#27412818

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #23622

Subgroups

	subgroup	G/H
center	C₄	#16810546
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#413	C₂
socle	C₂×C₅₃	C₂×#396
prime radical	C₂²×#41×C₅₃	C₂²
hypercenter	C₂×#413	D₅₃
hypocenter	C₅₃	#23622
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#413

960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 198294);