C₃₇×#2179 #17990997

order	4736
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#2179

Subgroups

	subgroup	G/H
center	C₄×C₃₇	C₂×D₈
commutator	C₈	C₂²×C₄×C₃₇
Frattini, cosocle	C₂×C₈	C₂³×C₃₇
socle	C₂×C₃₇	C₄×D₈
prime radical	C₃₇×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₇ ◁ C₂×C₈×C₃₇ ◁ C₄×C₈×C₃₇ ◁ G

144 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)(38,39,43,48,44,50,60,49)(40,51,53,66,55,68,54,62)(41,47,56,65,57,46,42,58)(45,61,63,69,64,59,67,52);(38,40)(39,45)(41,52)(42,59)(43,53)(44,55)(46,62)(47,66)(48,63)(49,67)(50,64)(51,58)(54,60)(56,61)(57,69)(65,68);(38,41,43,56,44,57,60,42)(39,46,48,58,50,47,49,65)(40,52,53,61,55,69,54,59)(45,62,63,51,64,66,67,68)

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#7389, C₃₇×#12068, C₃₇×#13657, …
maximal normal subgroup of	C₃₇×#7389, C₃₇×#12068, C₃₇×#13657, …
maximal quotient of	C₃₇×#12043, C₃₇×#12044, C₃₇×#12045, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 911);

C37×#2179 #17990997