#27400009

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 ⋊ #22456

Subgroups

	subgroup	G/H
center	C₂	C₂²⋊C₄×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1757	C₂
socle	C₂×C₅₃	D₄×C₂²⋊C₄
prime radical	C₂×#16811435	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1757	D₅₃
hypocenter	C₅₃	#22456
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757

1336 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)(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)(56,66)(58,73)(60,75)(62,72)(64,76)(65,80)(67,68)(70,85)(74,79)(77,84)(78,83)(81,82);(54,57)(55,61)(56,65)(58,70)(59,71)(60,74)(62,77)(63,69)(64,78)(66,80)(67,81)(68,82)(72,84)(73,85)(75,79)(76,83);(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)(54,56,59,68)(55,60,63,76)(57,65,71,82)(58,66,73,67)(61,74,69,83)(62,75,72,64)(70,80,85,81)(77,79,84,78);(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)(56,64)(57,69,71,61)(58,72)(59,63)(60,67)(62,73)(65,79,82,78)(66,76)(68,75)(70,77,85,84)(74,81,83,80)

Polycyclic presentation:

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

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 185485);