Abstract group 144.162: $C_{12}:D_6$

• Label: 144.162
• Order: \( 2^{4} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $10$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_{12}:D_6$
Order:	\(144\)\(\medspace = 2^{4} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_{12}:C_2^4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	23	8	8	76	28	144
Conjugacy classes	1	7	5	2	23	7	45
Divisions	1	7	3	2	13	4	30
Autjugacy classes	1	4	3	2	8	4	22

Dimension	1	2	4	8
Irr. complex chars.	24	18	3	0	45
Irr. rational chars.	8	14	7	1	30

Minimal presentations

Permutation degree:	$10$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$4368$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	8
Arbitrary	4	6	6

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{6}=c^{12}=[a,b]=1, c^{a}=c^{5}, c^{b}=c^{7} \rangle$
Permutation group:	Degree $10$ $\langle(1,2)(3,5)(4,6)(7,8)(9,10), (7,9)(8,10), (9,10), (1,3,4)(2,5,6), (7,8)(9,10), (1,4,3)(2,5,6)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rr} 4 & 7 \\ 7 & 11 \end{array}\right), \left(\begin{array}{rr} 1 & 7 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 0 & 9 \\ 3 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/14\Z)$
Transitive group:	24T208	36T112			more information
Direct product:	$C_3$ $\, \times\, $ $S_3$ $\, \times\, $ $D_4$
Semidirect product:	$D_{12}$ $\,\rtimes\,$ $C_6$	$C_{12}$ $\,\rtimes\,$ $D_6$	$C_6^2$ $\,\rtimes\,$ $C_2^2$ (2)	$(C_4\times S_3)$ $\,\rtimes\,$ $C_6$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_6$ . $(C_2\times C_6)$	$C_6$ . $(C_2\times D_6)$	$C_2$ . $(C_6\times D_6)$	$(C_6\times S_3)$ . $C_2^2$	all 6

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

Homology

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

Subgroups

There are 260 subgroups in 116 conjugacy classes, 50 normal (26 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_2\times D_6$
Commutator:	$G' \simeq$ $C_6$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_6\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_3^2$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{12}:D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
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

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 144: $C_{12}:D_6$

Order 72: $C_6\times D_6$ x 2, $C_6\wr C_2$ x 2, $D_4\times C_3^2$, $C_3\times D_{12}$, $S_3\times C_{12}$

Order 48: $C_6\times D_4$, $S_3\times D_4$

Order 36: $C_6\times S_3$ x 7, $C_6^2$ x 2, $C_3\times C_{12}$, $C_3:C_{12}$

Order 24: $C_3\times D_4$ x 6, $C_3:D_4$ x 2, $C_2^2\times C_6$ x 2, $C_2\times D_6$ x 2, $C_2\times C_{12}$, $D_{12}$, $C_4\times S_3$

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

Order 16: $C_2\times D_4$

Order 12: $C_2\times C_6$ x 13, $D_6$ x 7, $C_{12}$ x 4, $C_3:C_4$

Order 9: $C_3^2$

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

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

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 144: $C_{12}:D_6$

Order 72: $C_6\times D_6$, $D_4\times C_3^2$, $C_6\wr C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$

Order 48: $C_6\times D_4$, $S_3\times D_4$

Order 36: $C_6\times S_3$ x 3, $C_3\times C_{12}$, $C_3:C_{12}$, $C_6^2$

Order 24: $C_3\times D_4$ x 5, $C_2\times C_{12}$, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_2^2\times C_6$, $C_2\times D_6$

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

Order 16: $C_2\times D_4$

Order 12: $C_2\times C_6$ x 6, $C_{12}$ x 4, $D_6$ x 3, $C_3:C_4$

Order 9: $C_3^2$

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

Order 6: $C_6$ x 8, $S_3$ x 2

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 144: $C_{12}:D_6$ ($C_1$)

Order 72: $C_6\times D_6$ ($C_2$) x 2, $C_6\wr C_2$ ($C_2$) x 2, $D_4\times C_3^2$ ($C_2$), $C_3\times D_{12}$ ($C_2$), $S_3\times C_{12}$ ($C_2$)

Order 48: $S_3\times D_4$ ($C_3$)

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

Order 24: $C_3:D_4$ ($C_6$) x 2, $C_2\times D_6$ ($C_6$) x 2, $D_{12}$ ($C_6$), $C_4\times S_3$ ($C_6$), $C_3\times D_4$ ($C_6$), $C_3\times D_4$ ($S_3$)

Order 18: $C_3\times S_3$ ($D_4$) x 2, $C_3\times C_6$ ($C_2^3$)

Order 12: $D_6$ ($C_2\times C_6$) x 3, $C_2\times C_6$ ($C_2\times C_6$) x 2, $C_2\times C_6$ ($D_6$) x 2, $C_{12}$ ($C_2\times C_6$), $C_{12}$ ($D_6$), $C_3:C_4$ ($C_2\times C_6$)

Order 9: $C_3^2$ ($C_2\times D_4$)

Order 8: $D_4$ ($C_3\times S_3$)

Order 6: $S_3$ ($C_3\times D_4$) x 2, $C_6$ ($C_2^2\times C_6$), $C_6$ ($C_2\times D_6$)

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

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

Order 2: $C_2$ ($C_6\times D_6$)

Order 1: $C_1$ ($C_{12}:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 144: $C_{12}:D_6$ ($C_1$)

Order 72: $C_6\times D_6$ ($C_2$), $D_4\times C_3^2$ ($C_2$), $C_6\wr C_2$ ($C_2$), $C_3\times D_{12}$ ($C_2$), $S_3\times C_{12}$ ($C_2$)

Order 48: $S_3\times D_4$ ($C_3$)

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

Order 24: $C_3:D_4$ ($C_6$), $D_{12}$ ($C_6$), $C_4\times S_3$ ($C_6$), $C_2\times D_6$ ($C_6$), $C_3\times D_4$ ($C_6$), $C_3\times D_4$ ($S_3$)

Order 18: $C_3\times C_6$ ($C_2^3$), $C_3\times S_3$ ($D_4$)

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

Order 9: $C_3^2$ ($C_2\times D_4$)

Order 8: $D_4$ ($C_3\times S_3$)

Order 6: $C_6$ ($C_2^2\times C_6$), $C_6$ ($C_2\times D_6$), $S_3$ ($C_3\times D_4$)

Order 4: $C_2^2$ ($C_6\times S_3$), $C_4$ ($C_6\times S_3$)

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

Order 2: $C_2$ ($C_6\times D_6$)

Order 1: $C_1$ ($C_{12}:D_6$)

Series

Derived series

$C_{12}:D_6$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$C_{12}:D_6$

$\rhd$

$S_3\times C_{12}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{12}:D_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_3\times D_4$

Supergroups

This group is a maximal subgroup of 78 larger groups in the database.

This group is a maximal quotient of 86 larger groups in the database.

Character theory

Complex character table

See the $45 \times 45$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.