#27431372

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#473 ⋊ C₂
Sylow	C₅₃ ⋊ #25142

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#473	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#16811130	C₂²
hypercenter	C₂×#473	D₅₃
hypocenter	C₅₃	#25142
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#473

986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 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)(55,57);(58,61,63,74)(59,65,67,81)(60,68,70,72)(62,75,76,73)(64,77,79,80)(66,82,83,71)(69,86,87,85)(78,88,89,84);(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)(55,57)(58,60,62,69,63,70,76,87)(59,64,66,78,67,79,83,89)(61,72,73,86,74,68,75,85)(65,80,71,88,81,77,82,84);(54,55)(56,57)(58,59,62,66,63,67,76,83)(60,64,69,78,70,79,87,89)(61,71,73,81,74,82,75,65)(68,84,85,80,72,88,86,77)

Polycyclic presentation:

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

Conjugacy classes & character table

875 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 216848);