C₂×C₃⋊#19128628 #44544871

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

Aliases

direct product	C₂×C₃⋊#19128628
unambiguous semidirect product	(C₂×C₃)⋊#19128628

Splittings

Fitting, socle	C₂×C₃×C₇×C₁₂₇ ⋊ C₂×C₃
derived	C₃×C₇×C₁₂₇ ⋊ C₂²×C₃
Hall	C₃×#14581888 ⋊ C₂²
Hall	C₇×C₁₂₇ ⋊ C₂×C₃×D₃
Sylow	C₁₂₇ ⋊ C₂×C₃⋊AGL(1,𝔽₇)
Sylow	C₇ ⋊ C₂×D₃×C₁₂₇⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊#19128628
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)(7,10,11,8,9,12)(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)(50,65,81)(53,67,78)(56,70,75)(59,73,72)(62,76,69)(64,79,66)(68,84,97)(71,86,94)(74,89,91)(77,92,88)(80,95,85)(82,98,83)(87,101,111)(90,103,108)(93,106,105)(96,109,102)(99,112,100)(104,115,122)(107,117,119)(110,120,116)(113,123,114)(118,126,130)(121,128,127)(124,131,125)(129,134,135)(132,136,133)(137,139,138);(1,2)(3,4,5)(6,7,9,12,11,10,8)(13,14,16,20,27,37,50,65,82,99,113,124,132,137,136,120,92,58,32,17,21,28,38,51,66,83,84,49,64,80,96,110,121,129,131,109,76,45,24,31,41,54,69,85,100,101,67,36,48,62,77,93,107,118,123,95,61,34,44,57,72,88,102,114,115,86,52,26,35,46,59,74,90,104,112,79,47,60,75,91,105,116,125,126,103,70,39,19,25,33,43,56,71,87,98,63,78,94,108,119,127,133,134,117,89,55,29,15,18,23,30,40,53,68,81,97,111,122,130,135,138,139,128,106,73,42,22)

Conjugacy classes & character table

1032 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32004, 253);

C2×C3⋊#19128628 #44544871