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, C3, C4, C22, C6, C8, C2×C4, C32, Dic3, C12, C2×C6, C4⋊C4, C2×C8, C3×C6, C3×C6, C2×Dic3, C2×C12, C4.Q8, C3×Dic3, C3⋊Dic3, C62, Dic3⋊C4, C32⋊2C8, C6×Dic3, C2×C3⋊Dic3, C62.C22, C2×C32⋊2C8, C62.6D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C4⋊C4, SD16, C4.Q8, S3≀C2, C3⋊S3.Q8, C32⋊2SD16, 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 55)(2 16 26 56 78 88)(3 49)(4 82 80 50 28 10)(5 51)(6 12 30 52 74 84)(7 53)(8 86 76 54 32 14)(9 79)(11 73)(13 75)(15 77)(17 89 44 61 35 71)(18 62)(19 65 37 63 46 91)(20 64)(21 93 48 57 39 67)(22 58)(23 69 33 59 42 95)(24 60)(25 87)(27 81)(29 83)(31 85)(34 96)(36 90)(38 92)(40 94)(41 68)(43 70)(45 72)(47 66)
(1 73 25 5 77 29)(2 6)(3 31 79 7 27 75)(4 8)(9 53 81 13 49 85)(10 14)(11 87 51 15 83 55)(12 16)(17 21)(18 41 36 22 45 40)(19 23)(20 34 47 24 38 43)(26 30)(28 32)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(57 61)(58 72 94 62 68 90)(59 63)(60 92 70 64 96 66)(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 62 55 18)(2 57 56 21)(3 60 49 24)(4 63 50 19)(5 58 51 22)(6 61 52 17)(7 64 53 20)(8 59 54 23)(9 43 79 70)(10 46 80 65)(11 41 73 68)(12 44 74 71)(13 47 75 66)(14 42 76 69)(15 45 77 72)(16 48 78 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)| (1,55)(2,16,26,56,78,88)(3,49)(4,82,80,50,28,10)(5,51)(6,12,30,52,74,84)(7,53)(8,86,76,54,32,14)(9,79)(11,73)(13,75)(15,77)(17,89,44,61,35,71)(18,62)(19,65,37,63,46,91)(20,64)(21,93,48,57,39,67)(22,58)(23,69,33,59,42,95)(24,60)(25,87)(27,81)(29,83)(31,85)(34,96)(36,90)(38,92)(40,94)(41,68)(43,70)(45,72)(47,66), (1,73,25,5,77,29)(2,6)(3,31,79,7,27,75)(4,8)(9,53,81,13,49,85)(10,14)(11,87,51,15,83,55)(12,16)(17,21)(18,41,36,22,45,40)(19,23)(20,34,47,24,38,43)(26,30)(28,32)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(57,61)(58,72,94,62,68,90)(59,63)(60,92,70,64,96,66)(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), (1,62,55,18)(2,57,56,21)(3,60,49,24)(4,63,50,19)(5,58,51,22)(6,61,52,17)(7,64,53,20)(8,59,54,23)(9,43,79,70)(10,46,80,65)(11,41,73,68)(12,44,74,71)(13,47,75,66)(14,42,76,69)(15,45,77,72)(16,48,78,67)(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( (1,55)(2,16,26,56,78,88)(3,49)(4,82,80,50,28,10)(5,51)(6,12,30,52,74,84)(7,53)(8,86,76,54,32,14)(9,79)(11,73)(13,75)(15,77)(17,89,44,61,35,71)(18,62)(19,65,37,63,46,91)(20,64)(21,93,48,57,39,67)(22,58)(23,69,33,59,42,95)(24,60)(25,87)(27,81)(29,83)(31,85)(34,96)(36,90)(38,92)(40,94)(41,68)(43,70)(45,72)(47,66), (1,73,25,5,77,29)(2,6)(3,31,79,7,27,75)(4,8)(9,53,81,13,49,85)(10,14)(11,87,51,15,83,55)(12,16)(17,21)(18,41,36,22,45,40)(19,23)(20,34,47,24,38,43)(26,30)(28,32)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(57,61)(58,72,94,62,68,90)(59,63)(60,92,70,64,96,66)(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), (1,62,55,18)(2,57,56,21)(3,60,49,24)(4,63,50,19)(5,58,51,22)(6,61,52,17)(7,64,53,20)(8,59,54,23)(9,43,79,70)(10,46,80,65)(11,41,73,68)(12,44,74,71)(13,47,75,66)(14,42,76,69)(15,45,77,72)(16,48,78,67)(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([[(1,55),(2,16,26,56,78,88),(3,49),(4,82,80,50,28,10),(5,51),(6,12,30,52,74,84),(7,53),(8,86,76,54,32,14),(9,79),(11,73),(13,75),(15,77),(17,89,44,61,35,71),(18,62),(19,65,37,63,46,91),(20,64),(21,93,48,57,39,67),(22,58),(23,69,33,59,42,95),(24,60),(25,87),(27,81),(29,83),(31,85),(34,96),(36,90),(38,92),(40,94),(41,68),(43,70),(45,72),(47,66)], [(1,73,25,5,77,29),(2,6),(3,31,79,7,27,75),(4,8),(9,53,81,13,49,85),(10,14),(11,87,51,15,83,55),(12,16),(17,21),(18,41,36,22,45,40),(19,23),(20,34,47,24,38,43),(26,30),(28,32),(33,37),(35,39),(42,46),(44,48),(50,54),(52,56),(57,61),(58,72,94,62,68,90),(59,63),(60,92,70,64,96,66),(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)], [(1,62,55,18),(2,57,56,21),(3,60,49,24),(4,63,50,19),(5,58,51,22),(6,61,52,17),(7,64,53,20),(8,59,54,23),(9,43,79,70),(10,46,80,65),(11,41,73,68),(12,44,74,71),(13,47,75,66),(14,42,76,69),(15,45,77,72),(16,48,78,67),(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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