#27538813

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2228 ⋊ C₂
Sylow	C₅₃ ⋊ #33634

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2228	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#13133236	C₂²
hypercenter	#2228	D₅₃
hypocenter	C₅₃	#33634
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2228

534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 93 points:

(1,2)(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)(62,64,67,73)(63,69,72,83)(65,74,78,87)(66,75,77,88)(68,80,82,91)(70,84,81,92)(71,85,79,90)(76,89,86,93);(63,68)(65,76)(66,77)(69,80)(70,71)(72,82)(74,89)(75,88)(78,86)(79,81)(83,91)(84,85)(87,93)(90,92);(54,56,57,61)(55,58,59,60)(64,73)(69,83)(74,87)(75,88)(80,91)(84,92)(85,90)(89,93);(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)(54,55,57,59)(56,60,61,58)(62,63,67,72)(64,69,73,83)(65,68,78,82)(66,79,77,71)(70,86,81,76)(74,80,87,91)(75,90,88,85)(84,93,92,89)

Polycyclic presentation:

2^8,53|a^2=eh,b^2=e,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,i^a=i^52,d^b=dh,f^c=fg,g^c=gh,i^d=i^52

Conjugacy classes & character table

800 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324289);