C₃₇×#1552 #17990370

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×#1552

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

208 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 96 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)(39,43)(40,41)(44,45)(46,47,49,53,50,52,58,60)(48,54,55,61,56,51,57,59);(38,39,42,44)(40,43,41,45)(46,48,50,56)(47,51,52,54)(49,57,58,55)(53,61,60,59);(38,40,42,41)(39,43,44,45)(47,52)(51,54)(53,60)(59,61)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#8457, C₃₇×#8458, C₃₇×#8460, …
maximal normal subgroup of	C₃₇×#8457, C₃₇×#8458, C₃₇×#8460, …
maximal quotient of	C₃₇×#7579, C₃₇×#7585, C₃₇×#7613, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 284);