#27349909

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1459 ⋊ C₂
Sylow	C₅₃ ⋊ #17350

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1459	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂²×D₄×D₅₃	C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1459	D₅₃
hypocenter	C₅₃	#17350
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1459

1418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 77 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)(65,73)(68,75)(71,77)(74,76);(62,65)(63,68)(64,71)(66,73)(67,74)(69,75)(70,76)(72,77);(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)(62,64)(63,67)(65,71)(66,72)(68,74)(69,70)(73,77)(75,76);(54,55,56,58,57,59,60,61)(62,63)(64,70,72,67)(65,68)(66,69)(71,76,77,74)(73,75)

Polycyclic presentation:

2^8,53|a^2=f,f^2=h,b^a=be,e^a=eg,f^b=fg,i^b=i^52,d^c=dg,i^d=i^52

Conjugacy classes & character table

1226 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 135385);