C₂²×#16810463 #27542214

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

Aliases

direct product

C₂²×#16810463

Splittings

Sylow

C₅₃ ⋊ C₂²×Q₆₄

Subgroups

	subgroup	G/H
center	C₂³	#13133158
commutator, cosocle	C₁₆×C₅₃	C₂⁴
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₂²×Q₃₂×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	D₁₆
hypercenter	C₂²×Q₃₂	D₅₃
hypocenter	C₅₃	C₂²×Q₆₄
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×Q₃₂

462 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 171 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) = 121
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 4 generators on 121 points:

(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)(58,60,64,74)(59,65,69,90)(61,76,80,110)(62,82,84,116)(63,86,83,73)(66,92,94,119)(67,96,97,121)(68,98,85,89)(70,99,101,75)(71,91,87,78)(72,95,104,77)(79,114,112,109)(81,88,115,117)(93,120,106,103)(100,107,105,113)(102,111,108,118);(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)(58,59,64,69)(60,70,74,101)(61,75,80,99)(62,81,84,115)(63,85,83,68)(65,87,90,71)(66,91,94,78)(67,95,97,77)(72,103,104,120)(73,105,86,100)(76,96,110,121)(79,113,112,107)(82,92,116,119)(88,114,117,109)(89,102,98,108)(93,111,106,118);(56,57);(54,55)

Conjugacy classes & character table

1220 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 327690);