#27431386

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#452 ⋊ C₂
Sylow	C₅₃ ⋊ #25143

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133188
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#452	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#20283957	C₂
hypercenter	C₂×#452	D₅₃
hypocenter	C₅₃	#25143
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#452

996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 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)(55,57)(58,62,64,63)(59,67,69,68)(60,71,73,72)(61,76,77,78)(65,80,82,81)(66,74,84,85)(70,87,75,88)(79,86,83,89);(58,61,64,77)(59,66,69,84)(60,70,73,75)(62,76,63,78)(65,79,82,83)(67,74,68,85)(71,87,72,88)(80,86,81,89);(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)(55,57)(58,60,63,72,64,73,62,71)(59,65,68,81,69,82,67,80)(61,75,76,88,77,70,78,87)(66,83,74,89,84,79,85,86);(54,55)(56,57)(58,59,63,68,64,69,62,67)(60,65,72,81,73,82,71,80)(61,74,76,84,77,85,78,66)(70,86,87,83,75,89,88,79)

Polycyclic presentation:

2^8,53|a^2=f,b^2=f,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,i^b=i^52,f^c=fh

Conjugacy classes & character table

1031 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 216862);