#27440465

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3142 ⋊ C₂
Sylow	C₅₃ ⋊ #25879

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3142	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288495	C₂
hypercenter	#3142	D₅₃
hypocenter	C₅₃	#25879
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3142

1178 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)(54,57,58,56)(55,61,60,59)(65,78)(70,88)(73,90)(76,89)(77,91)(79,82)(83,92)(87,93);(55,60)(59,61)(62,65,66,79,68,78,67,82)(63,70,71,76,72,88,80,89)(64,73,74,91,75,90,81,77)(69,83,84,87,86,92,85,93);(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)(54,56,58,57)(55,59,60,61)(62,64)(63,69)(65,77)(66,74)(67,81)(68,75)(70,87)(71,84)(72,86)(73,79)(76,92)(78,91)(80,85)(82,90)(83,89)(88,93);(54,55)(56,61)(57,59)(58,60)(62,63,68,72)(64,69,75,86)(65,76,78,89)(66,80,67,71)(70,82,88,79)(73,87,90,93)(74,85,81,84)(77,92,91,83)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 225941);