C₃₇×#3573 #17992391

order	4736
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₃₇×#3573

Subgroups

	subgroup	G/H
center	C₈×C₃₇	C₂⁴
commutator	C₂	C₂⁴×C₄×C₃₇
Frattini, cosocle	C₄	C₂⁵×C₃₇
socle	C₂×C₃₇	C₂⁴×C₄
prime radical	C₃₇×#585	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₃₇ ◁ G

1424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 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)(38,39,42,46,43,47,53,59)(40,44,50,56,49,55,62,66)(41,45,52,58,51,57,63,67)(48,54,61,65,60,64,68,69);(38,40)(39,44)(41,48)(42,50)(43,49)(45,54)(46,56)(47,55)(51,60)(52,61)(53,62)(57,64)(58,65)(59,66)(63,68)(67,69);(38,41)(39,45)(40,48)(42,52)(43,51)(44,54)(46,58)(47,57)(49,60)(50,61)(53,63)(55,64)(56,65)(59,67)(62,68)(66,69);(41,51)(45,57)(48,60)(52,63)(54,64)(58,67)(61,68)(65,69);(40,49)(44,55)(48,60)(50,62)(54,64)(56,66)(61,68)(65,69)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#12684, C₃₇×#12685, C₃₇×#21477, …
maximal normal subgroup of	C₃₇×#12684, C₃₇×#12685, C₃₇×#21477, …
maximal quotient of	C₃₇×#60058, C₃₇×#60065, C₃₇×#60100, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2305);

C37×#3573 #17992391