Abstract group 416.175: $C_2\times C_4\times C_{52}$

• Label: 416.175
• Order: \( 2^{5} \cdot 13 \)
• Exponent: \( 2^{2} \cdot 13 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \)
• Perm deg.: $23$
• Trans deg.: $416$
• Rank: $3$

Group information

Description:	$C_2\times C_4\times C_{52}$
Order:	\(416\)\(\medspace = 2^{5} \cdot 13 \)
Exponent:	\(52\)\(\medspace = 2^{2} \cdot 13 \)
Automorphism group:	$C_{12}\times C_2^6.S_4$, of order \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Composition factors:	$C_2$ x 5, $C_{13}$
Nilpotency class:	$1$
Derived length:	$1$

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

Group statistics

Order	1	2	4	13	26	52
Elements	1	7	24	12	84	288	416
Conjugacy classes	1	7	24	12	84	288	416
Divisions	1	7	12	1	7	12	40
Autjugacy classes	1	2	1	1	2	1	8

Dimension	1	2	12	24
Irr. complex chars.	416	0	0	0	416
Irr. rational chars.	8	12	8	12	40

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$416$
Rank:	$3$
Inequivalent generating triples:	$1281$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	5	16

Constructions

Presentation:	Abelian group $\langle a, b, c \mid a^{2}=b^{4}=c^{52}=1 \rangle$
Permutation group:	Degree $23$ $\langle(3,6,4,5), (7,10,8,9), (1,2), (11,23,22,21,20,19,18,17,16,15,14,13,12), (3,4)(5,6), (7,8)(9,10)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 13 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 31 & 0 \\ 0 & 31 \end{array}\right), \left(\begin{array}{rr} 1 & 26 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/52\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_4$ ${}^2$ $\, \times\, $ $C_{13}$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^2\times C_{52})$ . $C_2$ (3)	$(C_2\times C_{52})$ . $C_2^2$ (6)	$(C_2\times C_{26})$ . $C_2^3$	$C_{26}$ . $(C_2^2\times C_4)$ (3)	all 12
Aut. group:	$\Aut(C_{795})$	$\Aut(C_{848})$	$\Aut(C_{1060})$	$\Aut(C_{1590})$

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

Homology

Primary decomposition:	$C_{2} \times C_{4}^{2} \times C_{13}$
Schur multiplier:	$C_{2}^{2} \times C_{4}$
Commutator length:	$0$

Subgroups

There are 108 subgroups, all normal (8 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 416: $C_2\times C_4\times C_{52}$

Order 208: $C_4\times C_{52}$ x 4, $C_2^2\times C_{52}$ x 3

Order 104: $C_2\times C_{52}$ x 18, $C_2^2\times C_{26}$

Order 52: $C_{52}$ x 12, $C_2\times C_{26}$ x 7

Order 32: $C_2\times C_4^2$

Order 26: $C_{26}$ x 7

Order 16: $C_4^2$ x 4, $C_2^2\times C_4$ x 3

Order 13: $C_{13}$

Order 8: $C_2\times C_4$ x 18, $C_2^3$

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 416: $C_2\times C_4\times C_{52}$

Order 208: $C_2^2\times C_{52}$, $C_4\times C_{52}$

Order 104: $C_2\times C_{52}$ x 2, $C_2^2\times C_{26}$

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

Order 32: $C_2\times C_4^2$

Order 26: $C_{26}$ x 2

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

Order 13: $C_{13}$

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

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 416: $C_2\times C_4\times C_{52}$ ($C_1$)

Order 208: $C_4\times C_{52}$ ($C_2$) x 4, $C_2^2\times C_{52}$ ($C_2$) x 3

Order 104: $C_2\times C_{52}$ ($C_4$) x 12, $C_2\times C_{52}$ ($C_2^2$) x 6, $C_2^2\times C_{26}$ ($C_2^2$)

Order 52: $C_{52}$ ($C_2\times C_4$) x 12, $C_2\times C_{26}$ ($C_2\times C_4$) x 6, $C_2\times C_{26}$ ($C_2^3$)

Order 32: $C_2\times C_4^2$ ($C_{13}$)

Order 26: $C_{26}$ ($C_4^2$) x 4, $C_{26}$ ($C_2^2\times C_4$) x 3

Order 16: $C_4^2$ ($C_{26}$) x 4, $C_2^2\times C_4$ ($C_{26}$) x 3

Order 13: $C_{13}$ ($C_2\times C_4^2$)

Order 8: $C_2\times C_4$ ($C_{52}$) x 12, $C_2\times C_4$ ($C_2\times C_{26}$) x 6, $C_2^3$ ($C_2\times C_{26}$)

Order 4: $C_4$ ($C_2\times C_{52}$) x 12, $C_2^2$ ($C_2\times C_{52}$) x 6, $C_2^2$ ($C_2^2\times C_{26}$)

Order 2: $C_2$ ($C_4\times C_{52}$) x 4, $C_2$ ($C_2^2\times C_{52}$) x 3

Order 1: $C_1$ ($C_2\times C_4\times C_{52}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 416: $C_2\times C_4\times C_{52}$ ($C_1$)

Order 208: $C_2^2\times C_{52}$ ($C_2$), $C_4\times C_{52}$ ($C_2$)

Order 104: $C_2\times C_{52}$ ($C_2^2$), $C_2\times C_{52}$ ($C_4$), $C_2^2\times C_{26}$ ($C_2^2$)

Order 52: $C_2\times C_{26}$ ($C_2^3$), $C_2\times C_{26}$ ($C_2\times C_4$), $C_{52}$ ($C_2\times C_4$)

Order 32: $C_2\times C_4^2$ ($C_{13}$)

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

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

Order 13: $C_{13}$ ($C_2\times C_4^2$)

Order 8: $C_2^3$ ($C_2\times C_{26}$), $C_2\times C_4$ ($C_2\times C_{26}$), $C_2\times C_4$ ($C_{52}$)

Order 4: $C_2^2$ ($C_2\times C_{52}$), $C_2^2$ ($C_2^2\times C_{26}$), $C_4$ ($C_2\times C_{52}$)

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

Order 1: $C_1$ ($C_2\times C_4\times C_{52}$)

Series

Derived series

$C_2\times C_4\times C_{52}$

$\rhd$

$C_1$

Chief series

$C_2\times C_4\times C_{52}$

$\rhd$

$C_2^2\times C_{52}$

$\rhd$

$C_2^2\times C_{26}$

$\rhd$

$C_2\times C_{26}$

$\rhd$

$C_{26}$

$\rhd$

$C_{13}$

$\rhd$

$C_1$

Lower central series

$C_2\times C_4\times C_{52}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4\times C_{52}$

Supergroups

This group is a maximal subgroup of 77 larger groups in the database.

This group is a maximal quotient of 13 larger groups in the database.

Character theory

Complex character table

See the $416 \times 416$ 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 $40 \times 40$ rational character table.