#27418378

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #24149

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133186
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3510	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₅₃×#535	C₂²
hypercenter	#3510	D₅₃
hypocenter	C₅₃	#24149
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3510

1038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

2^8,53|c^2=h,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,g^c=gh,i^c=i^52

Conjugacy classes & character table

1187 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 203854);