non-abelian, soluble, monomial
Aliases: C62.6D4, C32⋊2C8⋊3C4, C3⋊Dic3.2Q8, (C3×C6).6SD16, C32⋊3(C4.Q8), C22.10S3≀C2, C62.C22.2C2, C2.3(C32⋊2SD16), (C3×C6).4(C4⋊C4), C2.4(C3⋊S3.Q8), C3⋊Dic3.12(C2×C4), (C2×C32⋊2C8).8C2, (C2×C3⋊Dic3).4C22, SmallGroup(288,390)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.6D4
G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3, ab=ba, cac-1=a3b4, dad-1=a-1, cbc-1=a2b3, bd=db, dcd-1=c3 >
Subgroups: 264 in 62 conjugacy classes, 19 normal (11 characteristic)
C1, C2, C2 [×2], C3 [×2], C4 [×4], C22, C6 [×6], C8 [×2], C2×C4 [×3], C32, Dic3 [×6], C12 [×2], C2×C6 [×2], C4⋊C4 [×2], C2×C8, C3×C6, C3×C6 [×2], C2×Dic3 [×4], C2×C12 [×2], C4.Q8, C3×Dic3 [×2], C3⋊Dic3 [×2], C62, Dic3⋊C4 [×2], C32⋊2C8 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C62.C22 [×2], C2×C32⋊2C8, C62.6D4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4, Q8, C4⋊C4, SD16 [×2], C4.Q8, S3≀C2, C3⋊S3.Q8, C32⋊2SD16 [×2], C62.6D4
Character table of C62.6D4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 12 | 12 | 18 | 18 | 4 | 4 | 4 | 4 | 4 | 4 | 18 | 18 | 18 | 18 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | -i | i | i | i | i | -i | -i | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | 1 | -i | -i | i | i | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | -i | -i | i | i | -i | i | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | 1 | 1 | i | i | -i | -i | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | i | i | -i | -i | i | -i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | i | -i | -i | -i | -i | i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ11 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | -2 | -√-2 | √-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ12 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | -2 | √-2 | -√-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ13 | 2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | -2 | 2 | -√-2 | -√-2 | √-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ14 | 2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | -2 | 2 | √-2 | √-2 | -√-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ15 | 4 | 4 | 4 | 4 | 1 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | orthogonal lifted from S3≀C2 |
| ρ16 | 4 | 4 | 4 | 4 | -2 | 1 | 0 | 2 | 2 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | 0 | orthogonal lifted from S3≀C2 |
| ρ17 | 4 | 4 | 4 | 4 | -2 | 1 | 0 | -2 | -2 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | orthogonal lifted from S3≀C2 |
| ρ18 | 4 | 4 | 4 | 4 | 1 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | 0 | -1 | -1 | 0 | 0 | 0 | -1 | orthogonal lifted from S3≀C2 |
| ρ19 | 4 | -4 | 4 | -4 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | √3 | 0 | 0 | √3 | -√3 | -√3 | 0 | symplectic lifted from C32⋊2SD16, Schur index 2 |
| ρ20 | 4 | 4 | -4 | -4 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | √3 | 0 | √3 | -√3 | 0 | 0 | 0 | -√3 | symplectic lifted from C32⋊2SD16, Schur index 2 |
| ρ21 | 4 | 4 | -4 | -4 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -√3 | 0 | -√3 | √3 | 0 | 0 | 0 | √3 | symplectic lifted from C32⋊2SD16, Schur index 2 |
| ρ22 | 4 | -4 | 4 | -4 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -√3 | 0 | 0 | -√3 | √3 | √3 | 0 | symplectic lifted from C32⋊2SD16, Schur index 2 |
