#27550779

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #34584

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2618	C₂
prime radical	C₂×#121×C₅₃	C₂²
hypercenter	#2618	D₅₃
hypocenter	C₅₃	#34584
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2618

1774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|b^2=g,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,c^b=ch,i^e=i^52

Conjugacy classes & character table

896 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 336255);