D₅₃×#1711 #27371766

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

Aliases

direct product

D₅₃×#1711

Splittings

Fitting	C₅₃×#1711 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1711

Subgroups

	subgroup	G/H
center	C₂²	#127×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1711	C₂
socle	C₂²×C₅₃	C₂×#127
prime radical	#127×D₅₃	C₂²
hypercenter	#1711	D₅₃
hypocenter	C₅₃	C₂×#1711
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1711

802 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 179 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) = 3392

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

Conjugacy classes & character table

1064 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 157242);