#27541841

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #33936

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2226	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₈×C₅₃	C₂³
hypercenter	#2226	D₅₃
hypocenter	C₅₃	#33936
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2226

474 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:

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

Polycyclic presentation:

2^8,53|b^2=eh,c^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,c^b=ch,d^b=de,i^b=i^52,f^c=fg,g^c=gh,i^c=i^52

Conjugacy classes & character table

797 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 327317);