4,6)(7,8,11,14)(9,15,16,12)(10,18,19,13)(17,20,22,21), (4,5) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 1 & 4 & 3 & 2 \\ 1 & 3 & 1 & 2 \\ 0 & 3 & 2 & 4 \\ 3 & 0 & 1 & 4 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 2 & 4 & 3 & 3 \\ 2 & 2 & 3 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 3 & 2 & 3 & 2 \\ 4 & 2 & 2 & 1 \\ 3 & 2 & 4 & 4 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 2 & 3 \\ 1 & 3 & 3 & 2 \\ 3 & 2 & 3 & 1 \\ 4 & 3 & 1 & 3 \end{array}\right), \left(\begin{array}{rrrr} 3 & 3 & 1 & 4 \\ 0 & 4 & 2 & 1 \\ 2 & 0 & 2 & 1 \\ 4 & 1 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 1 & 0 & 3 \\ 2 & 0 & 4 & 2 \\ 0 & 2 & 1 & 2 \\ 2 & 2 & 4 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6.D_6)$ . $C_4$ (2)	$(C_{12}.D_6)$ . $C_2$	$(C_{12}.D_6)$ . $C_2$	$(C_3\times C_{12})$ . $D_4$	all 11

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

Homology

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

Subgroups

There are 328 subgroups in 67 conjugacy classes, 15 normal (13 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3^2:C_4$
Commutator:	$G' \simeq$ $C_6:S_3$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $\SOPlus(4,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_{12}$	$G/\operatorname{Fit} \simeq$ $D_4$
Radical:	$R \simeq$ $C_4.\SOPlus(4,2)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 288: $C_4.\SOPlus(4,2)$

Order 144: $C_{12}.D_6$, $C_{12}.D_6$, $C_3^2:\OD_{16}$

Order 72: $C_6.D_6$ x 2, $C_{12}:S_3$, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_3^2:C_8$, $C_3:C_{24}$

Order 48: $C_{24}:C_2$, $S_3\times Q_8$

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

Order 32: $C_4.D_4$

Order 24: $C_4\times S_3$ x 4, $C_3:Q_8$ x 3, $C_{24}$, $C_3\times Q_8$, $C_3:C_8$

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

Order 16: $\OD_{16}$ x 2, $C_2\times Q_8$

Order 12: $C_{12}$ x 4, $C_3:C_4$ x 4, $D_6$ x 2

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 3, $Q_8$ x 2, $C_8$ x 2

Order 6: $S_3$ x 3, $C_6$ x 2

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 288: $C_4.\SOPlus(4,2)$

Order 144: $C_{12}.D_6$, $C_{12}.D_6$, $C_3^2:\OD_{16}$

Order 72: $C_{12}:S_3$, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_6.D_6$, $C_3^2:C_8$, $C_3:C_{24}$

Order 48: $C_{24}:C_2$, $S_3\times Q_8$

Order 36: $C_3\times C_{12}$, $C_3^2:C_4$, $C_3:C_{12}$, $C_6:S_3$

Order 32: $C_4.D_4$

Order 24: $C_4\times S_3$ x 3, $C_3:Q_8$ x 2, $C_{24}$, $C_3\times Q_8$, $C_3:C_8$

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

Order 16: $\OD_{16}$ x 2, $C_2\times Q_8$

Order 12: $C_{12}$ x 3, $C_3:C_4$ x 3, $D_6$ x 2

Order 9: $C_3^2$

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

Order 6: $C_6$ x 2, $S_3$ x 2

Order 4: $C_4$ x 3, $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 288: $C_4.\SOPlus(4,2)$ ($C_1$)

Order 144: $C_{12}.D_6$ ($C_2$), $C_{12}.D_6$ ($C_2$), $C_3^2:\OD_{16}$ ($C_2$)

Order 72: $C_6.D_6$ ($C_4$) x 2, $C_{12}:S_3$ ($C_2^2$)

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

Order 18: $C_3\times C_6$ ($C_2^2:C_4$)

Order 9: $C_3^2$ ($C_4.D_4$)

Order 4: $C_4$ ($\SOPlus(4,2)$)

Order 2: $C_2$ ($S_3^2:C_4$)

Order 1: $C_1$ ($C_4.\SOPlus(4,2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 288: $C_4.\SOPlus(4,2)$ ($C_1$)

Order 144: $C_{12}.D_6$ ($C_2$), $C_{12}.D_6$ ($C_2$), $C_3^2:\OD_{16}$ ($C_2$)

Order 72: $C_{12}:S_3$ ($C_2^2$), $C_6.D_6$ ($C_4$)

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

Order 18: $C_3\times C_6$ ($C_2^2:C_4$)

Order 9: $C_3^2$ ($C_4.D_4$)

Order 4: $C_4$ ($\SOPlus(4,2)$)

Order 2: $C_2$ ($S_3^2:C_4$)

Order 1: $C_1$ ($C_4.\SOPlus(4,2)$)

Series

Derived series

$C_4.\SOPlus(4,2)$

$\rhd$

$C_6:S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_4.\SOPlus(4,2)$

$\rhd$

$C_{12}.D_6$

$\rhd$

$C_{12}:S_3$

$\rhd$

$C_6:S_3$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_4.\SOPlus(4,2)$

$\rhd$

$C_6:S_3$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 25 larger groups in the database.

This group is a maximal quotient of 16 larger groups in the database.

Character theory

Complex character table

See the $24 \times 24$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $18 \times 18$ rational character table.