C₃₇×#2342 #17991160

order	4736
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₃₇×#2342

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₃₇
Frattini, cosocle	C₂³	C₂⁴×C₃₇
prime radical	C₂⁵×C₃₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ G

792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 1184

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

Conjugacy classes & character table

1850 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#22405, C₃₇×#22477, C₂×C₃₇×#2342, …
maximal normal subgroup of	C₃₇×#22405, C₃₇×#22477, C₂×C₃₇×#2342, …
maximal quotient of	C₂²⋊C₄²×C₃₇, C₃₇×#13799, C₃₇×#13802, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1074);

C37×#2342 #17991160