C₃₇×#2375 #17991193

order	4736
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₃₇×#2375

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₃₇
Frattini, cosocle	C₂³	C₂⁴×C₃₇
prime radical	C₂×#120×C₃₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ G

688 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 264 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) = 1184

A minimal faithful permutation representation — 4 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)(38,39,40,41)(43,46)(44,48)(50,51)(52,56)(53,54)(55,57);(43,46)(47,49)(51,54)(52,57);(42,43)(44,47)(45,46)(48,49)(50,51)(52,55)(53,54)(56,57);(42,44)(43,47)(45,48)(46,49)(50,52)(51,55)(53,57)(54,56)

Conjugacy classes & character table

1406 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#12960, C₃₇×#13043, C₂×C₃₇×#2375, …
maximal normal subgroup of	C₃₇×#12960, C₃₇×#13043, C₂×C₃₇×#2375, …
maximal quotient of	C₃₇×#13827, C₃₇×#13831, C₃₇×#13832, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1107);