Abstract group 1328.3: $C_2^2:C_{332}$

• Label: 1328.3
• Order: \( 2^{4} \cdot 83 \)
• Exponent: \( 2^{2} \cdot 83 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 83 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 83 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 41 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 41 \)
• Perm deg.: $91$
• Trans deg.: $664$
• Rank: $2$

Group information

Description:	$C_2^2:C_{332}$
Order:	\(1328\)\(\medspace = 2^{4} \cdot 83 \)
Exponent:	\(332\)\(\medspace = 2^{2} \cdot 83 \)
Automorphism group:	$C_{82}\times C_2^2\wr C_2$, of order \(2624\)\(\medspace = 2^{6} \cdot 41 \)
Composition factors:	$C_2$ x 4, $C_{83}$
Nilpotency class:	$2$
Derived length:	$2$

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

Group statistics

Order	1	2	4	83	166	332
Elements	1	7	8	82	574	656	1328
Conjugacy classes	1	5	4	82	410	328	830
Divisions	1	5	2	1	5	2	16
Autjugacy classes	1	3	1	1	3	1	10

Dimension	1	2	82	164
Irr. complex chars.	664	166	0	0	830
Irr. rational chars.	4	4	4	4	16

Minimal presentations

Permutation degree:	$91$
Transitive degree:	$664$
Rank:	$2$
Inequivalent generating pairs:	$252$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{4}=c^{166}=[a,c]=[b,c]=1, b^{a}=bc^{83} \rangle$
Permutation group:	Degree $91$ $\langle(1,6,2,5)(3,8,4,7), (1,2)(3,4)(5,8)(6,7), (1,3)(2,4)(5,7)(6,8), (1,2)(3,4) \!\cdots\! \rangle$
Direct product:	$C_{83}$ $\, \times\, $ $(C_2^2:C_4)$
Semidirect product:	$C_2^2$ $\,\rtimes\,$ $C_{332}$ (2)	$(C_2\times C_{332})$ $\,\rtimes\,$ $C_2$ (2)	$(C_2\times C_{166})$ $\,\rtimes\,$ $C_4$ (2)	$(C_2\times C_4)$ $\,\rtimes\,$ $C_{166}$ (2)	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{166}$ . $D_4$ (2)	$C_2^3$ . $C_{166}$	$C_2$ . $(D_4\times C_{83})$ (2)	$C_{166}$ . $(C_2\times C_4)$	all 8

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

Homology

Abelianization:	$C_{2} \times C_{332} \simeq C_{2} \times C_{4} \times C_{83}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 46 subgroups in 34 conjugacy classes, 22 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{166}$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2\times C_{332}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2\times C_{166}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2:C_{332}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^2:C_{332}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{166}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:C_4$
83-Sylow subgroup:	$P_{ 83 } \simeq$ $C_{83}$

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 1328: $C_2^2:C_{332}$

Order 664: $C_2\times C_{332}$ x 2, $C_2^2\times C_{166}$

Order 332: $C_2\times C_{166}$ x 5, $C_{332}$ x 2

Order 166: $C_{166}$ x 5

Order 83: $C_{83}$

Order 16: $C_2^2:C_4$

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

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

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1328: $C_2^2:C_{332}$

Order 664: $C_2\times C_{332}$, $C_2^2\times C_{166}$

Order 332: $C_2\times C_{166}$ x 3, $C_{332}$

Order 166: $C_{166}$ x 3

Order 83: $C_{83}$

Order 16: $C_2^2:C_4$

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

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1328: $C_2^2:C_{332}$ ($C_1$)

Order 664: $C_2\times C_{332}$ ($C_2$) x 2, $C_2^2\times C_{166}$ ($C_2$)

Order 332: $C_2\times C_{166}$ ($C_4$) x 2, $C_2\times C_{166}$ ($C_2^2$)

Order 166: $C_{166}$ ($D_4$) x 2, $C_{166}$ ($C_2\times C_4$)

Order 83: $C_{83}$ ($C_2^2:C_4$)

Order 16: $C_2^2:C_4$ ($C_{83}$)

