#27390069

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1757 ⋊ C₂
Sylow	C₅₃ ⋊ #21478

Subgroups

	subgroup	G/H
center	C₄	C₂×#13133187
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1757	C₂
socle	C₂×C₅₃	C₄×#120
prime radical	#20284068	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1757	D₅₃
hypocenter	C₅₃	#21478
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1757

1282 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)(54,58,60,74,61,75,59,63)(55,64,65,77,66,57,72,73)(56,69,70,85,71,82,76,78)(62,79,81,84,80,68,67,83);(54,57)(55,63)(56,68)(58,65)(59,77)(60,73)(61,64)(62,78)(66,74)(67,82)(69,81)(70,83)(71,79)(72,75)(76,84)(80,85);(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)(54,56)(55,62)(57,68)(58,69)(59,76)(60,70)(61,71)(63,78)(64,79)(65,81)(66,80)(67,72)(73,83)(74,85)(75,82)(77,84);(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)(54,55,60,65,61,66,59,72)(56,67,70,62,71,81,76,80)(57,63,73,58,64,74,77,75)(68,69,83,85,79,82,84,78)

Polycyclic presentation:

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

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, 175545);