C₃₇×H(7,𝔽₂) #17992412

order	4736
nilpotency class	2
isoclinism	isoclinic class of H(7,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₃₇ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₃₇×H(7,𝔽₂)

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₇	C₂⁶
commutator, Frattini, cosocle	C₂	C₂⁶×C₃₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₇ ◁ G

5992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 592

A minimal faithful permutation representation — 6 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)(41,48)(42,43)(45,49)(46,50)(47,53)(51,52);(39,43)(44,50)(45,48)(51,53);(40,46)(44,50)(47,52)(51,53);(38,40)(39,44)(41,47)(42,46)(43,50)(45,51)(48,53)(49,52);(38,39)(40,44)(41,45)(42,43)(46,50)(47,51)(48,49)(52,53);(38,41)(39,45)(40,47)(42,49)(43,48)(44,51)(46,52)(50,53)

Conjugacy classes & character table

2405 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#32894, C₃₇×#32896, C₃₇×#60388, …
maximal normal subgroup of	C₃₇×#32894, C₃₇×#32896, C₃₇×#60388, …
maximal quotient of	C₃₇×#62652, C₃₇×#62654, C₃₇×#62842, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2326);

C37×H(7,𝔽2) #17992412