#27538198

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3025 ⋊ C₂
Sylow	C₅₃ ⋊ #33556

Subgroups

	subgroup	G/H
center	C₂	C₂×#16810809
commutator, cosocle	C₂×D₄×C₅₃	C₂⁴
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#3025	C₂
socle	C₂×C₅₃	C₂×B(4,𝔽₂)
hypercenter	#3025	D₅₃
hypocenter	C₅₃	#33556
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3025

1134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

866 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323674);