#27385993

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #20998

Subgroups

	subgroup	G/H
center	C₄	#16810503
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×#575	C₂
socle	C₂×C₅₃	C₂×#386
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#575	D₅₃
hypocenter	C₅₃	#20998
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#575

1704 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)(54,58,60,59)(55,63,65,64)(56,68,70,69)(57,72,74,73)(61,76,77,66)(62,78,79,71)(67,81,83,82)(75,84,85,80);(54,57)(55,62)(56,67)(58,72)(59,73)(60,74)(61,75)(63,78)(64,71)(65,79)(66,80)(68,81)(69,82)(70,83)(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)(55,61,65,77)(57,67)(58,68)(59,69)(60,70)(62,75,79,85)(63,76,64,66)(71,80,78,84)(72,81)(73,82)(74,83);(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)(54,55,59,64,60,65,58,63)(56,66,69,77,70,76,68,61)(57,71,72,62,74,78,73,79)(67,75,81,84,83,85,82,80)

Polycyclic presentation:

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

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, 171469);