#27540950

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2236 ⋊ C₂
Sylow	C₅₃ ⋊ #33864

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2236	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	#2236	D₅₃
hypocenter	C₅₃	#33864
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2236

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

809 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326426);