C₃₇×#2206 #17991024

order	4736
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#2206

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

186 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 68 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)(42,43)(44,47)(45,54)(46,50)(48,51)(49,52)(53,56)(55,57);(38,39,40,41);(43,47)(44,51)(45,52)(48,49)(50,56)(53,57)(54,55)

Conjugacy classes & character table

1073 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#12316, C₃₇×#12319, C₃₇×#12324, …
maximal normal subgroup of	C₃₇×#12316, C₃₇×#12319, C₃₇×#12324, …
maximal quotient of	C₃₇×#12416, C₃₇×#12424, C₃₇×#12445, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 938);