Dic₃×C₁₂₁ #13089702

order	1452
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₂₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	Dic₃×C₁₂₁
unambiguous semidirect product	C₃⋊(C₄×C₁₂₁)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₂₁	D₃
commutator, hypocenter	C₃	C₄×C₁₂₁
Frattini	C₂×C₁₁	D₃×C₁₁
Fitting	C₂×C₃×C₁₂₁	C₂
socle, prime radical, cosocle	C₂×C₃×C₁₁	C₂×C₁₁
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₁₂₁

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);(1,3,2,4)(6,7)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	Dic₉×C₁₂₁, C₃×Dic₃×C₁₂₁, C₃⋊Dic₃×C₁₂₁, …
maximal normal subgroup of	C₃×Dic₃×C₁₂₁, Dic₃×C₁₃×C₁₂₁, Dic₃×C₁₇×C₁₂₁, …
maximal quotient of	Dic₉×C₁₂₁, C₃×Dic₃×C₁₂₁, C₃⋊Dic₃×C₁₂₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1452, 1);

Dic3×C121 #13089702