| ρ23 | 4 | -4 | 4 | -4 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | √-3 | 0 | -√-3 | √-3 | 0 | 0 | 0 | -√-3 | complex lifted from C32⋊2SD16 |
| ρ24 | 4 | 4 | -4 | -4 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -√-3 | 0 | 0 | √-3 | -√-3 | √-3 | 0 | complex lifted from C32⋊2SD16 |
| ρ25 | 4 | 4 | -4 | -4 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | √-3 | 0 | 0 | -√-3 | √-3 | -√-3 | 0 | complex lifted from C32⋊2SD16 |
| ρ26 | 4 | -4 | 4 | -4 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | -√-3 | 0 | √-3 | -√-3 | 0 | 0 | 0 | √-3 | complex lifted from C32⋊2SD16 |
| ρ27 | 4 | -4 | -4 | 4 | 1 | -2 | 2i | 0 | 0 | -2i | 0 | 0 | 2 | 2 | -2 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | i | 0 | -i | -i | 0 | 0 | 0 | i | complex lifted from C3⋊S3.Q8 |
| ρ28 | 4 | -4 | -4 | 4 | -2 | 1 | 0 | -2i | 2i | 0 | 0 | 0 | -1 | -1 | 1 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | i | 0 | 0 | -i | -i | i | 0 | complex lifted from C3⋊S3.Q8 |
| ρ29 | 4 | -4 | -4 | 4 | 1 | -2 | -2i | 0 | 0 | 2i | 0 | 0 | 2 | 2 | -2 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | -i | 0 | i | i | 0 | 0 | 0 | -i | complex lifted from C3⋊S3.Q8 |
| ρ30 | 4 | -4 | -4 | 4 | -2 | 1 | 0 | 2i | -2i | 0 | 0 | 0 | -1 | -1 | 1 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -i | 0 | 0 | i | i | -i | 0 | complex lifted from C3⋊S3.Q8 |
►On 96 points
Generators in S96
(1 79)(2 16 26 80 52 88)(3 73)(4 82 54 74 28 10)(5 75)(6 12 30 76 56 84)(7 77)(8 86 50 78 32 14)(9 53)(11 55)(13 49)(15 51)(17 89 59 46 35 71)(18 47)(19 65 37 48 61 91)(20 41)(21 93 63 42 39 67)(22 43)(23 69 33 44 57 95)(24 45)(25 87)(27 81)(29 83)(31 85)(34 96)(36 90)(38 92)(40 94)(58 70)(60 72)(62 66)(64 68)
(1 55 25 5 51 29)(2 6)(3 31 53 7 27 49)(4 8)(9 77 81 13 73 85)(10 14)(11 87 75 15 83 79)(12 16)(17 21)(18 64 36 22 60 40)(19 23)(20 34 62 24 38 58)(26 30)(28 32)(33 37)(35 39)(41 96 66 45 92 70)(42 46)(43 72 94 47 68 90)(44 48)(50 54)(52 56)(57 61)(59 63)(65 69)(67 71)(74 78)(76 80)(82 86)(84 88)(89 93)(91 95)
(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 47 79 18)(2 42 80 21)(3 45 73 24)(4 48 74 19)(5 43 75 22)(6 46 76 17)(7 41 77 20)(8 44 78 23)(9 58 53 70)(10 61 54 65)(11 64 55 68)(12 59 56 71)(13 62 49 66)(14 57 50 69)(15 60 51 72)(16 63 52 67)(25 90 87 36)(26 93 88 39)(27 96 81 34)(28 91 82 37)(29 94 83 40)(30 89 84 35)(31 92 85 38)(32 95 86 33)
G:=sub<Sym(96)| (1,79)(2,16,26,80,52,88)(3,73)(4,82,54,74,28,10)(5,75)(6,12,30,76,56,84)(7,77)(8,86,50,78,32,14)(9,53)(11,55)(13,49)(15,51)(17,89,59,46,35,71)(18,47)(19,65,37,48,61,91)(20,41)(21,93,63,42,39,67)(22,43)(23,69,33,44,57,95)(24,45)(25,87)(27,81)(29,83)(31,85)(34,96)(36,90)(38,92)(40,94)(58,70)(60,72)(62,66)(64,68), (1,55,25,5,51,29)(2,6)(3,31,53,7,27,49)(4,8)(9,77,81,13,73,85)(10,14)(11,87,75,15,83,79)(12,16)(17,21)(18,64,36,22,60,40)(19,23)(20,34,62,24,38,58)(26,30)(28,32)(33,37)(35,39)(41,96,66,45,92,70)(42,46)(43,72,94,47,68,90)(44,48)(50,54)(52,56)(57,61)(59,63)(65,69)(67,71)(74,78)(76,80)(82,86)(84,88)(89,93)(91,95), (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), (1,47,79,18)(2,42,80,21)(3,45,73,24)(4,48,74,19)(5,43,75,22)(6,46,76,17)(7,41,77,20)(8,44,78,23)(9,58,53,70)(10,61,54,65)(11,64,55,68)(12,59,56,71)(13,62,49,66)(14,57,50,69)(15,60,51,72)(16,63,52,67)(25,90,87,36)(26,93,88,39)(27,96,81,34)(28,91,82,37)(29,94,83,40)(30,89,84,35)(31,92,85,38)(32,95,86,33)>;
