#27538218

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2121 ⋊ C₂
Sylow	C₅₃ ⋊ #33558

Subgroups

	subgroup	G/H
center	C₂	C₂×#16810811
commutator, cosocle	C₂×D₄×C₅₃	C₂⁴
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#2121	C₂
socle	C₂×C₅₃	C₂×B(4,𝔽₂)
hypercenter	#2121	D₅₃
hypocenter	C₅₃	#33558
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ #2121

1158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 points:

(3,13)(4,15)(7,20)(8,21)(11,25)(14,28)(16,17)(22,30)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,4)(2,8)(3,12)(5,17)(6,15)(7,19)(9,22)(10,21)(11,24)(13,27)(14,29)(16,18)(20,31)(23,30)(25,32)(26,28)(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)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85);(1,3,6,13)(2,7,10,20)(4,12,15,27)(5,14,18,28)(8,19,21,31)(9,11,23,25)(16,26,17,29)(22,24,30,32);(1,2)(3,11)(4,8)(5,9)(6,10)(7,14)(12,24)(13,25)(15,21)(16,22)(17,30)(18,23)(19,26)(20,28)(27,32)(29,31)(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)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)

Polycyclic presentation:

2^8,53|b^2=h,b^a=be,c^a=cf,g^a=gh,i^a=i^52,d^b=dh,f^b=fgh,d^c=dh,e^c=eg,i^c=i^52,f^e=fh

Conjugacy classes & character table

632 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323694);