Q₈×C₁₆×C₃₇ #17991000

order	4736
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

Q₈×C₁₆×C₃₇

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₃₇	C₂²
commutator	C₂	C₂²×C₁₆×C₃₇
Frattini, cosocle	C₂×C₈	C₂³×C₃₇
socle, prime radical	C₂²×C₃₇	C₂²×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₃₇ ◁ G

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 4736

A minimal faithful permutation representation — 3 generators on 61 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×Q₈⋊C₃₂, C₁₆×SD₁₆×C₃₇, C₃₇×#12050, …
maximal normal subgroup of	C₃₇×Q₈⋊C₃₂, C₁₆×SD₁₆×C₃₇, C₃₇×#12050, …
maximal quotient of	C₁₆×C₄⋊C₄×C₃₇, C₃₇×#13104, C₃₇×#13159, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 914);

Q8×C16×C37 #17991000