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G = C6×CSU2(𝔽3)  order 288 = 25·32

Direct product of C6 and CSU2(𝔽3)

direct product, non-abelian, soluble

Aliases: C6×CSU2(𝔽3), C2.5(C6×S4), C6.42(C2×S4), (C2×C6).19S4, Q8.1(S3×C6), (C6×Q8).8S3, C22.4(C3×S4), (C3×Q8).19D6, (C6×SL2(𝔽3)).5C2, SL2(𝔽3).1(C2×C6), (C2×SL2(𝔽3)).2C6, (C3×SL2(𝔽3)).13C22, (C2×Q8).2(C3×S3), SmallGroup(288,899)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — C6×CSU2(𝔽3)
C1C2Q8SL2(𝔽3)C3×SL2(𝔽3)C3×CSU2(𝔽3) — C6×CSU2(𝔽3)
SL2(𝔽3) — C6×CSU2(𝔽3)
C1C2×C6

Generators and relations for C6×CSU2(𝔽3)
 G = < a,b,c,d,e | a6=b4=d3=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=bc, ebe-1=b2c, dcd-1=b, ede-1=d-1 >

Subgroups: 270 in 85 conjugacy classes, 24 normal (16 characteristic)
C1, C2, C2 [×2], C3, C3 [×2], C4 [×4], C22, C6, C6 [×2], C6 [×6], C8 [×2], C2×C4 [×2], Q8, Q8 [×4], C32, Dic3 [×2], C12 [×4], C2×C6, C2×C6 [×2], C2×C8, Q16 [×4], C2×Q8, C2×Q8, C3×C6 [×3], C24 [×2], SL2(𝔽3), SL2(𝔽3), C2×Dic3, C2×C12 [×2], C3×Q8, C3×Q8 [×4], C2×Q16, C3×Dic3 [×2], C62, C2×C24, C3×Q16 [×4], CSU2(𝔽3) [×2], C2×SL2(𝔽3), C2×SL2(𝔽3), C6×Q8, C6×Q8, C3×SL2(𝔽3), C6×Dic3, C6×Q16, C2×CSU2(𝔽3), C3×CSU2(𝔽3) [×2], C6×SL2(𝔽3), C6×CSU2(𝔽3)
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D6, C2×C6, C3×S3, S4, S3×C6, CSU2(𝔽3) [×2], C2×S4, C3×S4, C2×CSU2(𝔽3), C3×CSU2(𝔽3) [×2], C6×S4, C6×CSU2(𝔽3)

Smallest permutation representation of C6×CSU2(𝔽3)
On 96 points
Generators in S96
(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)
(1 70 44 87)(2 71 45 88)(3 72 46 89)(4 67 47 90)(5 68 48 85)(6 69 43 86)(7 34 91 77)(8 35 92 78)(9 36 93 73)(10 31 94 74)(11 32 95 75)(12 33 96 76)(13 60 40 26)(14 55 41 27)(15 56 42 28)(16 57 37 29)(17 58 38 30)(18 59 39 25)(19 84 61 52)(20 79 62 53)(21 80 63 54)(22 81 64 49)(23 82 65 50)(24 83 66 51)
(1 24 44 66)(2 19 45 61)(3 20 46 62)(4 21 47 63)(5 22 48 64)(6 23 43 65)(7 13 91 40)(8 14 92 41)(9 15 93 42)(10 16 94 37)(11 17 95 38)(12 18 96 39)(25 76 59 33)(26 77 60 34)(27 78 55 35)(28 73 56 36)(29 74 57 31)(30 75 58 32)(49 85 81 68)(50 86 82 69)(51 87 83 70)(52 88 84 71)(53 89 79 72)(54 90 80 67)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 28 32)(14 29 33)(15 30 34)(16 25 35)(17 26 36)(18 27 31)(19 50 67)(20 51 68)(21 52 69)(22 53 70)(23 54 71)(24 49 72)(37 59 78)(38 60 73)(39 55 74)(40 56 75)(41 57 76)(42 58 77)(43 47 45)(44 48 46)(61 82 90)(62 83 85)(63 84 86)(64 79 87)(65 80 88)(66 81 89)(91 93 95)(92 94 96)
(1 92 44 8)(2 93 45 9)(3 94 46 10)(4 95 47 11)(5 96 48 12)(6 91 43 7)(13 86 40 69)(14 87 41 70)(15 88 42 71)(16 89 37 72)(17 90 38 67)(18 85 39 68)(19 36 61 73)(20 31 62 74)(21 32 63 75)(22 33 64 76)(23 34 65 77)(24 35 66 78)(25 81 59 49)(26 82 60 50)(27 83 55 51)(28 84 56 52)(29 79 57 53)(30 80 58 54)

