#27411295

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#575 ⋊ C₂
Sylow	C₅₃ ⋊ #23438

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133189
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#575	C₂
socle	C₂²×C₅₃	C₂×#121
hypercenter	C₂×#575	D₅₃
hypocenter	C₅₃	#23438
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#575

1922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 196771);