#27437576

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2658 ⋊ C₂
Sylow	C₅₃ ⋊ #25596

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2658	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#16811190	C₂²
hypercenter	#2658	D₅₃
hypocenter	C₅₃	#25596
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2658

1064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 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)(54,56,57,61)(55,59,58,60)(63,70)(65,77)(68,72)(71,80)(73,89)(76,91)(78,92)(83,87);(62,65)(63,69)(64,73,74,89)(66,77)(67,78)(68,82,72,88)(70,85)(71,86)(75,76,81,91)(79,92)(80,93)(83,84,87,90);(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)(55,58)(59,60)(62,64)(63,68)(65,76)(66,74)(67,81)(69,84)(70,72)(71,87)(73,78)(75,79)(77,91)(80,83)(82,86)(85,90)(88,93)(89,92);(54,55)(56,60)(57,58)(59,61)(62,63,66,70)(64,72,74,68)(65,69,77,85)(67,80,79,71)(73,82,89,88)(75,83,81,87)(76,90,91,84)(78,93,92,86)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 223052);