#27421412

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2880 ⋊ C₂
Sylow	C₅₃ ⋊ #24403

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133191
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2880	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂×#16811421	C₂
hypercenter	#2880	D₅₃
hypocenter	C₅₃	#24403
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2880

1332 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)(55,57)(58,62,64,63)(59,67,69,68)(60,71,73,72)(61,76,77,78)(65,80,82,81)(66,74,84,85)(70,87,88,75)(79,86,89,83);(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)(60,70)(62,76)(63,78)(64,77)(65,79)(67,74)(68,85)(69,84)(71,87)(72,75)(73,88)(80,86)(81,83)(82,89);(55,57)(58,60)(59,65)(61,75)(62,71)(63,72)(64,73)(66,83)(67,80)(68,81)(69,82)(70,76)(74,79)(77,87)(78,88)(84,86)(85,89);(54,55)(56,57)(58,59,62,67,64,69,63,68)(60,65,71,80,73,82,72,81)(61,74,78,66,77,85,76,84)(70,86,75,79,88,83,87,89)

Polycyclic presentation:

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

Conjugacy classes & character table

1343 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 206888);