C₃₇×#2162 #17990980

order	4736
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₃₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₇×#2162

Subgroups

	subgroup	G/H
center	C₄×C₈×C₃₇	C₂²
commutator	C₂	C₂²×C₁₆×C₃₇
Frattini, cosocle	C₂×C₈	C₂³×C₃₇
socle	C₂²×C₃₇	C₂²×C₈
prime radical	C₂³×C₃₇	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₃₇ ◁ G

160 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 61
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 3 generators on 61 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)(40,43)(41,42)(44,45)(46,47,48,51,49,52,54,57,50,53,55,58,56,59,60,61);(39,42)(43,45);(38,40,41,44)(39,43,42,45)

Conjugacy classes & character table

2960 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#7142, C₃₇×#7144, C₃₇×#7145, …
maximal normal subgroup of	C₃₇×#7142, C₃₇×#7144, C₃₇×#7145, …
maximal quotient of	C₃₇×#12590, C₄×C₃₇×C₂²⋊C₁₆, C₄×C₃₇×C₄⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 894);