#27539242

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2211 ⋊ C₂
Sylow	C₅₃ ⋊ #33681

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2211	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#16810856	C₂²
hypercenter	#2211	D₅₃
hypocenter	C₅₃	#33681
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2211

532 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)(54,56,57,60)(55,59,58,61)(63,69)(68,81)(70,83)(71,84)(80,92)(82,87)(85,91)(86,93);(62,64,67,74)(63,68,69,81)(65,76,78,90)(66,79,77,73)(70,86,83,93)(71,87,84,82)(72,88,89,75)(80,91,92,85);(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)(55,58)(59,61)(62,63)(64,68)(65,70)(66,71)(67,69)(72,80)(73,82)(74,81)(75,85)(76,86)(77,84)(78,83)(79,87)(88,91)(89,92)(90,93);(54,55)(56,59)(57,58)(60,61)(64,72)(65,75)(66,77)(68,80)(70,85)(71,84)(73,90)(74,89)(76,79)(78,88)(81,92)(82,93)(83,91)(86,87)

Polycyclic presentation:

2^8,53|c^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,i^b=i^52,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, 324718);