Abstract group 288.642: $C_{12}^2:C_2$

• Label: 288.642
• Order: \( 2^{5} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{4} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$C_{12}^2:C_2$
Order:	\(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2\times C_2^4:C_3.D_4\times S_3$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Composition factors:	$C_2$ x 5, $C_3$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	15	8	48	48	168	288
Conjugacy classes	1	7	5	24	23	84	144
Divisions	1	7	3	12	13	24	60
Autjugacy classes	1	2	3	2	4	4	16

Dimension	1	2	4	8
Irr. complex chars.	96	48	0	0	144
Irr. rational chars.	8	24	22	6	60

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$96$
Rank:	$3$
Inequivalent generating triples:	$1456$

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:	$\langle a, b, c \mid a^{2}=b^{12}=c^{12}=[a,b]=[b,c]=1, c^{a}=c^{5} \rangle$
Permutation group:	Degree $14$ $\langle(1,2)(3,5)(4,6), (11,12,13,14), (7,8,9,10)(11,12,13,14), (1,3,4)(2,5,6), (7,9)(8,10)(11,13)(12,14), (11,13)(12,14), (1,4,3)(2,5,6)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrr} 6 & 8 & 9 \\ 3 & 6 & 3 \\ 7 & 11 & 2 \end{array}\right), \left(\begin{array}{rrr} 6 & 8 & 8 \\ 2 & 12 & 11 \\ 10 & 3 & 4 \end{array}\right), \left(\begin{array}{rrr} 3 & 9 & 9 \\ 12 & 0 & 1 \\ 8 & 5 & 4 \end{array}\right), \left(\begin{array}{rrr} 6 & 6 & 6 \\ 8 & 4 & 5 \\ 1 & 12 & 11 \end{array}\right), \left(\begin{array}{rrr} 12 & 0 & 0 \\ 0 & 12 & 0 \\ 0 & 0 & 12 \end{array}\right), \left(\begin{array}{rrr} 8 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrr} 0 & 6 & 11 \\ 8 & 5 & 3 \\ 4 & 2 & 9 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{13})$
Direct product:	$C_4$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $S_3$
Semidirect product:	$C_{12}^2$ $\,\rtimes\,$ $C_2$	$(C_{12}:C_4)$ $\,\rtimes\,$ $C_6$	$(C_4\times C_{12})$ $\,\rtimes\,$ $C_6$	$(C_{12}:C_{12})$ $\,\rtimes\,$ $C_2$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $C_2^3$	$(C_4\times D_6)$ . $C_6$	$C_6$ . $(C_4\times D_6)$	$(C_{12}\times D_6)$ . $C_2$	all 21
Aut. group:	$\Aut(C_2\times C_{130})$	$\Aut(S_3\times C_{65})$

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

Homology

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

Subgroups

There are 402 subgroups in 231 conjugacy classes, 138 normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4\times C_{12}$	$G/Z \simeq$ $S_3$
Commutator:	$G' \simeq$ $C_3$	$G/G' \simeq$ $C_2\times C_4\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_6\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}^2$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{12}^2:C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 288: $C_{12}^2:C_2$

Order 144: $C_{12}:C_{12}$ x 3, $C_{12}\times D_6$ x 3, $C_{12}^2$

Order 96: $S_3\times C_4^2$, $C_2\times C_4\times C_{12}$

Order 72: $S_3\times C_{12}$ x 12, $C_6\times C_{12}$ x 3, $C_6:C_{12}$ x 3, $C_6\times D_6$

Order 48: $C_4\times C_{12}$ x 6, $C_2^2\times C_{12}$ x 3, $C_4\times D_6$ x 3, $C_{12}:C_4$ x 3

Order 36: $C_3\times C_{12}$ x 6, $C_3:C_{12}$ x 6, $C_6\times S_3$ x 6, $C_6^2$

Order 32: $C_2\times C_4^2$

Order 24: $C_2\times C_{12}$ x 24, $C_4\times S_3$ x 12, $C_6:C_4$ x 3, $C_2^2\times C_6$, $C_2\times D_6$

Order 18: $C_3\times S_3$ x 4, $C_3\times C_6$ x 3

Order 16: $C_4^2$ x 4, $C_2^2\times C_4$ x 3

Order 12: $C_{12}$ x 24, $C_2\times C_6$ x 9, $D_6$ x 6, $C_3:C_4$ x 6

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 18, $C_2^3$

Order 6: $C_6$ x 13, $S_3$ x 4

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 288: $C_{12}^2:C_2$

Order 144: $C_{12}:C_{12}$, $C_{12}\times D_6$, $C_{12}^2$

Order 96: $S_3\times C_4^2$, $C_2\times C_4\times C_{12}$

Order 72: $C_6\times D_6$, $C_6\times C_{12}$, $C_6:C_{12}$, $S_3\times C_{12}$

Order 48: $C_4\times C_{12}$ x 4, $C_2^2\times C_{12}$, $C_4\times D_6$, $C_{12}:C_4$

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

Order 32: $C_2\times C_4^2$

Order 24: $C_2\times C_{12}$ x 5, $C_6:C_4$, $C_4\times S_3$, $C_2^2\times C_6$, $C_2\times D_6$

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

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

Order 12: $C_2\times C_6$ x 4, $C_{12}$ x 4, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

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

Order 6: $C_6$ x 4, $S_3$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 288: $C_{12}^2:C_2$ ($C_1$)

