#27390047

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄×#101×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #21476

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133187
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#101×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂×#13133265	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#101	D₅₃
hypocenter	C₅₃	#21476
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#101

1322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 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)(54,56,57,55)(61,72)(65,80)(69,86)(73,76)(78,89)(81,83)(84,85)(87,88);(58,61,63,72)(59,65,67,80)(60,69,71,86)(62,76,77,73)(64,78,79,89)(66,83,74,81)(68,84,82,85)(70,88,75,87);(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)(58,60)(59,64)(61,69)(62,75)(63,71)(65,78)(66,82)(67,79)(68,74)(70,77)(72,86)(73,88)(76,87)(80,89)(81,84)(83,85);(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)(54,55,57,56)(58,59,63,67)(60,68,71,82)(61,65,72,80)(62,74,77,66)(64,75,79,70)(69,85,86,84)(73,83,76,81)(78,88,89,87)

Polycyclic presentation:

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

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 175523);