#27382610

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1776 ⋊ C₂
Sylow	C₅₃ ⋊ #20504

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1776	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂²×D₂₁₂	C₂³
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1776	D₅₃
hypocenter	C₅₃	#20504
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1776

1130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,c^2=gh,d^2=f,e^2=h,b^a=be,e^a=eh,e^b=eh,i^b=i^52,d^c=dh,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, 168086);