Order 144: $C_{12}:C_{12}$ ($C_2$) x 3, $C_{12}\times D_6$ ($C_2$) x 3, $C_{12}^2$ ($C_2$)

Order 96: $S_3\times C_4^2$ ($C_3$)

Order 72: $S_3\times C_{12}$ ($C_4$) x 12, $C_6\times C_{12}$ ($C_2^2$) x 3, $C_6:C_{12}$ ($C_2^2$) x 3, $C_6\times D_6$ ($C_2^2$)

Order 48: $C_4\times D_6$ ($C_6$) x 3, $C_{12}:C_4$ ($C_6$) x 3, $C_4\times C_{12}$ ($C_6$), $C_4\times C_{12}$ ($S_3$)

Order 36: $C_3\times C_{12}$ ($C_2\times C_4$) x 6, $C_3:C_{12}$ ($C_2\times C_4$) x 6, $C_6\times S_3$ ($C_2\times C_4$) x 6, $C_6^2$ ($C_2^3$)

Order 24: $C_4\times S_3$ ($C_{12}$) x 12, $C_2\times C_{12}$ ($C_2\times C_6$) x 3, $C_2\times C_{12}$ ($D_6$) x 3, $C_6:C_4$ ($C_2\times C_6$) x 3, $C_2\times D_6$ ($C_2\times C_6$)

Order 18: $C_3\times S_3$ ($C_4^2$) x 4, $C_3\times C_6$ ($C_2^2\times C_4$) x 3

Order 16: $C_4^2$ ($C_3\times S_3$)

Order 12: $D_6$ ($C_2\times C_{12}$) x 6, $C_{12}$ ($C_2\times C_{12}$) x 6, $C_{12}$ ($C_4\times S_3$) x 6, $C_3:C_4$ ($C_2\times C_{12}$) x 6, $C_2\times C_6$ ($C_2^2\times C_6$), $C_2\times C_6$ ($C_2\times D_6$)

Order 9: $C_3^2$ ($C_2\times C_4^2$)

Order 8: $C_2\times C_4$ ($C_6\times S_3$) x 3

Order 6: $S_3$ ($C_4\times C_{12}$) x 4, $C_6$ ($C_2^2\times C_{12}$) x 3, $C_6$ ($C_4\times D_6$) x 3

Order 4: $C_4$ ($S_3\times C_{12}$) x 6, $C_2^2$ ($C_6\times D_6$)

Order 3: $C_3$ ($S_3\times C_4^2$), $C_3$ ($C_2\times C_4\times C_{12}$)

Order 2: $C_2$ ($C_{12}\times D_6$) x 3

Order 1: $C_1$ ($C_{12}^2:C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 288: $C_{12}^2:C_2$ ($C_1$)

Order 144: $C_{12}:C_{12}$ ($C_2$), $C_{12}\times D_6$ ($C_2$), $C_{12}^2$ ($C_2$)

Order 96: $S_3\times C_4^2$ ($C_3$)

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

Order 48: $C_4\times D_6$ ($C_6$), $C_4\times C_{12}$ ($C_6$), $C_4\times C_{12}$ ($S_3$), $C_{12}:C_4$ ($C_6$)

Order 36: $C_3\times C_{12}$ ($C_2\times C_4$), $C_3:C_{12}$ ($C_2\times C_4$), $C_6^2$ ($C_2^3$), $C_6\times S_3$ ($C_2\times C_4$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_6$), $C_2\times C_{12}$ ($D_6$), $C_6:C_4$ ($C_2\times C_6$), $C_4\times S_3$ ($C_{12}$), $C_2\times D_6$ ($C_2\times C_6$)

Order 18: $C_3\times C_6$ ($C_2^2\times C_4$), $C_3\times S_3$ ($C_4^2$)

Order 16: $C_4^2$ ($C_3\times S_3$)

Order 12: $C_2\times C_6$ ($C_2^2\times C_6$), $C_2\times C_6$ ($C_2\times D_6$), $D_6$ ($C_2\times C_{12}$), $C_{12}$ ($C_2\times C_{12}$), $C_{12}$ ($C_4\times S_3$), $C_3:C_4$ ($C_2\times C_{12}$)

Order 9: $C_3^2$ ($C_2\times C_4^2$)

Order 8: $C_2\times C_4$ ($C_6\times S_3$)

Order 6: $C_6$ ($C_2^2\times C_{12}$), $C_6$ ($C_4\times D_6$), $S_3$ ($C_4\times C_{12}$)

Order 4: $C_2^2$ ($C_6\times D_6$), $C_4$ ($S_3\times C_{12}$)

Order 3: $C_3$ ($S_3\times C_4^2$), $C_3$ ($C_2\times C_4\times C_{12}$)

Order 2: $C_2$ ($C_{12}\times D_6$)

Order 1: $C_1$ ($C_{12}^2:C_2$)

Series

Derived series

$C_{12}^2:C_2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_{12}^2:C_2$

$\rhd$

$C_{12}\times D_6$

$\rhd$

$C_6\times C_{12}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{12}^2:C_2$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_4\times C_{12}$

Supergroups

This group is a maximal subgroup of 92 larger groups in the database.

This group is a maximal quotient of 29 larger groups in the database.

Character theory

Complex character table

See the $144 \times 144$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $60 \times 60$ rational character table.