#27289585

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#344 ⋊ C₂
Sylow	C₅₃ ⋊ #10778

Subgroups

	subgroup	G/H
center	C₄	#16810478
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#344	C₂
socle	C₂×C₅₃	C₂²⋊#95
prime radical	C₂²×#11719945	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#344	D₅₃
hypocenter	C₅₃	#10778
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#344

1164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 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)(54,57)(55,62)(56,67)(58,74)(59,72)(60,73)(61,75)(63,78)(64,79)(65,71)(66,80)(68,84)(69,83)(70,82)(76,81)(77,85);(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)(54,56,59,70,60,69,58,68)(55,61,65,77,63,66,64,76)(57,67,72,82,73,83,74,84)(62,75,71,85,78,80,79,81);(54,55)(56,66,69,61)(57,71,73,79)(58,64)(59,65)(60,63)(62,72,78,74)(67,81)(68,77,70,76)(75,82)(80,84)(83,85)

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^b=fh,e^c=eh,f^c=fh,i^c=i^52

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 75061);