#27289523

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#344 ⋊ C₂
Sylow	C₅₃ ⋊ #10764

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#344	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂²×#11719945	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#344	D₅₃
hypocenter	C₅₃	#10764
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#344

1194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 74999);