#27420096

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#444 ⋊ C₂
Sylow	C₅₃ ⋊ #24293

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₂₁₂
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#444	C₂
socle	C₂²×C₅₃	D₄²
hypercenter	C₂×#444	D₅₃
hypocenter	C₅₃	#24293
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#444

1290 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)(58,61,77,89,63,79,62,78)(59,66,69,85,68,84,67,75)(60,71,74,88,73,76,72,87)(64,80,65,82,81,86,70,83);(55,57)(58,60)(59,65)(61,76)(62,72)(63,73)(64,67)(66,83)(68,70)(69,81)(71,79)(74,77)(75,86)(78,88)(80,85)(82,84)(87,89);(58,60,62,72,63,73,77,74)(59,64,67,70,68,81,69,65)(61,71,78,87,79,76,89,88)(66,80,75,83,84,86,85,82);(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)(56,57)(58,59)(60,70)(61,75)(62,69)(63,68)(64,72)(65,73)(66,78)(67,77)(71,86)(74,81)(76,80)(79,85)(82,88)(83,87)(84,89)

Polycyclic presentation:

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

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 205572);