Abstract group 832.270: $C_{52}.(C_2\times C_8)$

• Label: 832.270
• Order: \( 2^{6} \cdot 13 \)
• Exponent: \( 2^{4} \cdot 13 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $45$
• Trans deg.: $416$
• Rank: $2$

Group information

Description:	$C_{52}.(C_2\times C_8)$
Order:	\(832\)\(\medspace = 2^{6} \cdot 13 \)
Exponent:	\(208\)\(\medspace = 2^{4} \cdot 13 \)
Automorphism group:	$C_{26}.(C_2^2\times C_4\times C_{12})$, of order \(4992\)\(\medspace = 2^{7} \cdot 3 \cdot 13 \)
Composition factors:	$C_2$ x 6, $C_{13}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	13	16	26	52
Elements	1	7	8	208	12	416	84	96	832
Conjugacy classes	1	3	4	8	3	12	12	15	58
Divisions	1	3	3	3	1	2	3	3	19
Autjugacy classes	1	3	4	6	1	4	3	4	26

Dimension	1	2	4	8	12	16	24	48
Irr. complex chars.	16	12	24	6	0	0	0	0	58
Irr. rational chars.	4	4	3	0	4	1	2	1	19

Minimal presentations

Permutation degree:	$45$
Transitive degree:	$416$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	192
Arbitrary	6	8	28

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{16}=c^{26}=1, b^{a}=b^{5}c^{13}, c^{a}=b^{8}c, c^{b}=c^{21} \rangle$
Permutation group:	Degree $45$ $\langle(1,2,4,9,6,10,17,25,7,11,18,26,5,19,16,29)(3,12,13,27,14,8,21,22,15,23,28,31,20,24,30,32) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{26}:C_{16})$ $\,\rtimes\,$ $C_2$	$(C_{13}:\OD_{32})$ $\,\rtimes\,$ $C_2$	$C_{13}$ $\,\rtimes\,$ $(\OD_{32}:C_2)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(Q_8\times C_{13})$ . $C_8$	$Q_8$ . $(C_{13}:C_8)$	$(D_4:C_{26})$ . $C_4$	$(D_4\times C_{13})$ . $C_8$	all 20

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

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 298 subgroups in 56 conjugacy classes, 26 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $D_{26}:C_4$
Commutator:	$G' \simeq$ $C_{52}$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_{26}:C_4$
Fitting:	$\operatorname{Fit} \simeq$ $D_4:C_{26}$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $C_{52}.(C_2\times C_8)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{26}$	$G/\operatorname{soc} \simeq$ $C_2^2:C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{32}:C_2$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$

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 832: $C_{52}.(C_2\times C_8)$

Order 416: $C_{13}:\OD_{32}$, $C_{26}:C_{16}$, $C_{52}.C_2^3$

Order 208: $C_{26}:C_8$ x 2, $C_{13}:C_{16}$ x 2, $C_{13}:\OD_{16}$ x 2, $D_4:C_{26}$

Order 104: $C_{13}:C_8$ x 3, $C_2\times C_{52}$ x 2, $D_4\times C_{13}$ x 2, $Q_8\times C_{13}$

Order 64: $\OD_{32}:C_2$

Order 52: $C_{52}$ x 3, $C_2\times C_{26}$ x 2

Order 32: $\OD_{16}:C_2$, $\OD_{32}$, $C_2\times C_{16}$

Order 26: $C_{26}$ x 3

Order 16: $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_{16}$ x 2, $D_4:C_2$

Order 13: $C_{13}$

Order 8: $C_8$ x 3, $D_4$ x 2, $C_2\times C_4$ x 2, $Q_8$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 832: $C_{52}.(C_2\times C_8)$

Order 416: $C_{13}:\OD_{32}$, $C_{26}:C_{16}$, $C_{52}.C_2^3$

Order 208: $C_{26}:C_8$ x 2, $C_{13}:C_{16}$ x 2, $C_{13}:\OD_{16}$ x 2, $D_4:C_{26}$

Order 104: $C_{13}:C_8$ x 3, $C_2\times C_{52}$ x 2, $D_4\times C_{13}$ x 2, $Q_8\times C_{13}$

Order 64: $\OD_{32}:C_2$

Order 52: $C_{52}$ x 3, $C_2\times C_{26}$ x 2

Order 32: $\OD_{16}:C_2$, $\OD_{32}$, $C_2\times C_{16}$

Order 26: $C_{26}$ x 3

Order 16: $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_{16}$ x 2, $D_4:C_2$

