C₃₇×#3296 #17992114

order	4736
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#3296

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
prime radical	C₃₇×#546	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₄×D₄×C₃₇ ◁ G

408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 1184

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

Conjugacy classes & character table

1295 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₃₇×#3296, C₃₇×#61124, C₃₇×#61156, …
maximal normal subgroup of	C₂×C₃₇×#3296, C₃₇×#61124, C₃₇×#61156, …
maximal quotient of	C₃₇×#22414, C₃₇×#22486, C₃₇×#22570, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2028);

C37×#3296 #17992114