C₃₇×#2161 #17990979

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

Aliases

direct product

C₃₇×#2161

Subgroups

	subgroup	G/H
center	C₄×C₃₇	C₂×D₈
commutator	C₈	C₂²×C₄×C₃₇
Frattini, cosocle	C₂×C₈	C₂³×C₃₇
socle	C₂×C₃₇	C₂×#107
prime radical	C₂³×C₃₇	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ C₂²×C₈×C₃₇ ◁ G

140 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)(38,39,42,47,44,48,43,58)(40,49,53,65,51,63,54,68)(41,55,56,62,57,69,50,46)(45,59,61,67,60,66,64,52);(38,40,50,66,43,54,57,67,44,51,56,59,42,53,41,52)(39,45,55,65,47,61,62,63,48,60,69,68,58,64,46,49);(40,51)(45,60)(49,63)(52,67)(53,54)(59,66)(61,64)(65,68)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#12678, C₃₇×#12924, C₃₇×#12927, …
maximal normal subgroup of	C₃₇×#12678, C₃₇×#12924, C₃₇×#12927, …
maximal quotient of	C₃₇×#12009, C₃₇×#12025, C₃₇×#12032, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 893);

C37×#2161 #17990979