C₂×C₃⋊#19128633 #44544870

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

Aliases

direct product	C₂×C₃⋊#19128633
unambiguous semidirect product	(C₂×C₃)⋊#19128633

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×C₃⋊AGL(1,𝔽₇)
Sylow	C₇ ⋊ C₂×C₃⋊(D₁₂₇⋊C₃)

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊#19128633
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)(14,18,19,15,16,20)(17,26,27,21,22,36)(23,39,40,29,30,53)(24,38,34,32,28,25)(31,58,59,42,43,76)(33,56,57,45,41,50)(35,37,55,47,48,51)(44,81,82,61,62,105)(46,79,80,64,60,73)(49,54,78,66,67,74)(52,94,92,70,71,69)(63,110,86,84,85,112)(65,108,109,87,83,102)(68,77,107,89,90,103)(72,120,121,95,96,98)(75,125,118,99,100,93)(88,133,117,114,111,131)(91,106,113,115,116,132)(97,137,136,122,123,129)(101,139,138,126,127,124)(104,134,135,128,130,119);(1,2)(3,4,5)(6,7,9,12,11,10,8)(13,14,17,23,31,44,63,86,113,134,139,137,133,110,114,87,64,45,32,16,20,25,35,49,68,91,117,82,107,125,120,108,81,106,122,95,70,48,22,28,36,50,69,93,119,136,109,59,78,94,79,58,77,104,126,99,67,30,41,51,53,73,98,124,138,121,80,40,55,56,39,54,75,101,128,90,43,60,71,74,76,102,129,135,118,92,57,27,38,26,37,52,72,97,116,62,83,96,100,103,105,131,115,89,66,47,34,19,18,24,33,46,65,88,85,111,123,127,130,132,112,84,61,42,29,21,15)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32004, 252);

C2×C3⋊#19128633 #44544870