#27544160

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2316 ⋊ C₂
Sylow	C₅₃ ⋊ #34179

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Fitting	C₅₃×#2316	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂⁴×D₅₃	C₂³
hypercenter	#2316	D₅₃
hypocenter	C₅₃	#34179
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2316

2390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 5 generators on 81 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)(67,75)(72,76)(73,74)(80,81);(66,70,71,69)(67,74,75,73)(68,77,78,79)(72,80,76,81);(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)(59,63)(62,65)(66,69,71,70)(67,73,75,74)(68,77,78,79)(72,80,76,81);(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)(58,60,61,64)(59,62,63,65)(66,68)(67,72)(69,77)(70,79)(71,78)(73,80)(74,81)(75,76);(54,55,56,57)(58,59)(60,62)(61,63)(64,65)(66,67)(68,76)(69,73)(70,74)(71,75)(72,78)(77,81)(79,80)

Polycyclic presentation:

2^8,53|a^2=h,b^2=g,c^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df,i^b=i^52,i^c=i^52

Conjugacy classes & character table

1232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 329636);