#27468192

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #27833

Subgroups

	subgroup	G/H
center	C₄	#16811338
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1764	C₂
socle	C₂×C₅₃	#2495
prime radical	C₂×#41×C₅₃	C₂³
hypercenter	#1764	D₅₃
hypocenter	C₅₃	#27833
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1764

442 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)(54,58,59,71,61,70,60,56)(55,64,65,62,67,68,66,78)(57,73,74,84,75,85,76,69)(63,79,72,77,81,82,80,83);(54,57)(55,63)(56,69)(58,73)(59,74)(60,76)(61,75)(62,77)(64,79)(65,72)(66,80)(67,81)(68,82)(70,85)(71,84)(78,83);(54,56,60,70,61,71,59,58)(55,62,66,64,67,78,65,68)(57,73,74,84,75,85,76,69)(63,79,72,77,81,82,80,83);(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)(54,55,60,65,61,67,59,66)(56,68,70,78,71,64,58,62)(57,72,76,81,75,80,74,63)(69,83,85,79,84,77,73,82)

Polycyclic presentation:

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

Conjugacy classes & character table

1502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 253668);