#27367258

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1546 ⋊ C₂
Sylow	C₅₃ ⋊ #19199

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133190
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1546	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#16811173	C₂²
hypercenter	#1546	D₅₃
hypocenter	C₅₃	#19199
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1546

650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 152734);