G:=sub<Sym(96)| (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), (1,70,44,87)(2,71,45,88)(3,72,46,89)(4,67,47,90)(5,68,48,85)(6,69,43,86)(7,34,91,77)(8,35,92,78)(9,36,93,73)(10,31,94,74)(11,32,95,75)(12,33,96,76)(13,60,40,26)(14,55,41,27)(15,56,42,28)(16,57,37,29)(17,58,38,30)(18,59,39,25)(19,84,61,52)(20,79,62,53)(21,80,63,54)(22,81,64,49)(23,82,65,50)(24,83,66,51), (1,24,44,66)(2,19,45,61)(3,20,46,62)(4,21,47,63)(5,22,48,64)(6,23,43,65)(7,13,91,40)(8,14,92,41)(9,15,93,42)(10,16,94,37)(11,17,95,38)(12,18,96,39)(25,76,59,33)(26,77,60,34)(27,78,55,35)(28,73,56,36)(29,74,57,31)(30,75,58,32)(49,85,81,68)(50,86,82,69)(51,87,83,70)(52,88,84,71)(53,89,79,72)(54,90,80,67), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,28,32)(14,29,33)(15,30,34)(16,25,35)(17,26,36)(18,27,31)(19,50,67)(20,51,68)(21,52,69)(22,53,70)(23,54,71)(24,49,72)(37,59,78)(38,60,73)(39,55,74)(40,56,75)(41,57,76)(42,58,77)(43,47,45)(44,48,46)(61,82,90)(62,83,85)(63,84,86)(64,79,87)(65,80,88)(66,81,89)(91,93,95)(92,94,96), (1,92,44,8)(2,93,45,9)(3,94,46,10)(4,95,47,11)(5,96,48,12)(6,91,43,7)(13,86,40,69)(14,87,41,70)(15,88,42,71)(16,89,37,72)(17,90,38,67)(18,85,39,68)(19,36,61,73)(20,31,62,74)(21,32,63,75)(22,33,64,76)(23,34,65,77)(24,35,66,78)(25,81,59,49)(26,82,60,50)(27,83,55,51)(28,84,56,52)(29,79,57,53)(30,80,58,54)>;

G:=Group( (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), (1,70,44,87)(2,71,45,88)(3,72,46,89)(4,67,47,90)(5,68,48,85)(6,69,43,86)(7,34,91,77)(8,35,92,78)(9,36,93,73)(10,31,94,74)(11,32,95,75)(12,33,96,76)(13,60,40,26)(14,55,41,27)(15,56,42,28)(16,57,37,29)(17,58,38,30)(18,59,39,25)(19,84,61,52)(20,79,62,53)(21,80,63,54)(22,81,64,49)(23,82,65,50)(24,83,66,51), (1,24,44,66)(2,19,45,61)(3,20,46,62)(4,21,47,63)(5,22,48,64)(6,23,43,65)(7,13,91,40)(8,14,92,41)(9,15,93,42)(10,16,94,37)(11,17,95,38)(12,18,96,39)(25,76,59,33)(26,77,60,34)(27,78,55,35)(28,73,56,36)(29,74,57,31)(30,75,58,32)(49,85,81,68)(50,86,82,69)(51,87,83,70)(52,88,84,71)(53,89,79,72)(54,90,80,67), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,28,32)(14,29,33)(15,30,34)(16,25,35)(17,26,36)(18,27,31)(19,50,67)(20,51,68)(21,52,69)(22,53,70)(23,54,71)(24,49,72)(37,59,78)(38,60,73)(39,55,74)(40,56,75)(41,57,76)(42,58,77)(43,47,45)(44,48,46)(61,82,90)(62,83,85)(63,84,86)(64,79,87)(65,80,88)(66,81,89)(91,93,95)(92,94,96), (1,92,44,8)(2,93,45,9)(3,94,46,10)(4,95,47,11)(5,96,48,12)(6,91,43,7)(13,86,40,69)(14,87,41,70)(15,88,42,71)(16,89,37,72)(17,90,38,67)(18,85,39,68)(19,36,61,73)(20,31,62,74)(21,32,63,75)(22,33,64,76)(23,34,65,77)(24,35,66,78)(25,81,59,49)(26,82,60,50)(27,83,55,51)(28,84,56,52)(29,79,57,53)(30,80,58,54) );

