C₄×Q₃₂×C₃₇ #17990992

order	4736
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₄×Q₃₂×C₃₇

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	D₈
commutator	C₈	C₂²×C₄×C₃₇
Frattini, cosocle	C₂×C₈	C₂³×C₃₇
socle, prime radical	C₂²×C₃₇	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₄²×C₃₇ ◁ C₄×C₈×C₃₇ ◁ G

150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 4736

A minimal faithful permutation representation — 3 generators on 73 points:

(1,2,3,4)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22);(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(9,25,17,33)(10,26,18,34)(11,27,19,35)(12,28,20,36)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#7384, C₃₇×#7386, C₈×Q₃₂×C₃₇, …
maximal normal subgroup of	C₃₇×#7384, C₃₇×#7386, C₈×Q₃₂×C₃₇, …
maximal quotient of	C₈×Q₃₂×C₃₇, C₃₇×#12072, C₃₇×#12083, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 906);

C4×Q32×C37 #17990992