C₃₇×#3192 #17992010

order	4736
nilpotency class	3
isoclinism	isoclinic class of #3192
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃₇×#3192

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂³×C₄×C₃₇ ◁ G

514 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 176 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) = 1184

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)(40,44)(41,43)(42,45)(46,47)(48,50)(49,51)(52,53);(38,40)(39,42)(41,45)(43,44)(47,50)(51,52);(40,45)(42,44)(47,51)(49,53)(50,52);(39,43)(42,44)(47,50)(51,52)

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#9343, C₃₇×#9930, C₃₇×#9932, …
maximal normal subgroup of	C₃₇×#9343, C₃₇×#9930, C₃₇×#9932, …
maximal quotient of	C₃₇×#21764, C₃₇×#21942, C₃₇×#22046, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1924);