Abstract group 32.24: $C_4^2:C_2$

• Label: 32.24
• Order: \( 2^{5} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $12$
• Trans deg.: $16$
• Rank: $3$

Group information

Description:	$C_4^2:C_2$
Order:	\(32\)\(\medspace = 2^{5} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2^5:D_4$, of order \(256\)\(\medspace = 2^{8} \)
Composition factors:	$C_2$ x 5
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.

Group statistics

Order	1	2	4
Elements	1	7	24	32
Conjugacy classes	1	5	14	20
Divisions	1	5	8	14
Autjugacy classes	1	3	3	7

Dimension	1	2	4
Irr. complex chars.	16	4	0	20
Irr. rational chars.	8	4	2	14

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$16$
Rank:	$3$
Inequivalent generating triples:	$42$

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:	$\langle a, b, c \mid a^{2}=b^{4}=c^{4}=[a,c]=[b,c]=1, b^{a}=bc^{2} \rangle$
Permutation group:	Degree $12$ $\langle(2,5)(6,8)(9,10,11,12), (1,2)(3,6)(4,5)(7,8)(9,11)(10,12), (1,3,4,7)(2,6,5,8)(9,11)(10,12), (1,4)(2,5)(3,7)(6,8), (9,11)(10,12)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 4 & 1 & 1 \\ 1 & 3 & 4 \end{array}\right), \left(\begin{array}{rrr} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 2 & 1 & 3 \end{array}\right), \left(\begin{array}{rrr} 4 & 0 & 0 \\ 1 & 1 & 0 \\ 3 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ 1 & 3 & 0 \\ 3 & 0 & 3 \end{array}\right), \left(\begin{array}{rrr} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{5})$
Transitive group:	16T17	32T3			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_4^2$ $\,\rtimes\,$ $C_2$ (2)	$(C_4:C_4)$ $\,\rtimes\,$ $C_2$ (2)	$(C_2\times C_4)$ $\,\rtimes\,$ $C_4$ (4)		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^3$ . $C_2^2$	$C_2^2$ . $C_2^3$	$C_2$ . $(D_4:C_2)$ (2)	$C_4$ . $(C_2\times C_4)$ (4)	all 9

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

Homology

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

Subgroups

There are 46 subgroups in 38 conjugacy classes, 30 normal (8 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 32: $C_4^2:C_2$

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

Order 8: $C_2\times C_4$ x 10, $C_2^3$

Order 4: $C_4$ x 8, $C_2^2$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 32: $C_4^2:C_2$

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

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

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 32: $C_4^2:C_2$ ($C_1$)

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

Order 8: $C_2\times C_4$ ($C_2^2$) x 6, $C_2\times C_4$ ($C_4$) x 4, $C_2^3$ ($C_2^2$)

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

Order 2: $C_2$ ($D_4:C_2$) x 2, $C_2$ ($C_2^2\times C_4$)

Order 1: $C_1$ ($C_4^2:C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 32: $C_4^2:C_2$ ($C_1$)

Order 16: $C_4:C_4$ ($C_2$), $C_2^2:C_4$ ($C_2$), $C_4^2$ ($C_2$), $C_2^2\times C_4$ ($C_2$)

Order 8: $C_2\times C_4$ ($C_2^2$) x 3, $C_2^3$ ($C_2^2$), $C_2\times C_4$ ($C_4$)

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

Order 2: $C_2$ ($D_4:C_2$), $C_2$ ($C_2^2\times C_4$)

Order 1: $C_1$ ($C_4^2:C_2$)

Series

Derived series

$C_4^2:C_2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_4^2:C_2$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4^2:C_2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_4^2:C_2$

Supergroups

This group is a maximal subgroup of 131 larger groups in the database.

This group is a maximal quotient of 118 larger groups in the database.

Character theory

Complex character table

See the $20 \times 20$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H
	Size	1	1	1	1	2	2	2	2	2	2	4	4	4	4
	2 P	1A	1A	1A	1A	1A	1A	2A	2A	2A	2A	2B	2C	2C	2B
32.24.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.24.1b		$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$1$
32.24.1c		$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$1$
32.24.1d		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$
32.24.1e		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
32.24.1f		$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$
32.24.1g		$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
32.24.1h		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$
32.24.1i		$2$	$2$	$-2$	$-2$	$-2$	$2$	$-2$	$2$	$0$	$0$	$0$	$-2$	$0$	$2$
32.24.1j		$2$	$2$	$-2$	$-2$	$-2$	$2$	$2$	$-2$	$0$	$0$	$0$	$2$	$0$	$-2$
32.24.1k		$2$	$2$	$-2$	$-2$	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$2$	$0$	$-2$
32.24.1l		$2$	$2$	$-2$	$-2$	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$-2$	$0$	$2$
32.24.2a		$4$	$-4$	$-4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.24.2b		$4$	$-4$	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$