C₃₇×#538 #14557203

order	2368
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#538

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ G

250 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 53
transitive degree	μ_t(G) = 592

A minimal faithful permutation representation — 4 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)(40,46)(41,45)(42,44)(43,47)(48,50)(49,52)(51,53);(38,40,42,48)(39,43,44,51)(41,47,49,53)(45,46,52,50);(39,44)(40,47)(43,50)(45,52)(46,51)(48,53);(39,45)(43,46)(44,52)(50,51)

Conjugacy classes & character table

814 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#1600, C₃₇×#1606, C₃₇×#1612, …
maximal normal subgroup of	C₃₇×#1600, C₃₇×#1606, C₃₇×#1612, …
maximal quotient of	C₃₇×#2312, C₃₇×#2318, C₃₇×#2321, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 219);