D₅₃×#1663 #27365279

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₅₃×#1663

Splittings

Fitting	C₅₃×#1663 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1663

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1663	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂²×D₅₃	#120
hypercenter	#1663	D₅₃
hypocenter	C₅₃	C₂×#1663
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1663

830 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 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)(54,56)(55,59)(57,66)(58,63)(60,69)(61,68)(62,64)(65,67)(70,73,75,71)(72,79,81,76)(74,77,84,82)(78,83,85,80);(54,55,58,61)(56,59,63,68)(57,65,64,60)(62,69,66,67)(70,72,75,81)(71,76,73,79)(74,83,84,80)(77,85,82,78);(56,62)(57,64)(59,67)(60,65)(63,66)(68,69)(70,71,75,73)(72,78,81,85)(74,82,84,77)(76,83,79,80);(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)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150755);