#27383441

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#413 ⋊ C₂
Sylow	C₅₃ ⋊ #20623

Subgroups

	subgroup	G/H
center	C₄	#16810478
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#413	C₂
socle	C₂×C₅₃	C₂×#379
prime radical	C₂×#16811459	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#413	D₅₃
hypocenter	C₅₃	#20623
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#413

3644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 85 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)(54,58,60,59)(55,63,64,73)(56,67,68,74)(57,70,72,71)(61,76,77,65)(62,78,69,84)(66,81,82,85)(75,80,83,79);(54,57)(55,62)(56,66)(58,70)(59,71)(60,72)(61,75)(63,78)(64,69)(65,79)(67,81)(68,82)(73,84)(74,85)(76,80)(77,83);(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)(54,56)(55,61)(57,66)(58,67)(59,74)(60,68)(62,75)(63,76)(64,77)(65,73)(69,83)(70,81)(71,85)(72,82)(78,80)(79,84);(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)(56,65)(57,69,72,62)(58,63)(59,73)(60,64)(61,67)(66,80,82,79)(68,76)(70,84,71,78)(74,77)(75,81,83,85)

Polycyclic presentation:

2^8,53|a^2=g,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,i^a=i^52,e^b=eh,i^b=i^52,g^c=gh

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, 168917);