C₃₇×C₂²⋊C₃₂ #17990217

order	4736
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₃₇×C₂²⋊C₃₂
unambiguous semidirect product	C₂²⋊(C₃₂×C₃₇)

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₃₇	C₂²
commutator	C₂	C₂×C₃₂×C₃₇
Frattini, cosocle	C₂×C₁₆	C₂²×C₃₇
socle	C₂²×C₃₇	C₂×C₁₆
prime radical	C₂³×C₃₇	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₃₇ ◁ G

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 2 generators on 73 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)(42,43,44,48,45,49,52,58,46,50,53,59,55,61,64,68,47,51,54,60,56,62,65,69,57,63,66,70,67,71,72,73);(39,41)

Conjugacy classes & character table

2960 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#7334, C₃₇×#7335, C₃₇×#7362, …
maximal normal subgroup of	C₃₇×#7334, C₃₇×#7335, C₃₇×#7362, …
maximal quotient of	C₃₇×#7334, C₃₇×#7362, C₃₇×D₄⋊C₃₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 131);

C37×C22⋊C32 #17990217