Order 13: $C_{13}$

Order 8: $C_8$ x 3, $D_4$ x 2, $C_2\times C_4$ x 2, $Q_8$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 832: $C_{52}.(C_2\times C_8)$ ($C_1$)

Order 416: $C_{13}:\OD_{32}$ ($C_2$), $C_{26}:C_{16}$ ($C_2$), $C_{52}.C_2^3$ ($C_2$)

Order 208: $C_{26}:C_8$ ($C_2^2$), $D_4:C_{26}$ ($C_4$), $C_{13}:\OD_{16}$ ($C_4$)

Order 104: $C_{13}:C_8$ ($D_4$) x 2, $C_2\times C_{52}$ ($C_2\times C_4$), $Q_8\times C_{13}$ ($C_8$), $D_4\times C_{13}$ ($C_8$)

Order 52: $C_2\times C_{26}$ ($\OD_{16}$), $C_{52}$ ($C_2\times C_8$), $C_{52}$ ($C_2^2:C_4$)

Order 26: $C_{26}$ ($C_2^2:C_8$)

Order 16: $D_4:C_2$ ($C_{13}:C_4$)

Order 13: $C_{13}$ ($\OD_{32}:C_2$)

Order 8: $Q_8$ ($C_{13}:C_8$), $D_4$ ($C_{13}:C_8$), $C_2\times C_4$ ($C_{26}:C_4$)

Order 4: $C_2^2$ ($C_{13}:\OD_{16}$), $C_4$ ($D_{26}:C_4$), $C_4$ ($C_{26}:C_8$)

Order 2: $C_2$ ($C_{26}.\OD_{16}$)

Order 1: $C_1$ ($C_{52}.(C_2\times C_8)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 832: $C_{52}.(C_2\times C_8)$ ($C_1$)

Order 416: $C_{13}:\OD_{32}$ ($C_2$), $C_{26}:C_{16}$ ($C_2$), $C_{52}.C_2^3$ ($C_2$)

Order 208: $C_{26}:C_8$ ($C_2^2$), $D_4:C_{26}$ ($C_4$), $C_{13}:\OD_{16}$ ($C_4$)

Order 104: $C_{13}:C_8$ ($D_4$) x 2, $C_2\times C_{52}$ ($C_2\times C_4$), $Q_8\times C_{13}$ ($C_8$), $D_4\times C_{13}$ ($C_8$)

Order 52: $C_2\times C_{26}$ ($\OD_{16}$), $C_{52}$ ($C_2\times C_8$), $C_{52}$ ($C_2^2:C_4$)

Order 26: $C_{26}$ ($C_2^2:C_8$)

Order 16: $D_4:C_2$ ($C_{13}:C_4$)

Order 13: $C_{13}$ ($\OD_{32}:C_2$)

Order 8: $Q_8$ ($C_{13}:C_8$), $D_4$ ($C_{13}:C_8$), $C_2\times C_4$ ($C_{26}:C_4$)

Order 4: $C_2^2$ ($C_{13}:\OD_{16}$), $C_4$ ($D_{26}:C_4$), $C_4$ ($C_{26}:C_8$)

Order 2: $C_2$ ($C_{26}.\OD_{16}$)

Order 1: $C_1$ ($C_{52}.(C_2\times C_8)$)

Series

Derived series

$C_{52}.(C_2\times C_8)$

$\rhd$

$C_{52}$

$\rhd$

$C_1$

Chief series

$C_{52}.(C_2\times C_8)$

$\rhd$

$C_{52}.C_2^3$

$\rhd$

$C_{26}:C_8$

$\rhd$

$C_{13}:C_8$

$\rhd$

$C_{52}$

$\rhd$

$C_{26}$

$\rhd$

$C_{13}$

$\rhd$

$C_1$

Lower central series

$C_{52}.(C_2\times C_8)$

$\rhd$

$C_{52}$

$\rhd$

$C_{26}$

$\rhd$

$C_{13}$

Upper central series

$C_1$

$\lhd$

$C_4$

$\lhd$

$C_2\times C_4$

$\lhd$

$D_4:C_2$

Character theory

Complex character table

See the $58 \times 58$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $19 \times 19$ rational character table.