#27491554

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1967 ⋊ C₂
Sylow	C₅₃ ⋊ #29821

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133190
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1967	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#20275823	C₂
hypercenter	#1967	D₅₃
hypocenter	C₅₃	#29821
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1967

800 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:

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

Polycyclic presentation:

2^8,53|a^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,i^a=i^52,c^b=cg,g^b=gh,i^b=i^52,d^c=de,f^c=fh,g^c=gh

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 277030);