#27278115

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1570 ⋊ C₂
Sylow	C₅₃ ⋊ #8862

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133185
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1570	C₂
socle	C₂²×C₅₃	#385
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1570	D₅₃
hypocenter	C₅₃	#8862
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1570

420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(62,65)(63,69)(64,72)(66,76)(67,75)(68,78)(70,82)(71,81)(73,85)(74,86)(77,88)(79,84)(80,90)(83,92)(87,89)(91,93);(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)(55,58)(56,61)(59,60)(62,64,66,74,67,73,77,87)(63,68,70,80,71,79,83,91)(65,72,76,86,75,85,88,89)(69,78,82,90,81,84,92,93);(54,55,57,60)(56,58,61,59)(62,63,66,70,67,71,77,83)(64,68,74,80,73,79,87,91)(65,69,76,82,75,81,88,92)(72,84,86,93,85,78,89,90)

Polycyclic presentation:

2^8,53|a^2=f,b^2=fg,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=eh,d^c=dh,i^c=i^52

Conjugacy classes & character table

1034 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 63591);