#27351445

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1519 ⋊ C₂
Sylow	C₅₃ ⋊ #17572

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1519	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂×#16811133	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1519	D₅₃
hypocenter	C₅₃	#17572
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1519

1470 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:

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

Polycyclic presentation:

2^8,53|a^2=f,d^2=h,e^2=h,b^a=be,c^a=cgh,e^a=eg,e^b=eh,f^b=fgh,i^d=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, 136921);