#27538921

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2235 ⋊ C₂
Sylow	C₅₃ ⋊ #33649

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2235	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×D₅₃	D₈
hypercenter	#2235	D₅₃
hypocenter	C₅₃	#33649
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2235

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 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)(62,65,68,67)(63,70,73,72)(64,75,77,83)(66,78,82,81)(69,85,86,90)(71,87,89,79)(74,88,91,92)(76,80,84,93);(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)(54,56,57,60)(55,59,58,61)(62,64)(63,69)(65,75)(66,80)(67,83)(68,77)(70,85)(71,88)(72,90)(73,86)(74,79)(76,81)(78,84)(82,93)(87,91)(89,92);(55,58)(59,61);(54,55)(56,59)(57,58)(60,61)(62,63,68,73)(64,74,77,91)(65,72,67,70)(66,79,82,87)(69,84,86,76)(71,78,89,81)(75,92,83,88)(80,90,93,85)

Polycyclic presentation:

2^8,53|a^2=h,c^2=e,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fg,g^a=gh,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, 324397);