D₅₃×#1407 #27304373

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

D₅₃×#1407

Splittings

Fitting	C₅₃×#1407 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1407

Subgroups

	subgroup	G/H
center	C₂	#353×D₅₃
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	#124	C₂²×D₅₃
Fitting	C₅₃×#1407	C₂
socle	C₂×C₅₃	C₂×#353
prime radical	C₂×D₄×D₅₃	D₄
cosocle	#124×C₅₃	C₂³
hypercenter	#1407	D₅₃
hypocenter	C₅₃	C₂×#1407
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #1407

416 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 1696

A minimal faithful permutation representation — 3 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)(54,56,59,67)(55,60,63,76)(57,65,69,83)(58,72,73,66)(61,77,78,64)(62,74,71,75)(68,80,84,79)(70,81,85,82);(54,55)(56,64,66,79)(57,68,69,84)(58,71,73,62)(59,63)(60,65,74,81)(61,70)(67,77,72,80)(75,82,76,83)(78,85);(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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89849);