C₃₇×#464 #14557129

order	2368
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#464

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃₇
prime radical	C₂×D₄×C₃₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₄²×C₃₇ ◁ G

112 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 3 generators on 57 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)(39,41)(42,43)(44,50)(45,47)(46,54)(48,52)(49,53)(51,57)(55,56);(38,39,40,41)(43,47)(48,56)(49,54)(50,57);(42,44,45,51)(43,48,47,56)(46,55,53,52)(49,57,54,50)

Conjugacy classes & character table

703 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₇×#464, C₃₇×#3035, C₃₇×#3037, …
maximal normal subgroup of	C₂×C₃₇×#464, C₃₇×#3035, C₃₇×#3037, …
maximal quotient of	C₃₇×#1678, C₃₇×#1680, C₃₇×#1682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 145);