C₃₇×#1378 #17990196

order	4736
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#1378

Subgroups

	subgroup	G/H
center	C₈×C₃₇	C₂²⋊C₄
commutator	C₂²	C₄×C₈×C₃₇
Frattini, cosocle	C₂²×C₈	C₂²×C₃₇
socle	C₂×C₃₇	#336
prime radical	C₂³×C₃₇	C₂×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₃₇ ◁ C₂²×C₈×C₃₇ ◁ G

132 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 69
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 2 generators on 69 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)(38,39,41,45,42,47,52,59,43,46,53,58,54,60,65,68)(40,48,50,61,51,62,64,69,49,44,55,56,63,57,66,67);(38,40)(39,44,46,48)(41,50)(42,51)(43,49)(45,56,58,61)(47,57,60,62)(52,64)(53,55)(54,63)(59,67,68,69)(65,66)

Conjugacy classes & character table

1628 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₃₇×#1378, C₃₇×#12597, C₃₇×#12609, …
maximal normal subgroup of	C₂×C₃₇×#1378, C₃₇×#12597, C₃₇×#12609, …
maximal quotient of	C₃₇×#7140, C₃₇×#7144, C₃₇×#7147, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 110);