#27410676

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #23362

Subgroups

	subgroup	G/H
center	C₂	#20277892
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3153	C₂
socle	C₂×C₅₃	#2380
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#3153	D₅₃
hypocenter	C₅₃	#23362
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3153

706 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)(55,62,64,63)(56,66)(60,75,76,80)(61,78,79,68)(65,82)(67,73)(69,83)(74,77,84,85);(54,57,70,81,58,71,59,72)(55,61,63,68,64,79,62,78)(56,65,73,69,66,82,67,83)(60,74,75,77,76,84,80,85);(54,56)(55,60)(57,69)(58,66)(59,73)(61,77)(62,75)(63,80)(64,76)(65,81)(67,70)(68,74)(71,83)(72,82)(78,84)(79,85);(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)(56,60,66,76)(57,68,71,78)(58,64)(59,63)(61,72,79,81)(62,70)(65,77)(67,75,73,80)(69,74)(82,85)(83,84)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 196152);