Q₆₄×Dic₅₃ #27304678

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

Aliases

direct product

Q₆₄×Dic₅₃

Splittings

Sylow

C₅₃ ⋊ C₄×Q₆₄

Subgroups

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

190 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 75 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 — 3 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)(54,55,56,57);(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)(58,60,64,74)(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);(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)

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, 90154);