#27538157

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2197 ⋊ C₂
Sylow	C₅₃ ⋊ #33552

Subgroups

	subgroup	G/H
center	C₂	B(4,𝔽₂)×D₅₃
commutator, cosocle	C₂×D₄×C₅₃	C₂⁴
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#2197	C₂
socle	C₂×C₅₃	C₂×B(4,𝔽₂)
hypercenter	#2197	D₅₃
hypocenter	C₅₃	#33552
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #2197

1156 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)(7,20)(11,25)(12,26)(14,28)(19,31)(24,32)(27,29)(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);(1,4,6,16)(2,8,10,22)(3,12,13,26)(5,17,18,15)(7,19,20,31)(9,21,23,30)(11,24,25,32)(14,29,28,27)(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);(1,3)(2,7)(4,12)(5,14)(6,13)(8,19)(9,11)(10,20)(15,27)(16,26)(17,29)(18,28)(21,24)(22,31)(23,25)(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)(19,27)(20,28)(26,32)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

554 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323633);