C₈×#13133123 #27271418

order	13568
automorphism	?
homology	Abelianization (H₁) = C₈×C₃₂ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₈×#13133123

Splittings

derived, Sylow

C₅₃ ⋊ C₈×C₃₂

Subgroups

	subgroup	G/H
center, hypercenter	C₈×C₁₆	D₅₃
commutator, hypocenter	C₅₃	C₈×C₃₂
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	C₈×C₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₄×C₁₆
cosocle	C₄×C₁₆×C₅₃	C₂²
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₈×C₁₆

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 2 generators on 93 points:

(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)(44,45)(46,47)(48,49)(50,51)(52,53)(62,63,64,68,65,69,72,78,66,70,73,79,75,81,84,88,67,71,74,80,76,82,85,89,77,83,86,90,87,91,92,93);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,55,56,58,57,59,60,61)

Conjugacy classes & character table

3584 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56894);

C8×#13133123 #27271418