C₃₇×#1554 #17990372

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#1554

Subgroups

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

198 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 2368

A minimal faithful permutation representation — 3 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)(38,39,42,45,43,46,41,50)(40,47,48,52,49,44,51,53)(54,55)(56,62)(57,65)(58,60)(59,63)(61,67)(64,69)(66,68);(38,40,43,49)(39,44,46,47)(41,48,42,51)(45,52,50,53)(54,56,58,64)(55,59,60,68)(57,66,67,63)(61,69,65,62);(38,41,43,42)(39,45,46,50)(40,48,49,51)(44,52,47,53)(55,60)(59,68)(61,65)(62,69)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#8490, C₃₇×#8501, C₃₇×#8503, …
maximal normal subgroup of	C₃₇×#8490, C₃₇×#8501, C₃₇×#8503, …
maximal quotient of	C₃₇×#7581, C₃₇×#7583, C₃₇×#7611, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 286);