C₃₇×#3407 #17992225

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×#3407

Subgroups

	subgroup	G/H
center	C₈×C₃₇	C₂⁴
commutator	C₂	C₂³×C₈×C₃₇
Frattini, cosocle	C₈	C₂⁴×C₃₇
socle	C₂×C₃₇	C₂³×C₈
prime radical	C₂×#41×C₃₇	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₃₇ ◁ G

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 4 generators on 69 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)(38,39,41,46,42,47,51,59,43,45,52,57,53,58,63,67)(40,44,49,55,50,56,62,66,48,54,60,64,61,65,68,69);(38,40)(39,44)(41,49)(42,50)(43,48)(45,54)(46,55)(47,56)(51,62)(52,60)(53,61)(57,64)(58,65)(59,66)(63,68)(67,69);(40,48)(44,54)(49,60)(50,61)(55,64)(56,65)(62,68)(66,69);(39,45)(44,54)(46,57)(47,58)(55,64)(56,65)(59,67)(66,69)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#12618, C₃₇×#12833, C₃₇×#12915, …
maximal normal subgroup of	C₃₇×#12618, C₃₇×#12833, C₃₇×#12915, …
maximal quotient of	C₃₇×#33324, C₃₇×#33326, C₃₇×#33333, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2139);

C37×#3407 #17992225