Abstract group $C_4:C_4$

• Label: 16.4
• Order: \( 2^{4} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $8$
• Trans deg.: $16$
• Rank: $2$

Group information

Description:	$C_4:C_4$
Order:	\(16\)\(\medspace = 2^{4} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) (generators)
Outer automorphisms:	$D_4$, of order \(8\)\(\medspace = 2^{3} \)
Composition factors:	$C_2$ x 4
Nilpotency class:	$2$
Derived length:	$2$

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

Group statistics

Order	1	2	4
Elements	1	3	12	16
Conjugacy classes	1	3	6	10
Divisions	1	3	4	8
Autjugacy classes	1	3	2	6

Dimension	1	2
Irr. complex chars.	8	2	10
Irr. rational chars.	4	4	8

Minimal Presentations

Permutation degree:	$8$
Transitive degree:	$16$
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	4	4

Constructions

Presentation:	$\langle a, b \mid a^{4}=b^{4}=1, b^{a}=b^{3} \rangle$
Permutation group:	$\langle(2,4)(5,6,7,8), (1,2,3,4)(5,7)(6,8), (1,3)(2,4), (5,7)(6,8)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\Z)$
	$\left\langle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 4 & 1 & 1 \\ 1 & 3 & 4 \end{array}\right), \left(\begin{array}{rrr} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 2 & 1 & 3 \end{array}\right), \left(\begin{array}{rrr} 4 & 0 & 0 \\ 1 & 1 & 0 \\ 3 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{5})$
Transitive group:	16T8				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_4$ $\,\rtimes\,$ $C_4$ (2)				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $Q_8$	$C_2$ . $D_4$	$C_2^2$ . $C_2^2$	$(C_2\times C_4)$ . $C_2$ (3)	all 5

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

Homology

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

Subgroups

There are 15 subgroups in 13 conjugacy classes, 11 normal (7 characteristic).

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

Special subgroups

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

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 16: $C_4:C_4$

Order 8: $C_2\times C_4$ x 3

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 16: $C_4:C_4$

Order 8: $C_2\times C_4$ x 2

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 16: $C_4:C_4$ ($C_1$)

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

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

Order 2: $C_2$ ($Q_8$), $C_2$ ($D_4$), $C_2$ ($C_2\times C_4$)

Order 1: $C_1$ ($C_4:C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 16: $C_4:C_4$ ($C_1$)

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

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

Order 2: $C_2$ ($Q_8$), $C_2$ ($D_4$), $C_2$ ($C_2\times C_4$)

Order 1: $C_1$ ($C_4:C_4$)

Series

Derived series	$C_4:C_4$	$\rhd$	$C_2$	$\rhd$	$C_1$
Chief series	$C_4:C_4$	$\rhd$	$C_2\times C_4$	$\rhd$	$C_2^2$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$C_4:C_4$	$\rhd$	$C_2$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_2^2$	$\lhd$	$C_4:C_4$

Supergroups

This group is a maximal subgroup of 126 larger groups in the database.

This group is a maximal quotient of 142 larger groups in the database.

Character theory

Complex character table

		1A	2A	2B	2C	4A	4B	4C1	4C-1	4D1	4D-1
	Size	1	1	1	1	2	2	2	2	2	2
	2 P	1A	1A	1A	1A	2B	2C	2C	2C	2B	2C
	Type
16.4.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.4.1b	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
16.4.1c	R	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$
16.4.1d	R	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$
16.4.1e1	C	$1$	$-1$	$1$	$-1$	$-1$	$-i$	$-i$	$i$	$1$	$i$
16.4.1e2	C	$1$	$-1$	$1$	$-1$	$-1$	$i$	$i$	$-i$	$1$	$-i$
16.4.1f1	C	$1$	$-1$	$1$	$-1$	$1$	$-i$	$i$	$-i$	$-1$	$i$
16.4.1f2	C	$1$	$-1$	$1$	$-1$	$1$	$i$	$-i$	$i$	$-1$	$-i$
16.4.2a	R	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
16.4.2b	S	$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$

Rational character table

		1A	2A	2B	2C	4A	4B	4C	4D
	Size	1	1	1	1	2	2	4	4
	2 P	1A	1A	1A	1A	2B	2C	2C	2B
	Schur
16.4.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.4.1b		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$
16.4.1c		$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
16.4.1d		$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$
16.4.1e		$2$	$-2$	$2$	$-2$	$-2$	$0$	$0$	$2$
16.4.1f		$2$	$-2$	$2$	$-2$	$2$	$0$	$0$	$-2$
16.4.2a		$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$
16.4.2b	2	$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$