#27445701

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1979 ⋊ C₂
Sylow	C₅₃ ⋊ #26271

Subgroups

	subgroup	G/H
center	C₂²	#16811257
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1979	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#13133239	C₂²
hypercenter	#1979	D₅₃
hypocenter	C₅₃	#26271
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1979

752 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 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)(54,56,57,61)(55,59,58,60)(62,66,86,93,68,82,67,76)(63,71,73,91,74,87,72,85)(64,77,79,84,80,83,78,65)(69,70,89,92,88,90,75,81);(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)(62,65,67,83,68,84,86,77)(63,70,72,81,74,90,73,92)(64,76,78,82,80,93,79,66)(69,87,75,91,88,71,89,85);(55,58)(59,60)(62,64,67,78,68,80,86,79)(63,69,72,75,74,88,73,89)(65,82,83,93,84,66,77,76)(70,85,81,87,90,91,92,71);(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)(56,60)(57,58)(59,61)(62,63)(64,75)(65,81)(66,85)(67,73)(68,74)(69,78)(70,83)(71,76)(72,86)(77,90)(79,88)(80,89)(82,91)(84,92)(87,93)

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 231177);