#27547430

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2356 ⋊ C₂
Sylow	C₅₃ ⋊ #34405

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Fitting	C₅₃×#2356	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂×#13133192	C₂²
hypercenter	#2356	D₅₃
hypocenter	C₅₃	#34405
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2356

1990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 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)(59,65)(63,66)(64,69)(72,73)(74,78,79,77)(75,82,81,86)(76,85,83,84)(80,87,88,89);(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)(59,65)(60,68)(64,69)(67,70)(74,77,79,78)(75,82,81,86)(76,84,83,85)(80,87,88,89);(58,61,62,71)(59,64,65,69)(60,67,68,70)(63,72,66,73)(75,81)(76,83)(82,86)(84,85);(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)(59,63)(61,70)(62,68)(64,73)(65,66)(67,71)(69,72)(74,76)(75,80)(77,84)(78,85)(79,83)(81,88)(82,87)(86,89);(54,55,56,57)(58,59)(60,66)(61,69)(62,65)(63,68)(64,71)(67,72)(70,73)(74,75)(76,80)(77,86)(78,82)(79,81)(83,88)(84,89)(85,87)

Polycyclic presentation:

2^8,53|a^2=h,c^2=f,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,i^b=i^52,i^d=i^52

Conjugacy classes & character table

1376 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 332906);