#27358064

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1517 ⋊ C₂
Sylow	C₅₃ ⋊ #18300

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133187
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1517	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#1517	D₅₃
hypocenter	C₅₃	#18300
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1517

1312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

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

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, 143540);