C₃₇×#2875 #17991693

order	4736
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄²×C₃₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₃₇×#2875

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂⁴
commutator	C₂	C₂²×C₄²×C₃₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
socle	C₂²×C₃₇	C₂³×C₄
prime radical	C₂×#41×C₃₇	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ G

556 conjugacy classes of subgroups; only the 504 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 504 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) = 2368

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)(42,43,45,49,46,47,52,54)(44,48,51,55,50,53,56,57);(38,39,40,41)(43,47)(48,53)(49,54)(55,57);(42,44)(43,48)(45,51)(46,50)(47,53)(49,55)(52,56)(54,57);(44,50)(48,53)(51,56)(55,57)

Conjugacy classes & character table

2516 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#9311, C₃₇×#9312, C₃₇×#12609, …
maximal normal subgroup of	C₃₇×#9311, C₃₇×#9312, C₃₇×#12609, …
maximal quotient of	C₃₇×#17316, C₃₇×#17320, C₃₇×#17329, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1607);