C₈×Q₈×C₅₃ #16810200

order	3392
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈×Q₈×C₅₃

Subgroups

	subgroup	G/H
center	C₂×C₈×C₅₃	C₂²
commutator	C₂	C₂²×C₈×C₅₃
Frattini, cosocle	C₂×C₄	C₂³×C₅₃
socle, prime radical	C₂²×C₅₃	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₃ ◁ G

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5,6,7,8)(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);(9,10,11,12)(13,16,15,14);(9,13,11,15)(10,14,12,16)

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	Q₈⋊#13133267, #28299005, C₈×#13133282, …
maximal subgroup of	C₅₃×Q₈⋊C₁₆, C₈×SD₁₆×C₅₃, C₈×Q₁₆×C₅₃, …
maximal normal subgroup of	C₅₃×Q₈⋊C₁₆, C₈×SD₁₆×C₅₃, C₈×Q₁₆×C₅₃, …
maximal quotient of	C₈×C₄⋊C₄×C₅₃, C₅₃×#1916, C₅₃×#1939, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3392, 126);

C8×Q8×C53 #16810200