#27350677

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2988 ⋊ C₂
Sylow	C₅₃ ⋊ #17469

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2988	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	#16811201	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2988	D₅₃
hypocenter	C₅₃	#17469
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2988

920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,e^2=g,f^2=h,b^a=be,e^a=eg,c^b=ch,e^b=eg,i^b=i^52,d^c=dg

Conjugacy classes & character table

1832 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 136153);