Q₈⋊C₈×C₇₁ #17985457

order	4544
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₇₁ · Schur multiplier (H₂) = C₂

Aliases

direct product	Q₈⋊C₈×C₇₁
unambiguous semidirect product	Q₈⋊(C₈×C₇₁)

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₁	D₄
commutator	C₄	C₂×C₈×C₇₁
Frattini, cosocle	C₄²	C₂²×C₇₁
socle, prime radical	C₂²×C₇₁	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₄²×C₇₁ ◁ G

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 4544

A minimal faithful permutation representation — 2 generators on 87 points:

(2,5)(3,7)(6,8)(9,10,11,13,12,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(1,2,4,6)(3,8,7,5)

Conjugacy classes & character table

1988 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×Q₈⋊C₈×C₇₁, C₇₁×#1476, C₇₁×#1477, …
maximal normal subgroup of	C₂×Q₈⋊C₈×C₇₁, C₇₁×#1476, C₇₁×#1477, …
maximal quotient of	C₇₁×#1271, C₇₁×#1272, C₇₁×#1279, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4544, 7);

Q8⋊C8×C71 #17985457