#27289493

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1742 ⋊ C₂
Sylow	C₅₃ ⋊ #10757

Subgroups

	subgroup	G/H
center	C₄	#16810501
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1742	C₂
socle	C₂×C₅₃	C₂²⋊#95
prime radical	C₂²×#11719945	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1742	D₅₃
hypocenter	C₅₃	#10757
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #1742

1168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 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)(54,57)(55,61)(56,65)(58,70)(59,69)(60,72)(62,77)(63,75)(64,76)(66,80)(67,81)(68,82)(71,83)(73,79)(74,84)(78,85);(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)(54,56,58,68,59,66,71,67)(55,60,64,74,62,73,63,78)(57,65,70,82,69,80,83,81)(61,72,76,84,77,79,75,85);(54,55,58,63,59,62,71,64)(56,60,67,74,66,73,68,78)(57,61,70,75,69,77,83,76)(65,79,81,85,80,72,82,84)

Polycyclic presentation:

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

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, 74969);