#27484038

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #29153

Subgroups

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2327	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#2327	D₅₃
hypocenter	C₅₃	#29153
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #2327

686 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 13568

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

Polycyclic presentation:

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

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, 269514);