Abstract group 1312.8: $C_{164}.D_4$

• Label: 1312.8
• Order: \( 2^{5} \cdot 41 \)
• Exponent: \( 2^{3} \cdot 41 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 41 \)
• $\card{Z(G)}$: \( 2 \cdot 41 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 5 \)
• Perm deg.: $57$
• Trans deg.: $656$
• Rank: $2$

Group information

Description:	$C_{164}.D_4$
Order:	\(1312\)\(\medspace = 2^{5} \cdot 41 \)
Exponent:	\(328\)\(\medspace = 2^{3} \cdot 41 \)
Automorphism group:	$C_{40}\times D_4^2.C_2$, of order \(5120\)\(\medspace = 2^{10} \cdot 5 \)
Composition factors:	$C_2$ x 5, $C_{41}$
Nilpotency class:	$3$
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	8	41	82	164	328
Elements	1	3	12	16	40	120	480	640	1312
Conjugacy classes	1	2	4	4	40	80	160	160	451
Divisions	1	2	4	2	1	2	4	2	18
Autjugacy classes	1	2	2	1	1	2	2	1	12

Dimension	1	2	4	40	80	160
Irr. complex chars.	328	82	41	0	0	0	451
Irr. rational chars.	4	4	1	4	4	1	18

Minimal presentations

Permutation degree:	$57$
Transitive degree:	$656$
Rank:	$2$
Inequivalent generating pairs:	$126$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	160
Arbitrary	4	8	48

Constructions

Presentation:	$\langle a, b, c \mid b^{4}=c^{82}=[b,c]=1, a^{4}=b^{2}, b^{a}=bc^{41}, c^{a}=b^{2}c \rangle$
Permutation group:	Degree $57$ $\langle(1,12,4,10,2,11,3,9)(5,15,7,14,6,16,8,13), (1,6,2,5)(3,8,4,7)(9,14,10,13) \!\cdots\! \rangle$
Direct product:	$C_{41}$ $\, \times\, $ $(C_4.D_4)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{164}$ . $D_4$ (2)	$\OD_{16}$ . $C_{82}$ (2)	$(Q_8\times C_{82})$ . $C_2$	$(C_2\times Q_8)$ . $C_{82}$	all 14

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

Homology

Abelianization:	$C_{2} \times C_{164} \simeq C_{2} \times C_{4} \times C_{41}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 52 subgroups in 38 conjugacy classes, 24 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{82}$	$G/Z \simeq$ $C_2^2:C_4$
Commutator:	$G' \simeq$ $C_2^2$	$G/G' \simeq$ $C_2\times C_{164}$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2\times C_{82}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{164}.D_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{164}.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{82}$	$G/\operatorname{soc} \simeq$ $C_2^2:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.D_4$
41-Sylow subgroup:	$P_{ 41 } \simeq$ $C_{41}$

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 1312: $C_{164}.D_4$

Order 656: $\OD_{16}\times C_{41}$ x 2, $Q_8\times C_{82}$

Order 328: $C_2\times C_{164}$ x 3, $C_{328}$ x 2, $Q_8\times C_{41}$ x 2

Order 164: $C_{164}$ x 4, $C_2\times C_{82}$

Order 82: $C_{82}$ x 2

Order 41: $C_{41}$

Order 32: $C_4.D_4$

Order 16: $\OD_{16}$ x 2, $C_2\times Q_8$

Order 8: $C_2\times C_4$ x 3, $Q_8$ x 2, $C_8$ x 2

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1312: $C_{164}.D_4$

Order 656: $Q_8\times C_{82}$, $\OD_{16}\times C_{41}$

Order 328: $C_2\times C_{164}$ x 2, $C_{328}$, $Q_8\times C_{41}$

Order 164: $C_{164}$ x 2, $C_2\times C_{82}$

Order 82: $C_{82}$ x 2

Order 41: $C_{41}$

Order 32: $C_4.D_4$

Order 16: $\OD_{16}$, $C_2\times Q_8$

Order 8: $C_2\times C_4$ x 2, $Q_8$, $C_8$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1312: $C_{164}.D_4$ ($C_1$)

Order 656: $\OD_{16}\times C_{41}$ ($C_2$) x 2, $Q_8\times C_{82}$ ($C_2$)

Order 328: $C_2\times C_{164}$ ($C_4$) x 2, $C_2\times C_{164}$ ($C_2^2$)

Order 164: $C_{164}$ ($D_4$) x 2, $C_2\times C_{82}$ ($C_2\times C_4$)

Order 82: $C_{82}$ ($C_2^2:C_4$)

Order 41: $C_{41}$ ($C_4.D_4$)

Order 32: $C_4.D_4$ ($C_{41}$)

Order 16: $\OD_{16}$ ($C_{82}$) x 2, $C_2\times Q_8$ ($C_{82}$)

Order 8: $C_2\times C_4$ ($C_{164}$) x 2, $C_2\times C_4$ ($C_2\times C_{82}$)

Order 4: $C_4$ ($D_4\times C_{41}$) x 2, $C_2^2$ ($C_2\times C_{164}$)

Order 2: $C_2$ ($C_2^2:C_{164}$)

Order 1: $C_1$ ($C_{164}.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1312: $C_{164}.D_4$ ($C_1$)

Order 656: $Q_8\times C_{82}$ ($C_2$), $\OD_{16}\times C_{41}$ ($C_2$)

Order 328: $C_2\times C_{164}$ ($C_2^2$), $C_2\times C_{164}$ ($C_4$)

Order 164: $C_2\times C_{82}$ ($C_2\times C_4$), $C_{164}$ ($D_4$)

Order 82: $C_{82}$ ($C_2^2:C_4$)

Order 41: $C_{41}$ ($C_4.D_4$)

Order 32: $C_4.D_4$ ($C_{41}$)

Order 16: $\OD_{16}$ ($C_{82}$), $C_2\times Q_8$ ($C_{82}$)

Order 8: $C_2\times C_4$ ($C_2\times C_{82}$), $C_2\times C_4$ ($C_{164}$)

Order 4: $C_2^2$ ($C_2\times C_{164}$), $C_4$ ($D_4\times C_{41}$)

Order 2: $C_2$ ($C_2^2:C_{164}$)

Order 1: $C_1$ ($C_{164}.D_4$)

Series

Derived series

$C_{164}.D_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_{164}.D_4$

$\rhd$

$\OD_{16}\times C_{41}$

$\rhd$

$C_2\times C_{164}$

$\rhd$

$C_{164}$

$\rhd$

$C_{82}$

$\rhd$

$C_{41}$

$\rhd$

$C_1$

Lower central series

$C_{164}.D_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{82}$

$\lhd$

$C_2\times C_{164}$

$\lhd$

$C_{164}.D_4$

Character theory

Complex character table

See the $451 \times 451$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $18 \times 18$ rational character table.