Abstract group 288.251: $C_{24}:C_{12}$

• Label: 288.251
• Order: \( 2^{5} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $18$
• Trans deg.: $96$
• Rank: $2$

Group information

Description:	$C_{24}:C_{12}$
Order:	\(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$(C_2\times C_{24}):C_2^5$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_3$ x 2
Derived length:	$2$

This group is nonabelian and metacyclic (hence solvable, supersolvable, monomial, and metabelian).

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	3	8	52	24	8	128	64	288
Conjugacy classes	1	3	5	6	15	4	24	32	90
Divisions	1	3	3	4	9	2	8	6	36
Autjugacy classes	1	2	3	3	6	1	7	3	26

Dimension	1	2	4	8	16
Irr. complex chars.	24	66	0	0	0	90
Irr. rational chars.	4	12	12	6	2	36

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$96$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	8

Constructions

Presentation:	$\langle a, b \mid a^{12}=b^{24}=1, b^{a}=b^{11} \rangle$
Permutation group:	Degree $18$ $\langle(1,2,4,6)(3,5,7,8)(9,10,11,12)(13,14)(15,17)(16,18), (1,3,5,6,4,7,8,2), (13,15,16) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 4 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 0 & 5 \\ 4 & 0 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 14 & 8 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 20 & 14 \\ 14 & 6 \end{array}\right), \left(\begin{array}{rr} 4 & 0 \\ 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/21\Z)$
Direct product:	$C_3$ $\, \times\, $ $(C_{24}:C_4)$
Semidirect product:	$C_{24}$ $\,\rtimes\,$ $C_{12}$ (2)	$(C_3\times C_{24})$ $\,\rtimes\,$ $C_4$ (2)	$C_{24}$ $\,\rtimes\,$ $(C_3:C_4)$ (2)	$C_8$ $\,\rtimes\,$ $(C_3:C_{12})$ (2)	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $D_4$	$(C_2\times C_{24})$ . $S_3$	$(C_3\times C_{12})$ . $Q_8$	$C_{12}$ . $(C_3:Q_8)$	all 30

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

Homology

Abelianization:	$C_{2} \times C_{12} \simeq C_{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 186 subgroups in 83 conjugacy classes, 50 normal (30 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $D_{12}$
Commutator:	$G' \simeq$ $C_{12}$	$G/G' \simeq$ $C_2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_6\times S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times C_{24}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{24}:C_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8:C_4$
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

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Classes of subgroups up to conjugation

Order 288: $C_{24}:C_{12}$

Order 144: $C_{12}:C_{12}$ x 2, $C_6\times C_{24}$

Order 96: $C_8:C_{12}$, $C_{24}:C_4$

Order 72: $C_6:C_{12}$ x 2, $C_3\times C_{24}$ x 2, $C_6\times C_{12}$

Order 48: $C_2\times C_{24}$ x 3, $C_4:C_{12}$ x 2, $C_{12}:C_4$ x 2

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

Order 32: $C_8:C_4$

Order 24: $C_{24}$ x 6, $C_2\times C_{12}$ x 5, $C_6:C_4$ x 2

Order 18: $C_3\times C_6$ x 3

Order 16: $C_4:C_4$ x 2, $C_2\times C_8$

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 3, $C_3:C_4$ x 2

Order 9: $C_3^2$

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

Order 6: $C_6$ x 9

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 288: $C_{24}:C_{12}$

Order 144: $C_{12}:C_{12}$, $C_6\times C_{24}$

Order 96: $C_8:C_{12}$, $C_{24}:C_4$

Order 72: $C_6\times C_{12}$, $C_6:C_{12}$, $C_3\times C_{24}$

Order 48: $C_2\times C_{24}$ x 3, $C_4:C_{12}$, $C_{12}:C_4$

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

Order 32: $C_8:C_4$

Order 24: $C_2\times C_{12}$ x 4, $C_{24}$ x 3, $C_6:C_4$

Order 18: $C_3\times C_6$ x 2

Order 16: $C_2\times C_8$, $C_4:C_4$

Order 12: $C_{12}$ x 7, $C_2\times C_6$ x 3, $C_3:C_4$

Order 9: $C_3^2$

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

Order 6: $C_6$ x 6

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 288: $C_{24}:C_{12}$ ($C_1$)

Order 144: $C_{12}:C_{12}$ ($C_2$) x 2, $C_6\times C_{24}$ ($C_2$)

Order 96: $C_{24}:C_4$ ($C_3$)

Order 72: $C_3\times C_{24}$ ($C_4$) x 2, $C_6\times C_{12}$ ($C_2^2$)

