#27302950

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13330

Subgroups

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×Q₃₂×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂²×Q₃₂	D₅₃
hypocenter	C₅₃	#13330
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(59,61)(62,65,68,64)(63,70,73,69)(66,82,85,81)(67,76,83,79)(71,91,92,90)(72,87,86,88)(74,80,77,93)(75,84,78,89);(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)(55,57)(58,59)(60,61)(62,64,68,65)(63,69,73,70)(66,81,85,82)(67,79,83,76)(71,90,92,91)(72,88,86,87)(74,93,77,80)(75,89,78,84);(54,55)(56,57)(62,63,68,73)(64,74,65,77)(66,80,85,93)(67,86,83,72)(69,78,70,75)(71,89,92,84)(76,91,79,90)(81,88,82,87)

Polycyclic presentation:

2^8,53|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,c^b=cf,d^b=dg,e^b=eg,g^b=gh,d^c=dg,e^c=eg,g^c=gh,i^c=i^52

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 88426);