G:=Group( (1,79)(2,16,26,80,52,88)(3,73)(4,82,54,74,28,10)(5,75)(6,12,30,76,56,84)(7,77)(8,86,50,78,32,14)(9,53)(11,55)(13,49)(15,51)(17,89,59,46,35,71)(18,47)(19,65,37,48,61,91)(20,41)(21,93,63,42,39,67)(22,43)(23,69,33,44,57,95)(24,45)(25,87)(27,81)(29,83)(31,85)(34,96)(36,90)(38,92)(40,94)(58,70)(60,72)(62,66)(64,68), (1,55,25,5,51,29)(2,6)(3,31,53,7,27,49)(4,8)(9,77,81,13,73,85)(10,14)(11,87,75,15,83,79)(12,16)(17,21)(18,64,36,22,60,40)(19,23)(20,34,62,24,38,58)(26,30)(28,32)(33,37)(35,39)(41,96,66,45,92,70)(42,46)(43,72,94,47,68,90)(44,48)(50,54)(52,56)(57,61)(59,63)(65,69)(67,71)(74,78)(76,80)(82,86)(84,88)(89,93)(91,95), (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), (1,47,79,18)(2,42,80,21)(3,45,73,24)(4,48,74,19)(5,43,75,22)(6,46,76,17)(7,41,77,20)(8,44,78,23)(9,58,53,70)(10,61,54,65)(11,64,55,68)(12,59,56,71)(13,62,49,66)(14,57,50,69)(15,60,51,72)(16,63,52,67)(25,90,87,36)(26,93,88,39)(27,96,81,34)(28,91,82,37)(29,94,83,40)(30,89,84,35)(31,92,85,38)(32,95,86,33) );
G=PermutationGroup([(1,79),(2,16,26,80,52,88),(3,73),(4,82,54,74,28,10),(5,75),(6,12,30,76,56,84),(7,77),(8,86,50,78,32,14),(9,53),(11,55),(13,49),(15,51),(17,89,59,46,35,71),(18,47),(19,65,37,48,61,91),(20,41),(21,93,63,42,39,67),(22,43),(23,69,33,44,57,95),(24,45),(25,87),(27,81),(29,83),(31,85),(34,96),(36,90),(38,92),(40,94),(58,70),(60,72),(62,66),(64,68)], [(1,55,25,5,51,29),(2,6),(3,31,53,7,27,49),(4,8),(9,77,81,13,73,85),(10,14),(11,87,75,15,83,79),(12,16),(17,21),(18,64,36,22,60,40),(19,23),(20,34,62,24,38,58),(26,30),(28,32),(33,37),(35,39),(41,96,66,45,92,70),(42,46),(43,72,94,47,68,90),(44,48),(50,54),(52,56),(57,61),(59,63),(65,69),(67,71),(74,78),(76,80),(82,86),(84,88),(89,93),(91,95)], [(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)], [(1,47,79,18),(2,42,80,21),(3,45,73,24),(4,48,74,19),(5,43,75,22),(6,46,76,17),(7,41,77,20),(8,44,78,23),(9,58,53,70),(10,61,54,65),(11,64,55,68),(12,59,56,71),(13,62,49,66),(14,57,50,69),(15,60,51,72),(16,63,52,67),(25,90,87,36),(26,93,88,39),(27,96,81,34),(28,91,82,37),(29,94,83,40),(30,89,84,35),(31,92,85,38),(32,95,86,33)])
Matrix representation of C62.6D4 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 45 | 18 | 1 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 9 | 27 | 1 | 0 |
| 0 | 0 | 9 | 27 | 0 | 1 |
| 19 | 60 | 0 | 0 | 0 | 0 |
| 34 | 42 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 25 | 46 | 19 |
| 0 | 0 | 26 | 25 | 19 | 46 |
| 0 | 0 | 47 | 68 | 72 | 23 |
| 0 | 0 | 20 | 68 | 72 | 23 |
| 24 | 16 | 0 | 0 | 0 | 0 |
| 5 | 49 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 40 | 16 | 27 | 27 |
| 0 | 0 | 66 | 41 | 0 | 46 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,45,0,0,0,0,72,18,0,0,0,0,0,1,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,9,9,0,0,72,72,27,27,0,0,0,0,1,0,0,0,0,0,0,1],[19,34,0,0,0,0,60,42,0,0,0,0,0,0,26,26,47,20,0,0,25,25,68,68,0,0,46,19,72,72,0,0,19,46,23,23],[24,5,0,0,0,0,16,49,0,0,0,0,0,0,27,0,40,66,0,0,0,27,16,41,0,0,0,0,27,0,0,0,0,0,27,46] >;
C62.6D4 in GAP, Magma, Sage, TeX
C_6^2._6D_4
% in TeX
G:=Group("C6^2.6D4"); // GroupNames label
G:=SmallGroup(288,390);
// by ID
G=gap.SmallGroup(288,390);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,85,176,422,219,100,2693,2028,691,797,2372]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^4=b^3,d^2=a^3,a*b=b*a,c*a*c^-1=a^3*b^4,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations
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