C₃₇×#2175 #17990993

order	4736
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#2175

Subgroups

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

160 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 80 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)(38,39,41,40)(42,43)(44,47)(45,54)(46,48)(49,51)(50,52)(53,55)(56,57);(43,47)(44,51)(45,52)(48,55)(49,50)(53,57)(54,56);(38,40,41,39)(43,48)(47,55)(49,56)(50,54)

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#12065, C₃₇×#12077, C₃₇×#12080, …
maximal normal subgroup of	C₃₇×#12065, C₃₇×#12077, C₃₇×#12080, …
maximal quotient of	C₃₇×#12043, C₃₇×#12074, C₃₇×#12075, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 907);