C₃₇×#2250 #17991068

order	4736
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₇×#2250

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₄²×C₃₇ ◁ C₄×C₈×C₃₇ ◁ G

198 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 2368

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)(43,47)(44,48)(45,51)(49,53)(50,57)(52,56)(54,55);(42,43,48,55,45,50,52,53,46,49,56,57,51,54,44,47);(38,39,40,41)(43,49)(47,53)(50,54)(55,57)

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#12324, C₃₇×#12325, C₃₇×#12336, …
maximal normal subgroup of	C₃₇×#12324, C₃₇×#12325, C₃₇×#12336, …
maximal quotient of	C₃₇×#12506, C₃₇×#12508, C₃₇×#12512, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 982);