#27536650

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#494 ⋊ C₂
Sylow	C₅₃ ⋊ #33309

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#494	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂²×D₄₂₄	C₂²
hypercenter	C₂×#494	D₅₃
hypocenter	C₅₃	#33309
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

1386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 93 points:

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

Polycyclic presentation:

2^8,53|a^2=h,b^2=eh,c^2=f,d^2=g,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,d^c=dh,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, 322126);