#27416928

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2038 ⋊ C₂
Sylow	C₅₃ ⋊ #24023

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2038	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288364	C₂
hypercenter	#2038	D₅₃
hypocenter	C₅₃	#24023
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2038

1214 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)(55,58)(59,61)(65,76)(69,86)(72,90)(75,88)(77,80)(82,85)(89,93)(91,92);(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)(62,65)(63,69)(64,72)(66,80)(67,76)(68,82)(70,88)(71,86)(73,92)(74,90)(75,78)(77,81)(79,91)(83,93)(84,85)(87,89);(55,58)(59,61)(62,64,67,74)(63,68,71,84)(65,72,76,90)(66,79,81,73)(69,85,86,82)(70,87,78,83)(75,89,88,93)(77,92,80,91);(54,55)(56,59)(57,58)(60,61)(62,63)(64,68)(65,75)(66,78)(67,71)(69,77)(70,81)(72,89)(73,87)(74,84)(76,88)(79,83)(80,86)(82,91)(85,92)(90,93)

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 202404);