#27400883

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1805 ⋊ C₂
Sylow	C₅₃ ⋊ #22541

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1805	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂²×D₂₁₂	C₂³
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1805	D₅₃
hypocenter	C₅₃	#22541
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1805

1074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 81 points:

(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)(58,61,62,60)(59,64,63,65);(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)(58,60,62,61)(59,64,63,65)(67,72)(69,77)(71,78)(73,74)(75,81)(79,80);(58,59,62,63)(60,64,61,65)(66,68,70,76)(67,71,74,80)(69,75,77,81)(72,78,73,79);(54,55,56,57)(59,63)(64,65)(66,67)(68,71)(69,72)(70,74)(73,77)(75,78)(76,80)(79,81)

Polycyclic presentation:

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

Conjugacy classes & character table

1058 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 186359);