C₃₇×#3092 #17991910

order	4736
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#3092

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
prime radical	C₃₇×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂³×C₄×C₃₇ ◁ G

458 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)(40,43)(41,42)(44,45)(46,47)(48,50)(49,51)(52,53);(39,42)(43,45);(38,40,41,44)(39,43,42,45)(47,50)(48,53)(51,52);(47,51)(50,52)

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#9521, C₃₇×#9522, C₃₇×#18140, …
maximal normal subgroup of	C₃₇×#9521, C₃₇×#9522, C₃₇×#18140, …
maximal quotient of	C₃₇×#20700, C₃₇×#20844, C₃₇×#20909, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1824);

C37×#3092 #17991910