C₃₇×#2220 #17991038

order	4736
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#2220

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃₇
prime radical	C₃₇×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ C₂²×C₈×C₃₇ ◁ G

160 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 64 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)(38,39)(40,41)(43,47)(44,51)(45,52)(48,53)(49,54)(50,57)(55,56);(42,43,48,57,45,50,44,47,46,49,51,56,52,55,53,54);(39,41)(43,49)(47,54)(50,55)(56,57)

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₇×#2220, C₃₇×#33604, C₃₇×#33605, …
maximal normal subgroup of	C₂×C₃₇×#2220, C₃₇×#33604, C₃₇×#33605, …
maximal quotient of	C₃₇×#12435, C₃₇×#12444, C₃₇×#12456, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 952);