G=PermutationGroup([(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)], [(1,70,44,87),(2,71,45,88),(3,72,46,89),(4,67,47,90),(5,68,48,85),(6,69,43,86),(7,34,91,77),(8,35,92,78),(9,36,93,73),(10,31,94,74),(11,32,95,75),(12,33,96,76),(13,60,40,26),(14,55,41,27),(15,56,42,28),(16,57,37,29),(17,58,38,30),(18,59,39,25),(19,84,61,52),(20,79,62,53),(21,80,63,54),(22,81,64,49),(23,82,65,50),(24,83,66,51)], [(1,24,44,66),(2,19,45,61),(3,20,46,62),(4,21,47,63),(5,22,48,64),(6,23,43,65),(7,13,91,40),(8,14,92,41),(9,15,93,42),(10,16,94,37),(11,17,95,38),(12,18,96,39),(25,76,59,33),(26,77,60,34),(27,78,55,35),(28,73,56,36),(29,74,57,31),(30,75,58,32),(49,85,81,68),(50,86,82,69),(51,87,83,70),(52,88,84,71),(53,89,79,72),(54,90,80,67)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,28,32),(14,29,33),(15,30,34),(16,25,35),(17,26,36),(18,27,31),(19,50,67),(20,51,68),(21,52,69),(22,53,70),(23,54,71),(24,49,72),(37,59,78),(38,60,73),(39,55,74),(40,56,75),(41,57,76),(42,58,77),(43,47,45),(44,48,46),(61,82,90),(62,83,85),(63,84,86),(64,79,87),(65,80,88),(66,81,89),(91,93,95),(92,94,96)], [(1,92,44,8),(2,93,45,9),(3,94,46,10),(4,95,47,11),(5,96,48,12),(6,91,43,7),(13,86,40,69),(14,87,41,70),(15,88,42,71),(16,89,37,72),(17,90,38,67),(18,85,39,68),(19,36,61,73),(20,31,62,74),(21,32,63,75),(22,33,64,76),(23,34,65,77),(24,35,66,78),(25,81,59,49),(26,82,60,50),(27,83,55,51),(28,84,56,52),(29,79,57,53),(30,80,58,54)])

48 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D6A···6F6G···6O8A8B8C8D12A12B12C12D12E12F12G12H24A···24H
order12223333344446···66···68888121212121212121224···24
size1111118886612121···18···866666666121212126···6

48 irreducible representations

dim111111222222333344
type+++++-++-
imageC1C2C2C3C6C6S3D6C3×S3S3×C6CSU2(𝔽3)C3×CSU2(𝔽3)S4C2×S4C3×S4C6×S4CSU2(𝔽3)C3×CSU2(𝔽3)
kernelC6×CSU2(𝔽3)C3×CSU2(𝔽3)C6×SL2(𝔽3)C2×CSU2(𝔽3)CSU2(𝔽3)C2×SL2(𝔽3)C6×Q8C3×Q8C2×Q8Q8C6C2C2×C6C6C22C2C6C2
# reps121242112248224424

Matrix representation of C6×CSU2(𝔽3) in GL3(𝔽73) generated by

900
0650
0065
,
100
0965
06564
,
100
0072
010
,
100
080
0164
,
7200
0239
07171
G:=sub<GL(3,GF(73))| [9,0,0,0,65,0,0,0,65],[1,0,0,0,9,65,0,65,64],[1,0,0,0,0,1,0,72,0],[1,0,0,0,8,1,0,0,64],[72,0,0,0,2,71,0,39,71] >;

C6×CSU2(𝔽3) in GAP, Magma, Sage, TeX

C_6\times {\rm CSU}_2({\mathbb F}_3)
% in TeX

G:=Group("C6xCSU(2,3)");
// GroupNames label

G:=SmallGroup(288,899);
// by ID

G=gap.SmallGroup(288,899);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,675,2524,655,172,1517,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^4=d^3=1,c^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*c*e^-1=b^-1,d*b*d^-1=b*c,e*b*e^-1=b^2*c,d*c*d^-1=b,e*d*e^-1=d^-1>;
// generators/relations

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