Dic₃×#14581888 #44544750

order	32004
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product	Dic₃×#14581888
unambiguous semidirect product	C₃⋊(C₄×#14581888)

Splittings

derived	C₃×C₇×C₁₂₇ ⋊ C₃×C₄
Hall	C₃×#14581888 ⋊ C₄
Hall	C₇×C₁₂₇ ⋊ C₃×Dic₃
Sylow	C₁₂₇ ⋊ Dic₃×ΓL(1,𝔽₈)
Sylow	C₇ ⋊ Dic₃×C₁₂₇⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×#14581888
commutator, hypocenter	C₃×C₇×C₁₂₇	C₃×C₄
Fitting, socle, cosocle	C₂×C₃×C₇×C₁₂₇	C₂×C₃
prime radical	C₂×C₃×#14581888	C₂
LCS	G ▷ C₃×C₇×C₁₂₇
UCS	C₁ ◁ C₂

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 10668

A minimal faithful permutation representation — 2 generators on 141 points:

(2,3)(4,5,6,7)(9,10,11)(12,14,13)(16,17,19)(18,21,26)(20,24,23)(22,28,36)(25,31,33)(27,34,30)(29,38,49)(32,41,46)(35,44,43)(37,47,40)(39,51,65)(42,54,62)(45,57,59)(48,60,56)(50,63,53)(52,67,68)(55,70,81)(58,73,78)(61,76,75)(64,79,72)(66,82,69)(71,87,85)(74,90,98)(77,93,95)(80,96,92)(83,99,89)(84,88,86)(91,104,102)(94,107,112)(97,110,109)(100,113,106)(101,105,103)(108,118,116)(111,121,123)(114,124,120)(115,119,117)(122,129,127)(125,132,131)(126,130,128)(133,137,135)(134,138,136)(139,141,140);(1,2,3)(8,9,11,13,10,12,14)(15,16,18,22,29,39,52,68,85,102,116,127,135,140,137,121,93,57,31,17,20,25,32,42,55,71,88,65,81,98,112,123,131,136,129,107,73,41,21,27,35,45,58,74,91,105,82,49,62,78,95,109,120,128,118,90,54,28,37,48,61,77,94,108,119,99,63,36,46,59,75,92,106,117,104,70,38,50,64,80,97,111,122,130,113,79,47,26,33,43,56,72,89,103,87,51,66,83,100,114,125,133,138,124,96,60,34,19,23,30,40,53,69,86,67,84,101,115,126,134,139,141,132,110,76,44,24)

Conjugacy classes & character table

1794 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32004, 132);

Dic3×#14581888 #44544750