C₃×Dic₁₂₁ #13089703

order	1452
isoclinism	isoclinic class of D₁₂₁
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×Dic₁₂₁
unambiguous semidirect product	C₁₂₁⋊(C₃×C₄)

Splittings

derived, Sylow	C₁₂₁ ⋊ C₃×C₄
Hall	C₃×C₁₂₁ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₁₂₁
commutator, hypocenter	C₁₂₁	C₃×C₄
Frattini	C₂×C₁₁	C₃×D₁₁
Fitting	C₂×C₃×C₁₂₁	C₂
socle, prime radical	C₂×C₃×C₁₁	D₁₁
cosocle	C₂×C₁₂₁	C₂×C₃
LCS	G ▷ C₁₂₁
UCS	C₁ ◁ C₂×C₃

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 1452

A minimal faithful permutation representation — 2 generators on 128 points:

(1,2,3)(4,5)(6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128);(4,6,5,7)(9,128)(10,127)(11,126)(12,125)(13,124)(14,123)(15,122)(16,121)(17,120)(18,119)(19,118)(20,117)(21,116)(22,115)(23,114)(24,113)(25,112)(26,111)(27,110)(28,109)(29,108)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₉×Dic₁₂₁, C₃²×Dic₁₂₁, C₃×Dic₃₆₃, …
maximal normal subgroup of	C₉×Dic₁₂₁, C₃²×Dic₁₂₁, C₃×C₁₃×Dic₁₂₁, …
maximal quotient of	C₉×Dic₁₂₁, C₃²×Dic₁₂₁, C₃×Dic₃₆₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1452, 2);

C3×Dic121 #13089703