C₃₇×#2153 #17990971

order	4736
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#2153

Subgroups

	subgroup	G/H
center	C₈×C₃₇	C₂×D₄
commutator	C₄	C₂²×C₈×C₃₇
Frattini, cosocle	C₂×C₈	C₂³×C₃₇
socle	C₂×C₃₇	C₂×C₄⋊C₈
prime radical	C₂³×C₃₇	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₃₇ ◁ C₂²×C₈×C₃₇ ◁ G

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 3 generators on 69 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)(38,39,42,47,43,46,56,61,44,48,57,62,41,54,55,67)(40,49,52,64,53,59,66,68,50,63,65,69,51,45,58,60);(38,40,41,51,44,50,43,53)(39,45,46,49,48,59,54,63)(42,52,55,58,57,65,56,66)(47,60,61,64,62,68,67,69);(40,50)(45,59)(49,63)(51,53)(52,65)(58,66)(60,68)(64,69)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#11989, C₃₇×#11990, C₃₇×#11991, …
maximal normal subgroup of	C₃₇×#11989, C₃₇×#11990, C₃₇×#11991, …
maximal quotient of	C₃₇×#11997, C₃₇×#12005, C₃₇×#12014, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 885);

C37×#2153 #17990971