#27274130

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8025

Subgroups

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1294	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1294	D₅₃
hypocenter	C₅₃	#8025
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1294

654 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:

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

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=h,d^2=g,b^a=bd,d^a=dg,f^a=fgh,c^b=cgh,d^b=dh,i^c=i^52

Conjugacy classes & character table

1052 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 59606);