Abstract group 48.11: $C_{12}:C_4$

• Label: 48.11
• Order: \( 2^{4} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $11$
• Trans deg.: $48$
• Rank: $2$

Group information

Description:	$C_{12}:C_4$
Order:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^4:D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \)
Composition factors:	$C_2$ x 4, $C_3$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	3	2	28	6	8	48
Conjugacy classes	1	3	1	12	3	4	24
Divisions	1	3	1	6	3	2	16
Autjugacy classes	1	2	1	2	2	1	9

Dimension	1	2	4
Irr. complex chars.	16	8	0	24
Irr. rational chars.	4	10	2	16

Minimal presentations

Permutation degree:	$11$
Transitive degree:	$48$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

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

Constructions

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

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

Homology

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

Subgroups

There are 44 subgroups in 30 conjugacy classes, 23 normal (9 characteristic).

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

Special subgroups

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

Order 48: $C_{12}:C_4$

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

Order 16: $C_4^2$

Order 12: $C_3:C_4$ x 4, $C_{12}$ x 2, $C_2\times C_6$

Order 8: $C_2\times C_4$ x 3

Order 6: $C_6$ x 3

Order 4: $C_4$ x 6, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 48: $C_{12}:C_4$

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

Order 16: $C_4^2$

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

Order 8: $C_2\times C_4$ x 2

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 48: $C_{12}:C_4$ ($C_1$)

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

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

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

Order 6: $C_6$ ($C_2\times C_4$) x 3

Order 4: $C_4$ ($C_3:C_4$) x 2, $C_2^2$ ($D_6$)

Order 3: $C_3$ ($C_4^2$)

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

Order 1: $C_1$ ($C_{12}:C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 48: $C_{12}:C_4$ ($C_1$)

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

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

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

Order 6: $C_6$ ($C_2\times C_4$) x 2

Order 4: $C_2^2$ ($D_6$), $C_4$ ($C_3:C_4$)

Order 3: $C_3$ ($C_4^2$)

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

Order 1: $C_1$ ($C_{12}:C_4$)

Series

Derived series

$C_{12}:C_4$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_{12}:C_4$

$\rhd$

$C_6:C_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{12}:C_4$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 87 larger groups in the database.

This group is a maximal quotient of 62 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

		1A	2A	2B	2C	3A	4A	4B	4C	4D	4E	4F	6A	6B	6C	12A	12B
	Size	1	1	1	1	2	2	2	6	6	6	6	2	2	2	4	4
	2 P	1A	1A	1A	1A	3A	2C	2C	2A	2A	2B	2B	3A	3A	3A	6C	6C
	3 P	1A	2A	2B	2C	1A	4A	4B	4C	4D	4E	4F	2A	2B	2C	4A	4B
	Schur
48.11.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
48.11.1b		$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
48.11.1c		$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
48.11.1d		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$
48.11.1e		$2$	$-2$	$-2$	$2$	$2$	$0$	$0$	$-2$	$2$	$-2$	$2$	$-2$	$2$	$-2$	$0$	$0$
48.11.1f		$2$	$-2$	$-2$	$2$	$2$	$0$	$0$	$2$	$-2$	$2$	$-2$	$-2$	$2$	$-2$	$0$	$0$
48.11.1g		$2$	$-2$	$2$	$-2$	$2$	$-2$	$2$	$0$	$0$	$0$	$0$	$2$	$-2$	$-2$	$2$	$-2$
48.11.1h		$2$	$-2$	$2$	$-2$	$2$	$2$	$-2$	$0$	$0$	$0$	$0$	$2$	$-2$	$-2$	$-2$	$2$
48.11.1i		$2$	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$-2$	$2$	$0$	$0$
48.11.1j		$2$	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$-2$	$2$	$0$	$0$
48.11.2a		$2$	$2$	$2$	$2$	$-1$	$2$	$2$	$0$	$0$	$0$	$0$	$-1$	$-1$	$-1$	$-1$	$-1$
48.11.2b		$2$	$2$	$2$	$2$	$-1$	$-2$	$-2$	$0$	$0$	$0$	$0$	$-1$	$-1$	$-1$	$1$	$1$
48.11.2c	2	$2$	$-2$	$2$	$-2$	$-1$	$-2$	$2$	$0$	$0$	$0$	$0$	$-1$	$1$	$1$	$-1$	$1$
48.11.2d	2	$2$	$-2$	$2$	$-2$	$-1$	$2$	$-2$	$0$	$0$	$0$	$0$	$-1$	$1$	$1$	$1$	$-1$
48.11.2e		$4$	$-4$	$-4$	$4$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$2$	$-2$	$2$	$0$	$0$
48.11.2f		$4$	$4$	$-4$	$-4$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$2$	$2$	$-2$	$0$	$0$