Order 8: $C_2\times C_4$ ($C_{166}$) x 2, $C_2^3$ ($C_{166}$)

Order 4: $C_2^2$ ($C_{332}$) x 2, $C_2^2$ ($C_2\times C_{166}$)

Order 2: $C_2$ ($D_4\times C_{83}$) x 2, $C_2$ ($C_2\times C_{332}$)

Order 1: $C_1$ ($C_2^2:C_{332}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1328: $C_2^2:C_{332}$ ($C_1$)

Order 664: $C_2\times C_{332}$ ($C_2$), $C_2^2\times C_{166}$ ($C_2$)

Order 332: $C_2\times C_{166}$ ($C_2^2$), $C_2\times C_{166}$ ($C_4$)

Order 166: $C_{166}$ ($D_4$), $C_{166}$ ($C_2\times C_4$)

Order 83: $C_{83}$ ($C_2^2:C_4$)

Order 16: $C_2^2:C_4$ ($C_{83}$)

Order 8: $C_2^3$ ($C_{166}$), $C_2\times C_4$ ($C_{166}$)

Order 4: $C_2^2$ ($C_2\times C_{166}$), $C_2^2$ ($C_{332}$)

Order 2: $C_2$ ($D_4\times C_{83}$), $C_2$ ($C_2\times C_{332}$)

Order 1: $C_1$ ($C_2^2:C_{332}$)

Series

Derived series

$C_2^2:C_{332}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_2^2:C_{332}$

$\rhd$

$C_2^2\times C_{166}$

$\rhd$

$C_2\times C_{166}$

$\rhd$

$C_{166}$

$\rhd$

$C_{83}$

$\rhd$

$C_1$

Lower central series

$C_2^2:C_{332}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{166}$

$\lhd$

$C_2^2:C_{332}$

Character theory

Complex character table

The $830 \times 830$ character table is not available for this group.

Rational character table

		1A	2A	2B	2C	2D	2E	4A	4B	83A	166A	166B	166C	166D	166E	332A	332B
	Size	1	1	1	1	2	2	4	4	82	82	82	82	164	164	328	328
	2 P	1A	1A	1A	1A	1A	1A	2B	2C	83A	83A	83A	83A	83A	83A	166B	166C
	83 P	1A	2A	2B	2C	2D	2E	4A	4B	1A	2A	2B	2C	2D	2E	4A	4B
1328.3.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1328.3.1b		$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$
1328.3.1c		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$
1328.3.1d		$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
1328.3.1e		$2$	$2$	$-2$	$-2$	$-2$	$2$	$0$	$0$	$2$	$2$	$-2$	$-2$	$-2$	$2$	$0$	$0$
1328.3.1f		$2$	$2$	$-2$	$-2$	$2$	$-2$	$0$	$0$	$2$	$2$	$-2$	$-2$	$2$	$-2$	$0$	$0$
1328.3.1g		$82$	$82$	$82$	$82$	$82$	$82$	$82$	$82$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
1328.3.1h		$82$	$82$	$82$	$82$	$-82$	$-82$	$-82$	$82$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$
1328.3.1i		$82$	$82$	$82$	$82$	$-82$	$-82$	$82$	$-82$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$1$
1328.3.1j		$82$	$82$	$82$	$82$	$82$	$82$	$-82$	$-82$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
1328.3.1k		$164$	$164$	$-164$	$-164$	$-164$	$164$	$0$	$0$	$-2$	$-2$	$2$	$2$	$2$	$-2$	$0$	$0$
1328.3.1l		$164$	$164$	$-164$	$-164$	$164$	$-164$	$0$	$0$	$-2$	$-2$	$2$	$2$	$-2$	$2$	$0$	$0$
1328.3.2a		$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$
1328.3.2b		$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$
1328.3.2c		$164$	$-164$	$-164$	$164$	$0$	$0$	$0$	$0$	$-2$	$2$	$2$	$-2$	$0$	$0$	$0$	$0$
1328.3.2d		$164$	$-164$	$164$	$-164$	$0$	$0$	$0$	$0$	$-2$	$2$	$-2$	$2$	$0$	$0$	$0$	$0$