#27392430

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #21676

Subgroups

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#1832	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1832	D₅₃
hypocenter	C₅₃	#21676
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1832

658 conjugacy classes of subgroups; only the 239 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

1232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 177906);