C₃₇×#2972 #17991790

order	4736
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈×C₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃₇×#2972

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂⁴
commutator	C₂	C₂³×C₈×C₃₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
socle	C₂²×C₃₇	C₂³×C₄
prime radical	C₂²×D₄×C₃₇	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ G

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 4 generators on 53 points:

(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)(38,39)(40,44)(41,43)(42,45)(46,47,48,50,49,51,52,53);(38,40)(39,42)(41,45)(43,44);(40,45)(42,44);(39,43)(42,44)

Conjugacy classes & character table

2516 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#8835, C₃₇×#8851, C₃₇×#8957, …
maximal normal subgroup of	C₃₇×#8835, C₃₇×#8851, C₃₇×#8957, …
maximal quotient of	C₃₇×#20523, C₃₇×#20533, C₃₇×#20555, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1704);

C37×#2972 #17991790