#27540627

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2175 ⋊ C₂
Sylow	C₅₃ ⋊ #33835

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2175	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	#2175	D₅₃
hypocenter	C₅₃	#33835
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2175

670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|a^2=h,b^2=h,c^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gh,i^a=i^52,d^b=de,e^b=eg,g^b=gh,i^c=i^52,e^d=eg,g^d=gh

Conjugacy classes & character table

875 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326103);