Abstract group 3844.d: $C_{31}\times D_{62}$

• Label: 3844.d
• Order: \( 2^{2} \cdot 31^{2} \)
• Exponent: \( 2 \cdot 31 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 31 \)
• $\card{Z(G)}$: \( 2 \cdot 31 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 31 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{2} \cdot 5^{2} \)
• Perm deg.: $64$
• Trans deg.: $124$
• Rank: $2$

Group information

Description:	$C_{31}\times D_{62}$
Order:	\(3844\)\(\medspace = 2^{2} \cdot 31^{2} \)
Exponent:	\(62\)\(\medspace = 2 \cdot 31 \)
Automorphism group:	$C_{31}.C_{15}^2.C_2^3$, of order \(55800\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 31 \)
Composition factors:	$C_2$ x 2, $C_{31}$ x 2
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.

Group statistics

Order	1	2	31	62
Elements	1	63	960	2820	3844
Conjugacy classes	1	3	495	555	1054
Divisions	1	3	17	19	40
Autjugacy classes	1	2	3	4	10

Dimension	1	2	30	60
Irr. complex chars.	124	930	0	0	1054
Irr. rational chars.	4	0	6	30	40

Minimal presentations

Permutation degree:	$64$
Transitive degree:	$124$
Rank:	$2$
Inequivalent generating pairs:	$96$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	60
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{62}=b^{62}=1, b^{a}=b^{61} \rangle$
Permutation group:	Degree $64$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 260 \end{array}\right), \left(\begin{array}{rr} 142 & 0 \\ 0 & 46 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{311})$
Direct product:	$C_2$ $\, \times\, $ $C_{31}$ $\, \times\, $ $D_{31}$
Semidirect product:	$C_{62}$ $\,\rtimes\,$ $C_{62}$	$C_{31}^2$ $\,\rtimes\,$ $C_2^2$	$(C_{31}\times C_{62})$ $\,\rtimes\,$ $C_2$	$C_{31}$ $\,\rtimes\,$ $(C_2\times C_{62})$	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:	$C_{2} \times C_{62} \simeq C_{2}^{2} \times C_{31}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 260 subgroups in 50 conjugacy classes, 14 normal (10 characteristic).

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

Special subgroups

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

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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Subgroup information

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Classes of subgroups up to conjugation

Order 3844: $C_{31}\times D_{62}$

Order 1922: $C_{31}\times D_{31}$ x 2, $C_{31}\times C_{62}$

Order 961: $C_{31}^2$

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

Order 62: $C_{62}$ x 19, $D_{31}$ x 2

Order 31: $C_{31}$ x 17

Order 4: $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3844: $C_{31}\times D_{62}$

Order 1922: $C_{31}\times C_{62}$, $C_{31}\times D_{31}$

Order 961: $C_{31}^2$

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

Order 62: $C_{62}$ x 4, $D_{31}$

Order 31: $C_{31}$ x 3

Order 4: $C_2^2$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3844: $C_{31}\times D_{62}$ ($C_1$)

Order 1922: $C_{31}\times D_{31}$ ($C_2$) x 2, $C_{31}\times C_{62}$ ($C_2$)

Order 961: $C_{31}^2$ ($C_2^2$)

Order 124: $D_{62}$ ($C_{31}$)

Order 62: $D_{31}$ ($C_{62}$) x 2, $C_{62}$ ($C_{62}$), $C_{62}$ ($D_{31}$)

Order 31: $C_{31}$ ($C_2\times C_{62}$), $C_{31}$ ($D_{62}$)

Order 2: $C_2$ ($C_{31}\times D_{31}$)

Order 1: $C_1$ ($C_{31}\times D_{62}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3844: $C_{31}\times D_{62}$ ($C_1$)

Order 1922: $C_{31}\times C_{62}$ ($C_2$), $C_{31}\times D_{31}$ ($C_2$)

Order 961: $C_{31}^2$ ($C_2^2$)

Order 124: $D_{62}$ ($C_{31}$)

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

Order 31: $C_{31}$ ($C_2\times C_{62}$), $C_{31}$ ($D_{62}$)

Order 2: $C_2$ ($C_{31}\times D_{31}$)

Order 1: $C_1$ ($C_{31}\times D_{62}$)

Series

Derived series	$C_{31}\times D_{62}$	$\rhd$	$C_{31}$	$\rhd$	$C_1$
Chief series	$C_{31}\times D_{62}$	$\rhd$	$C_{31}\times C_{62}$	$\rhd$	$C_{62}$	$\rhd$	$C_{31}$	$\rhd$	$C_1$
Lower central series	$C_{31}\times D_{62}$	$\rhd$	$C_{31}$
Upper central series	$C_1$	$\lhd$	$C_{62}$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 8 larger groups in the database.

Character theory

Complex character table

The $1054 \times 1054$ character table is not available for this group.

Rational character table

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