#27538178

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3028 ⋊ C₂
Sylow	C₅₃ ⋊ #33554

Subgroups

	subgroup	G/H
center	C₂	C₂×#16810809
commutator, cosocle	C₂×D₄×C₅₃	C₂⁴
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#3028	C₂
socle	C₂×C₅₃	C₂×B(4,𝔽₂)
prime radical	#20284107	C₂
hypercenter	#3028	D₅₃
hypocenter	C₅₃	#33554
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3028

1112 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

788 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323654);