C₃₇×#1778 #17990596

order	4736
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₃₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₇×#1778

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂×D₄
commutator	C₄	C₂×C₄²×C₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
socle	C₂²×C₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₃₇	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 2368

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

Conjugacy classes & character table

1628 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#12651, C₃₇×#12678, C₃₇×#13114, …
maximal normal subgroup of	C₃₇×#12651, C₃₇×#12678, C₃₇×#13114, …
maximal quotient of	C₃₇×#8275, C₃₇×#8276, C₃₇×#8277, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 510);

C37×#1778 #17990596