#27278725

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8955

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133185
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#137×C₅₃	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#137	D₅₃
hypocenter	C₅₃	#8955
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#137

488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(55,56)(60,61)(62,64)(63,69)(65,73)(66,75)(67,74)(68,71)(70,77)(72,76);(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)(54,56,58,61,57,55,59,60)(62,64,67,74)(63,68,71,69)(65,75,66,73)(70,77,76,72);(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)(54,55,58,60,57,56,59,61)(62,63,65,70,67,71,66,76)(64,72,73,69,74,77,75,68)

Polycyclic presentation:

2^8,53|a^2=f,b^2=df,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,e^a=eg,i^a=i^52,c^b=ch,d^b=dg,f^b=fg,i^b=i^52,e^c=eg

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 64201);