D₅₃×#1420 #27304561

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

D₅₃×#1420

Splittings

Fitting	C₅₃×#1420 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1420

Subgroups

	subgroup	G/H
center	C₂	#357×D₅₃
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1420	C₂
socle	C₂×C₅₃	C₂×#357
prime radical	C₂²×D₅₃	SD₃₂
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#1420	D₅₃
hypocenter	C₅₃	C₂×#1420
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1420

286 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 117
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 117 points:

(1,3,8,19)(2,9,13,35)(4,21,26,50)(5,18,7,16)(6,28,30,48)(10,37,40,46)(11,41,31,33)(12,42,43,44)(14,20,47,57)(15,24,51,61)(17,22,53,29)(23,59,25,58)(27,34,38,63)(32,36,49,60)(39,52,56,64)(45,62,54,55)(65,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,110,112,114,116,117,115,113,111,109,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67);(1,2,5,11,8,13,7,31)(3,14,16,45,19,47,18,54)(4,20,23,55,26,57,25,62)(6,27,22,43,30,38,29,12)(9,32,33,15,35,49,41,51)(10,36,39,61,40,60,56,24)(17,52,48,37,53,64,28,46)(21,42,58,63,50,44,59,34);(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)(114,115)(116,117)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90037);