#27352089

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2970 ⋊ C₂
Sylow	C₅₃ ⋊ #17653

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#11719921
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2970	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	#20288693	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2970	D₅₃
hypocenter	C₅₃	#17653
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #2970

964 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)(69,71)(74,75)(76,77);(62,65)(63,69)(64,71)(66,74)(67,75)(68,73)(70,76)(72,77);(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)(56,60)(57,59)(58,61)(62,64,68,63)(65,71,73,69)(66,70,67,72)(74,76,75,77);(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)(55,58,59,60)(62,63,67,70,68,64,66,72)(65,69,75,76,73,71,74,77)

Polycyclic presentation:

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

Conjugacy classes & character table

1361 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 137565);