Abstract group 48.36: $C_2\times D_{12}$

• Label: 48.36
• Order: \( 2^{4} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $9$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_2\times D_{12}$
Order:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2\wr C_2^2\times S_3$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \)
Composition factors:	$C_2$ x 4, $C_3$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	3	4	6	12
Elements	1	27	2	4	6	8	48
Conjugacy classes	1	7	1	2	3	4	18
Divisions	1	7	1	2	3	2	16
Autjugacy classes	1	3	1	1	2	1	9

Dimension	1	2	4
Irr. complex chars.	8	10	0	18
Irr. rational chars.	8	6	2	16

Minimal presentations

Permutation degree:	$9$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$84$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	3	4

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{12}=[a,b]=[b,c]=1, c^{a}=c^{11} \rangle$
Permutation group:	$\langle(2,3)(4,5)(7,9), (4,5), (6,7,8,9), (6,8)(7,9), (1,2,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrr} -1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\Z)$
	$\left\langle \left(\begin{array}{rrr} 6 & 0 & 10 \\ 3 & 1 & 6 \\ 2 & 0 & 5 \end{array}\right), \left(\begin{array}{rrr} 5 & 0 & 1 \\ 8 & 10 & 5 \\ 0 & 9 & 7 \end{array}\right), \left(\begin{array}{rrr} 10 & 1 & 7 \\ 0 & 4 & 2 \\ 8 & 2 & 7 \end{array}\right), \left(\begin{array}{rrr} 2 & 6 & 0 \\ 5 & 9 & 0 \\ 4 & 8 & 10 \end{array}\right), \left(\begin{array}{rrr} 10 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 10 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{11})$
Transitive group:	24T29				more information
Direct product:	$C_2$ $\, \times\, $ $D_{12}$
Semidirect product:	$C_6$ $\,\rtimes\,$ $D_4$ (2)	$C_4$ $\,\rtimes\,$ $D_6$ (2)	$D_6$ $\,\rtimes\,$ $C_2^2$ (4)	$C_{12}$ $\,\rtimes\,$ $C_2^2$ (2)	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^2$ . $D_6$	$C_6$ . $C_2^3$	$C_2$ . $(C_2\times D_6)$	$(C_2\times C_6)$ . $C_2^2$	more information

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

Homology

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

Subgroups

There are 124 subgroups in 54 conjugacy classes, 27 normal (9 characteristic).

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

Special subgroups

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

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 48: $C_2\times D_{12}$

Order 24: $D_{12}$ x 4, $C_2\times D_6$ x 2, $C_2\times C_{12}$

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 8, $C_{12}$ x 2, $C_2\times C_6$

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

Order 6: $S_3$ x 4, $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 48: $C_2\times D_{12}$

Order 24: $C_2\times C_{12}$, $D_{12}$, $C_2\times D_6$

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 2, $C_2\times C_6$, $C_{12}$

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

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

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 48: $C_2\times D_{12}$ ($C_1$)

Order 24: $D_{12}$ ($C_2$) x 4, $C_2\times D_6$ ($C_2$) x 2, $C_2\times C_{12}$ ($C_2$)

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

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

Order 6: $C_6$ ($D_4$) x 2, $C_6$ ($C_2^3$)

Order 4: $C_4$ ($D_6$) x 2, $C_2^2$ ($D_6$)

Order 3: $C_3$ ($C_2\times D_4$)

Order 2: $C_2$ ($D_{12}$) x 2, $C_2$ ($C_2\times D_6$)

Order 1: $C_1$ ($C_2\times D_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 48: $C_2\times D_{12}$ ($C_1$)

Order 24: $C_2\times C_{12}$ ($C_2$), $D_{12}$ ($C_2$), $C_2\times D_6$ ($C_2$)

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

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

Order 6: $C_6$ ($C_2^3$), $C_6$ ($D_4$)

Order 4: $C_2^2$ ($D_6$), $C_4$ ($D_6$)

Order 3: $C_3$ ($C_2\times D_4$)

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

Order 1: $C_1$ ($C_2\times D_{12}$)

Series

Derived series

$C_2\times D_{12}$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$C_2\times D_{12}$

$\rhd$

$C_2\times D_6$

$\rhd$

$D_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2\times D_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 87 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 $18 \times 18$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	2D	2E	2F	2G	3A	4A	4B	6A	6B	6C	12A	12B
	Size	1	1	1	1	6	6	6	6	2	2	2	2	2	2	4	4
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	3A	2C	2C	3A	3A	3A	6C	6C
	3 P	1A	2A	2B	2C	2D	2E	2F	2G	1A	4A	4B	2A	2B	2C	4A	4B
48.36.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
48.36.1b		$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$
48.36.1c		$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$
48.36.1d		$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$
48.36.1e		$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$
48.36.1f		$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
48.36.1g		$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
48.36.1h		$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
48.36.2a		$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$-1$	$2$	$2$	$-1$	$-1$	$-1$	$-1$	$-1$
48.36.2b		$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$2$	$0$	$0$	$2$	$-2$	$-2$	$0$	$0$
48.36.2c		$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$2$	$0$	$0$	$-2$	$2$	$-2$	$0$	$0$
48.36.2d		$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$-1$	$-2$	$2$	$1$	$1$	$-1$	$1$	$-1$
48.36.2e		$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$-1$	$2$	$-2$	$1$	$1$	$-1$	$-1$	$1$
48.36.2f		$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$-1$	$-2$	$-2$	$-1$	$-1$	$-1$	$1$	$1$
48.36.2g		$4$	$-4$	$-4$	$4$	$0$	$0$	$0$	$0$	$-2$	$0$	$0$	$-2$	$2$	$2$	$0$	$0$
48.36.2h		$4$	$4$	$-4$	$-4$	$0$	$0$	$0$	$0$	$-2$	$0$	$0$	$2$	$-2$	$2$	$0$	$0$