Abstract group 1924.5: $C_{37}:C_{52}$

• Label: 1924.5
• Order: \( 2^{2} \cdot 13 \cdot 37 \)
• Exponent: \( 2^{2} \cdot 13 \cdot 37 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 13 \)
• $\card{Z(G)}$: \( 13 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{3} \cdot 37 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{3} \)
• Perm deg.: $50$
• Trans deg.: $481$
• Rank: $2$

Group information

Description:	$C_{37}:C_{52}$
Order:	\(1924\)\(\medspace = 2^{2} \cdot 13 \cdot 37 \)
Exponent:	\(1924\)\(\medspace = 2^{2} \cdot 13 \cdot 37 \)
Automorphism group:	$C_{12}\times F_{37}$, of order \(15984\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 37 \)
Composition factors:	$C_2$ x 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	4	13	26	37	52	481
Elements	1	37	74	12	444	36	888	432	1924
Conjugacy classes	1	1	2	12	12	9	24	108	169
Divisions	1	1	1	1	1	1	1	1	8
Autjugacy classes	1	1	2	1	1	1	2	1	10

Dimension	1	2	4	12	24	36	432
Irr. complex chars.	52	0	117	0	0	0	0	169
Irr. rational chars.	2	1	0	2	1	1	1	8

Minimal presentations

Permutation degree:	$50$
Transitive degree:	$481$
Rank:	$2$
Inequivalent generating pairs:	$168$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	432
Arbitrary	4	6	48

Constructions

Presentation:	$\langle a, b \mid a^{4}=b^{481}=1, b^{a}=b^{339} \rangle$
Permutation group:	Degree $50$ $\langle(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,26)(14,21,19,22)(16,23,24,20)(25,30,31,29) \!\cdots\! \rangle$
Direct product:	$C_{13}$ $\, \times\, $ $(C_{37}:C_4)$
Semidirect product:	$C_{481}$ $\,\rtimes\,$ $C_4$	$C_{37}$ $\,\rtimes\,$ $C_{52}$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{37}$ . $C_{26}$	$(C_{13}\times D_{37})$ . $C_2$			more information

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

Homology

Abelianization:	$C_{52} \simeq C_{4} \times C_{13}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 156 subgroups in 12 conjugacy classes, 8 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{13}$	$G/Z \simeq$ $C_{37}:C_4$
Commutator:	$G' \simeq$ $C_{37}$	$G/G' \simeq$ $C_{52}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{37}:C_{52}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{481}$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $C_{37}:C_{52}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{481}$	$G/\operatorname{soc} \simeq$ $C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
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 1924: $C_{37}:C_{52}$

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

Order 481: $C_{481}$

Order 148: $C_{37}:C_4$

Order 74: $D_{37}$

Order 52: $C_{52}$

Order 37: $C_{37}$

Order 26: $C_{26}$

Order 13: $C_{13}$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1924: $C_{37}:C_{52}$

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

Order 481: $C_{481}$

Order 148: $C_{37}:C_4$

Order 74: $D_{37}$

Order 52: $C_{52}$

Order 37: $C_{37}$

Order 26: $C_{26}$

Order 13: $C_{13}$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1924: $C_{37}:C_{52}$ ($C_1$)

Order 962: $C_{13}\times D_{37}$ ($C_2$)

Order 481: $C_{481}$ ($C_4$)

Order 148: $C_{37}:C_4$ ($C_{13}$)

Order 74: $D_{37}$ ($C_{26}$)

Order 37: $C_{37}$ ($C_{52}$)

Order 13: $C_{13}$ ($C_{37}:C_4$)

Order 1: $C_1$ ($C_{37}:C_{52}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1924: $C_{37}:C_{52}$ ($C_1$)

Order 962: $C_{13}\times D_{37}$ ($C_2$)

Order 481: $C_{481}$ ($C_4$)

Order 148: $C_{37}:C_4$ ($C_{13}$)

Order 74: $D_{37}$ ($C_{26}$)

Order 37: $C_{37}$ ($C_{52}$)

Order 13: $C_{13}$ ($C_{37}:C_4$)

Order 1: $C_1$ ($C_{37}:C_{52}$)

Series

Derived series	$C_{37}:C_{52}$	$\rhd$	$C_{37}$	$\rhd$	$C_1$
Chief series	$C_{37}:C_{52}$	$\rhd$	$C_{13}\times D_{37}$	$\rhd$	$C_{481}$	$\rhd$	$C_{37}$	$\rhd$	$C_1$
Lower central series	$C_{37}:C_{52}$	$\rhd$	$C_{37}$
Upper central series	$C_1$	$\lhd$	$C_{13}$

Character theory

Complex character table

See the $169 \times 169$ 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	4A	13A	26A	37A	52A	481A
	Size	1	37	74	12	444	36	888	432
	2 P	1A	1A	2A	13A	13A	37A	26A	481A
	13 P	1A	2A	4A	13A	26A	37A	52A	481A
	37 P	1A	2A	4A	1A	2A	37A	4A	37A
1924.5.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1924.5.1b		$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$
1924.5.1c		$2$	$-2$	$0$	$2$	$-2$	$2$	$0$	$2$
1924.5.1d		$12$	$12$	$12$	$-1$	$-1$	$12$	$-1$	$-1$
1924.5.1e		$12$	$12$	$-12$	$-1$	$-1$	$12$	$1$	$-1$
1924.5.1f		$24$	$-24$	$0$	$-2$	$2$	$24$	$0$	$-2$
1924.5.4a		$36$	$0$	$0$	$36$	$0$	$-1$	$0$	$-1$
1924.5.4b		$432$	$0$	$0$	$-36$	$0$	$-12$	$0$	$1$