Order 48: $C_{12}:C_4$ ($C_6$) x 2, $C_2\times C_{24}$ ($C_6$), $C_2\times C_{24}$ ($S_3$)

Order 36: $C_3\times C_{12}$ ($Q_8$), $C_3\times C_{12}$ ($C_2\times C_4$), $C_6^2$ ($D_4$)

Order 24: $C_{24}$ ($C_{12}$) x 2, $C_{24}$ ($C_3:C_4$) x 2, $C_2\times C_{12}$ ($C_2\times C_6$), $C_2\times C_{12}$ ($D_6$)

Order 18: $C_3\times C_6$ ($\SD_{16}$) x 2, $C_3\times C_6$ ($C_4:C_4$)

Order 16: $C_2\times C_8$ ($C_3\times S_3$)

Order 12: $C_2\times C_6$ ($D_{12}$), $C_2\times C_6$ ($C_3\times D_4$), $C_{12}$ ($C_2\times C_{12}$), $C_{12}$ ($C_6:C_4$), $C_{12}$ ($C_3:Q_8$), $C_{12}$ ($C_3\times Q_8$)

Order 9: $C_3^2$ ($C_8:C_4$)

Order 8: $C_8$ ($C_3:C_{12}$) x 2, $C_2\times C_4$ ($C_6\times S_3$)

Order 6: $C_6$ ($C_{24}:C_2$) x 2, $C_6$ ($C_3\times \SD_{16}$) x 2, $C_6$ ($C_4:C_{12}$), $C_6$ ($C_{12}:C_4$)

Order 4: $C_2^2$ ($C_3\times D_{12}$), $C_4$ ($C_6:C_{12}$), $C_4$ ($C_3^2:Q_8$)

Order 3: $C_3$ ($C_8:C_{12}$), $C_3$ ($C_{24}:C_4$)

Order 2: $C_2$ ($C_{24}:C_6$) x 2, $C_2$ ($C_{12}:C_{12}$)

Order 1: $C_1$ ($C_{24}:C_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 288: $C_{24}:C_{12}$ ($C_1$)

Order 144: $C_{12}:C_{12}$ ($C_2$), $C_6\times C_{24}$ ($C_2$)

Order 96: $C_{24}:C_4$ ($C_3$)

Order 72: $C_6\times C_{12}$ ($C_2^2$), $C_3\times C_{24}$ ($C_4$)

Order 48: $C_2\times C_{24}$ ($C_6$), $C_2\times C_{24}$ ($S_3$), $C_{12}:C_4$ ($C_6$)

Order 36: $C_3\times C_{12}$ ($Q_8$), $C_3\times C_{12}$ ($C_2\times C_4$), $C_6^2$ ($D_4$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_6$), $C_2\times C_{12}$ ($D_6$), $C_{24}$ ($C_{12}$), $C_{24}$ ($C_3:C_4$)

Order 18: $C_3\times C_6$ ($\SD_{16}$), $C_3\times C_6$ ($C_4:C_4$)

Order 16: $C_2\times C_8$ ($C_3\times S_3$)

Order 12: $C_2\times C_6$ ($D_{12}$), $C_2\times C_6$ ($C_3\times D_4$), $C_{12}$ ($C_2\times C_{12}$), $C_{12}$ ($C_6:C_4$), $C_{12}$ ($C_3:Q_8$), $C_{12}$ ($C_3\times Q_8$)

Order 9: $C_3^2$ ($C_8:C_4$)

Order 8: $C_2\times C_4$ ($C_6\times S_3$), $C_8$ ($C_3:C_{12}$)

Order 6: $C_6$ ($C_{24}:C_2$), $C_6$ ($C_3\times \SD_{16}$), $C_6$ ($C_4:C_{12}$), $C_6$ ($C_{12}:C_4$)

Order 4: $C_2^2$ ($C_3\times D_{12}$), $C_4$ ($C_6:C_{12}$), $C_4$ ($C_3^2:Q_8$)

Order 3: $C_3$ ($C_8:C_{12}$), $C_3$ ($C_{24}:C_4$)

Order 2: $C_2$ ($C_{12}:C_{12}$), $C_2$ ($C_{24}:C_6$)

Order 1: $C_1$ ($C_{24}:C_{12}$)

Series

Derived series

$C_{24}:C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_1$

Chief series

$C_{24}:C_{12}$

$\rhd$

$C_6\times C_{24}$

$\rhd$

$C_3\times C_{24}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{24}:C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

$\lhd$

$C_2\times C_{12}$

$\lhd$

$C_2\times C_{24}$

Supergroups

This group is a maximal subgroup of 101 larger groups in the database.

This group is a maximal quotient of 15 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 $36 \times 36$ rational character table.