#27362726

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #18763

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133186
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#556	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂²×D₄×C₅₃	D₄
hypercenter	C₂×#556	D₅₃
hypocenter	C₅₃	#18763
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#556

910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 85 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)(71,78)(76,79)(77,85)(80,84);(54,57,59,58)(55,61,62,66)(56,64,65,67)(60,68,69,63)(70,72,75,83)(71,77,78,85)(73,82,74,81)(76,84,79,80);(54,56,59,65)(55,60,62,69)(57,64,58,67)(61,68,66,63)(71,76)(73,74)(77,84)(78,79)(80,85)(81,82);(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)(54,55,59,62)(56,63,65,68)(57,61,58,66)(60,67,69,64)(70,71,75,78)(72,80,83,84)(73,76,74,79)(77,81,85,82)

Polycyclic presentation:

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

Conjugacy classes & character table

1343 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 148202);