#27541718

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #33923

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2169	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₄×C₅₃	D₈
hypercenter	#2169	D₅₃
hypocenter	C₅₃	#33923
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #2169

470 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)(56,61)(59,60)(63,70)(64,74)(69,85)(71,86)(72,80)(75,90)(76,91)(92,93);(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)(54,57,58,55)(56,60,61,59)(63,69)(64,74)(66,79)(67,78)(68,81)(70,85)(71,72)(73,84)(75,92)(77,87)(80,86)(82,89)(83,88)(90,93);(54,56)(55,59)(57,60)(58,61)(62,64,65,74)(63,68,70,73)(66,75,77,90)(67,76,78,91)(69,84,85,81)(71,82,86,88)(72,83,80,89)(79,93,87,92);(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)(54,55,58,57)(56,60,61,59)(62,63,65,70)(64,73,74,68)(66,69,77,85)(67,80,78,72)(71,87,86,79)(75,81,90,84)(76,83,91,89)(82,93,88,92)

Polycyclic presentation:

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

Conjugacy classes & character table

1499 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 327194);