D₅₃×#1400 #27304291

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

direct product

D₅₃×#1400

Splittings

Fitting	C₅₃×#1400 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1400

Subgroups

	subgroup	G/H
center	C₈	C₂²⋊C₄×D₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1400	C₂
socle	C₂×C₅₃	C₂×C₂²⋊C₁₆
prime radical	C₂³×D₅₃	C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#1400	D₅₃
hypocenter	C₅₃	C₂×#1400
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #1400

306 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 101 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)(55,60)(56,64)(62,71)(63,72)(65,76)(66,77)(75,82)(78,84);(54,55,56,61,57,62,65,73,58,63,66,74,67,75,78,83,59,60,64,70,68,71,76,80,69,72,77,81,79,82,84,85);(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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89767);