Abstract group 962.1: $D_{13}\times C_{37}$

• Label: 962.1
• Order: \( 2 \cdot 13 \cdot 37 \)
• Exponent: \( 2 \cdot 13 \cdot 37 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 37 \)
• $\card{Z(G)}$: \( 37 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{3} \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{3} \)
• Perm deg.: $50$
• Trans deg.: $481$
• Rank: $2$

Group information

Description:	$D_{13}\times C_{37}$
Order:	\(962\)\(\medspace = 2 \cdot 13 \cdot 37 \)
Exponent:	\(962\)\(\medspace = 2 \cdot 13 \cdot 37 \)
Automorphism group:	$C_{39}.C_{12}^2$, of order \(5616\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 13 \)
Composition factors:	$C_2$, $C_{13}$, $C_{37}$
Derived length:	$2$

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

Group statistics

Order	1	2	13	37	74	481
Elements	1	13	12	36	468	432	962
Conjugacy classes	1	1	6	36	36	216	296
Divisions	1	1	1	1	1	1	6
Autjugacy classes	1	1	1	1	1	1	6

Dimension	1	2	12	36	432
Irr. complex chars.	74	222	0	0	0	296
Irr. rational chars.	2	0	1	2	1	6

Minimal presentations

Permutation degree:	$50$
Transitive degree:	$481$
Rank:	$2$
Inequivalent generating pairs:	$114$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	432
Arbitrary	2	4	48

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{481}=1, b^{a}=b^{38} \rangle$
Permutation group:	Degree $50$ $\langle(39,40)(41,42)(43,44)(45,46)(47,48)(49,50), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37) \!\cdots\! \rangle$
Direct product:	$C_{37}$ $\, \times\, $ $D_{13}$
Semidirect product:	$C_{481}$ $\,\rtimes\,$ $C_2$	$C_{13}$ $\,\rtimes\,$ $C_{74}$			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:	$C_{74} \simeq C_{2} \times C_{37}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 32 subgroups in 8 conjugacy classes, 6 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{37}$	$G/Z \simeq$ $D_{13}$
Commutator:	$G' \simeq$ $C_{13}$	$G/G' \simeq$ $C_{74}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_{13}\times C_{37}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{481}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{13}\times C_{37}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{481}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$
37-Sylow subgroup:	$P_{ 37 } \simeq$ $C_{37}$

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 962: $D_{13}\times C_{37}$

Order 481: $C_{481}$

Order 74: $C_{74}$

Order 37: $C_{37}$

Order 26: $D_{13}$

Order 13: $C_{13}$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 962: $D_{13}\times C_{37}$

Order 481: $C_{481}$

Order 74: $C_{74}$

Order 37: $C_{37}$

Order 26: $D_{13}$

Order 13: $C_{13}$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 962: $D_{13}\times C_{37}$ ($C_1$)

Order 481: $C_{481}$ ($C_2$)

Order 37: $C_{37}$ ($D_{13}$)

Order 26: $D_{13}$ ($C_{37}$)

Order 13: $C_{13}$ ($C_{74}$)

Order 1: $C_1$ ($D_{13}\times C_{37}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 962: $D_{13}\times C_{37}$ ($C_1$)

Order 481: $C_{481}$ ($C_2$)

Order 37: $C_{37}$ ($D_{13}$)

Order 26: $D_{13}$ ($C_{37}$)

Order 13: $C_{13}$ ($C_{74}$)

Order 1: $C_1$ ($D_{13}\times C_{37}$)

Series

Derived series	$D_{13}\times C_{37}$	$\rhd$	$C_{13}$	$\rhd$	$C_1$
Chief series	$D_{13}\times C_{37}$	$\rhd$	$C_{481}$	$\rhd$	$C_{37}$	$\rhd$	$C_1$
Lower central series	$D_{13}\times C_{37}$	$\rhd$	$C_{13}$
Upper central series	$C_1$	$\lhd$	$C_{37}$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $296 \times 296$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	13A	37A	74A	481A
	Size	1	13	12	36	468	432
	2 P	1A	1A	13A	37A	37A	481A
	13 P	1A	2A	13A	37A	74A	481A
	37 P	1A	2A	1A	37A	74A	37A
962.1.1a		$1$	$1$	$1$	$1$	$1$	$1$
962.1.1b		$1$	$-1$	$1$	$1$	$-1$	$1$
962.1.1c		$36$	$36$	$36$	$-1$	$-1$	$-1$
962.1.1d		$36$	$-36$	$36$	$-1$	$1$	$-1$
962.1.2a		$12$	$0$	$-1$	$12$	$0$	$-1$
962.1.2b		$432$	$0$	$-36$	$-12$	$0$	$1$