Abstract group 496.41: $C_2^2\times D_{62}$

• Label: 496.41
• Order: \( 2^{4} \cdot 31 \)
• Exponent: \( 2 \cdot 31 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Perm deg.: $37$
• Trans deg.: $248$
• Rank: $4$

Group information

Description:	$C_2^2\times D_{62}$
Order:	\(496\)\(\medspace = 2^{4} \cdot 31 \)
Exponent:	\(62\)\(\medspace = 2 \cdot 31 \)
Automorphism group:	$C_{31}.C_{30}.C_2^3.\PSL(2,7)$, of order \(1249920\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \)
Composition factors:	$C_2$ x 4, $C_{31}$
Derived length:	$2$

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

Group statistics

Order	1	2	31	62
Elements	1	255	30	210	496
Conjugacy classes	1	15	15	105	136
Divisions	1	15	1	7	24
Autjugacy classes	1	2	1	1	5

Dimension	1	2	30
Irr. complex chars.	16	120	0	136
Irr. rational chars.	16	0	8	24

Minimal presentations

Permutation degree:	$37$
Transitive degree:	$248$
Rank:	$4$
Inequivalent generating quadruples:	$14895$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{2}=d^{62}=[a,b]=[a,c]=[a,d]=[b,c]=[b,d]=1, d^{c}=d^{61} \rangle$
Permutation group:	Degree $37$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 32 & 0 \\ 0 & 32 \end{array}\right), \left(\begin{array}{rr} 1 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 32 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 61 & 0 \\ 0 & 61 \end{array}\right), \left(\begin{array}{rr} 32 & 3 \\ 0 & 92 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/93\Z)$
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $D_{31}$
Semidirect product:	$C_{62}$ $\,\rtimes\,$ $C_2^3$	$C_{31}$ $\,\rtimes\,$ $C_2^4$	$(C_2^2\times C_{62})$ $\,\rtimes\,$ $C_2$	$(C_2\times C_{62})$ $\,\rtimes\,$ $C_2^2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 1664 subgroups in 134 conjugacy classes, 83 normal (5 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $D_{31}$
Commutator:	$G' \simeq$ $C_{31}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^2\times D_{62}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{62}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_2^2\times D_{62}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{62}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4$
31-Sylow subgroup:	$P_{ 31 } \simeq$ $C_{31}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 496: $C_2^2\times D_{62}$

Order 248: $C_2\times D_{62}$ x 14, $C_2^2\times C_{62}$

Order 124: $D_{62}$ x 28, $C_2\times C_{62}$ x 7

Order 62: $D_{31}$ x 8, $C_{62}$ x 7

Order 31: $C_{31}$

Order 16: $C_2^4$

Order 8: $C_2^3$ x 15

Order 4: $C_2^2$ x 35

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 496: $C_2^2\times D_{62}$

Order 248: $C_2^2\times C_{62}$, $C_2\times D_{62}$

Order 124: $C_2\times C_{62}$, $D_{62}$

Order 62: $C_{62}$, $D_{31}$

Order 31: $C_{31}$

Order 16: $C_2^4$

Order 8: $C_2^3$ x 2

Order 4: $C_2^2$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 496: $C_2^2\times D_{62}$ ($C_1$)

Order 248: $C_2\times D_{62}$ ($C_2$) x 14, $C_2^2\times C_{62}$ ($C_2$)

Order 124: $D_{62}$ ($C_2^2$) x 28, $C_2\times C_{62}$ ($C_2^2$) x 7

Order 62: $D_{31}$ ($C_2^3$) x 8, $C_{62}$ ($C_2^3$) x 7

Order 31: $C_{31}$ ($C_2^4$)

Order 8: $C_2^3$ ($D_{31}$)

Order 4: $C_2^2$ ($D_{62}$) x 7

Order 2: $C_2$ ($C_2\times D_{62}$) x 7

Order 1: $C_1$ ($C_2^2\times D_{62}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 496: $C_2^2\times D_{62}$ ($C_1$)

Order 248: $C_2^2\times C_{62}$ ($C_2$), $C_2\times D_{62}$ ($C_2$)

Order 124: $C_2\times C_{62}$ ($C_2^2$), $D_{62}$ ($C_2^2$)

Order 62: $C_{62}$ ($C_2^3$), $D_{31}$ ($C_2^3$)

Order 31: $C_{31}$ ($C_2^4$)

Order 8: $C_2^3$ ($D_{31}$)

Order 4: $C_2^2$ ($D_{62}$)

Order 2: $C_2$ ($C_2\times D_{62}$)

Order 1: $C_1$ ($C_2^2\times D_{62}$)

Series

Derived series

$C_2^2\times D_{62}$

$\rhd$

$C_{31}$

$\rhd$

$C_1$

Chief series

$C_2^2\times D_{62}$

$\rhd$

$C_2\times D_{62}$

$\rhd$

$C_2\times C_{62}$

$\rhd$

$C_{62}$

$\rhd$

$C_{31}$

$\rhd$

$C_1$

Lower central series

$C_2^2\times D_{62}$

$\rhd$

$C_{31}$

Upper central series

$C_1$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 15 larger groups in the database.

This group is a maximal quotient of 19 larger groups in the database.

Character theory

Complex character table

See the $136 \times 136$ 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 $24 \times 24$ rational character table.