#27290830

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#419 ⋊ C₂
Sylow	C₅₃ ⋊ #11013

Subgroups

	subgroup	G/H
center	C₂²	#16810500
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#419	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂²×#11719945	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#419	D₅₃
hypocenter	C₅₃	#11013
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#419

1262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(58,61,63,74)(59,65,66,81)(60,69,71,87)(62,76,77,73)(64,78,68,86)(67,85,83,82)(70,88,75,72)(79,89,84,80);(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)(58,60)(59,64)(61,72)(62,75)(63,71)(65,80)(66,68)(67,84)(69,73)(70,77)(74,88)(76,87)(78,82)(79,83)(81,89)(85,86);(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)(60,68,71,64)(61,65)(62,67)(63,66)(69,86,87,78)(70,84,75,79)(72,89,88,80)(73,82)(74,81)(76,85)(77,83)

Polycyclic presentation:

2^8,53|a^2=g,c^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,i^a=i^52,c^b=cf,f^b=fh,g^b=gh,i^b=i^52,f^c=fh

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 76306);