C₂×D₃×#14581887 #44544878

order	32004
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₃×#14581887
unambiguous semidirect product	C₃⋊(C₂²×#14581887)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×#14581887)

Splittings

Fitting, socle	C₂×C₃×C₇×C₁₂₇ ⋊ C₂×C₃
derived	C₃×C₇×C₁₂₇ ⋊ C₂²×C₃
Hall	C₃×#14581887 ⋊ C₂²
Hall	C₇×C₁₂₇ ⋊ C₂×C₃×D₃
Sylow	C₁₂₇ ⋊ C₂×D₃×ΓL(1,𝔽₈)
Sylow	C₇ ⋊ C₂×D₃×C₁₂₇⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃×#14581887
commutator, cosocle, hypocenter	C₃×C₇×C₁₂₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇×C₁₂₇	C₂×C₃
LCS	G ▷ C₃×C₇×C₁₂₇
UCS	C₁ ◁ C₂

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 139
transitive degree	μ_t(G) = 5334

A minimal faithful permutation representation — 2 generators on 139 points:

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

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, 260);

C2×D3×#14581887 #44544878