#27392005

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1924 ⋊ C₂
Sylow	C₅₃ ⋊ #21635

Subgroups

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#1924	C₂
prime radical	C₂²×C₂²⋊D₅₃	C₂×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1924	D₅₃
hypocenter	C₅₃	#21635
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1924

890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 81 points:

(1,2)(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)(66,69,70,78,71,67,73,74)(68,76,77,81,75,72,79,80);(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)(54,56,57,55)(59,63)(62,65)(68,75)(72,76)(77,79)(80,81);(54,55,57,56)(58,60,61,64)(59,62,63,65)(66,68)(67,72)(69,76)(70,77)(71,75)(73,79)(74,80)(78,81);(55,56)(58,59)(60,62)(61,63)(64,65)(66,67,70,74,71,69,73,78)(68,72,77,80,75,76,79,81)

Polycyclic presentation:

2^8,53|a^2=g,b^2=f,c^2=e,d^2=g,g^2=h,b^a=be,c^a=cf,c^b=ch,i^c=i^52,i^d=i^52

Conjugacy classes & character table

1232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 177481);