non-abelian, soluble, monomial
Aliases: C62.4D4, (C3×C6).1Q16, C22.9S3≀C2, C32⋊2Q8⋊4C4, C3⋊Dic3.5D4, (C3×C6).5SD16, C2.1(C32⋊Q16), C32⋊3(Q8⋊C4), C62.C22.1C2, C2.2(C32⋊2SD16), C2.10(S32⋊C4), C3⋊Dic3.10(C2×C4), (C2×C32⋊2C8).1C2, (C2×C32⋊2Q8).1C2, (C3×C6).10(C22⋊C4), (C2×C3⋊Dic3).2C22, SmallGroup(288,388)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.4D4
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=a3c3 >
Subgroups: 328 in 74 conjugacy classes, 19 normal (17 characteristic)
C1, C2, C3, C4, C22, C6, C8, C2×C4, Q8, C32, Dic3, C12, C2×C6, C4⋊C4, C2×C8, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×C12, Q8⋊C4, C3×Dic3, C3⋊Dic3, C62, Dic3⋊C4, C2×Dic6, C32⋊2C8, C32⋊2Q8, C32⋊2Q8, C6×Dic3, C2×C3⋊Dic3, C62.C22, C2×C32⋊2C8, C2×C32⋊2Q8, C62.4D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, SD16, Q16, Q8⋊C4, S3≀C2, S32⋊C4, C32⋊2SD16, C32⋊Q16, C62.4D4
Character table of C62.4D4
| 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 | -1 | 1 | i | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | i | -i | i | -i | i | -1 | -i | -i | 1 | 1 | -1 | i | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | 1 | -i | 1 | -1 | i | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -i | i | -i | i | i | 1 | -i | -i | -1 | -1 | 1 | i | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | 1 | 1 | i | 1 | -1 | -i | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | i | -i | i | -i | -i | 1 | i | i | -1 | -1 | 1 | -i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | 1 | i | -1 | 1 | -i | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -i | i | -i | i | -i | -1 | i | i | 1 | 1 | -1 | -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 | orthogonal lifted from D4 |
| 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 | symplectic lifted from Q16, Schur index 2 |
| 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 | symplectic lifted from Q16, Schur index 2 |
| 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 | -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 S32⋊C4 |
| ρ19 | 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 |
| ρ20 | 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 S32⋊C4 |
| ρ21 | 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⋊Q16, 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 | symplectic lifted from C32⋊Q16, Schur index 2 |
| ρ24 | 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⋊Q16, Schur index 2 |
| 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 | symplectic lifted from C32⋊Q16, Schur index 2 |
| ρ26 | 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 |
| 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 S32⋊C4 |
| ρ28 | 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 S32⋊C4 |
| ρ29 | 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 |
| ρ30 | 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 |
►On 96 points
Generators in S96
(1 75)(2 24 94 76 35 60)(3 77)(4 62 37 78 96 18)(5 79)(6 20 90 80 39 64)(7 73)(8 58 33 74 92 22)(9 81 71 48 55 29)(10 41)(11 31 49 42 65 83)(12 43)(13 85 67 44 51 25)(14 45)(15 27 53 46 69 87)(16 47)(17 36)(19 38)(21 40)(23 34)(26 68)(28 70)(30 72)(32 66)(50 84)(52 86)(54 88)(56 82)(57 91)(59 93)(61 95)(63 89)
(1 38 93 5 34 89)(2 6)(3 91 36 7 95 40)(4 8)(9 13)(10 68 56 14 72 52)(11 15)(12 54 66 16 50 70)(17 73 61 21 77 57)(18 22)(19 59 79 23 63 75)(20 24)(25 29)(26 82 45 30 86 41)(27 31)(28 43 88 32 47 84)(33 37)(35 39)(42 46)(44 48)(49 53)(51 55)(58 62)(60 64)(65 69)(67 71)(74 78)(76 80)(81 85)(83 87)(90 94)(92 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 10 75 41)(2 44 76 13)(3 16 77 47)(4 42 78 11)(5 14 79 45)(6 48 80 9)(7 12 73 43)(8 46 74 15)(17 28 36 70)(18 65 37 31)(19 26 38 68)(20 71 39 29)(21 32 40 66)(22 69 33 27)(23 30 34 72)(24 67 35 25)(49 96 83 62)(50 57 84 91)(51 94 85 60)(52 63 86 89)(53 92 87 58)(54 61 88 95)(55 90 81 64)(56 59 82 93)
G:=sub<Sym(96)| (1,75)(2,24,94,76,35,60)(3,77)(4,62,37,78,96,18)(5,79)(6,20,90,80,39,64)(7,73)(8,58,33,74,92,22)(9,81,71,48,55,29)(10,41)(11,31,49,42,65,83)(12,43)(13,85,67,44,51,25)(14,45)(15,27,53,46,69,87)(16,47)(17,36)(19,38)(21,40)(23,34)(26,68)(28,70)(30,72)(32,66)(50,84)(52,86)(54,88)(56,82)(57,91)(59,93)(61,95)(63,89), (1,38,93,5,34,89)(2,6)(3,91,36,7,95,40)(4,8)(9,13)(10,68,56,14,72,52)(11,15)(12,54,66,16,50,70)(17,73,61,21,77,57)(18,22)(19,59,79,23,63,75)(20,24)(25,29)(26,82,45,30,86,41)(27,31)(28,43,88,32,47,84)(33,37)(35,39)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64)(65,69)(67,71)(74,78)(76,80)(81,85)(83,87)(90,94)(92,96), (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), (1,10,75,41)(2,44,76,13)(3,16,77,47)(4,42,78,11)(5,14,79,45)(6,48,80,9)(7,12,73,43)(8,46,74,15)(17,28,36,70)(18,65,37,31)(19,26,38,68)(20,71,39,29)(21,32,40,66)(22,69,33,27)(23,30,34,72)(24,67,35,25)(49,96,83,62)(50,57,84,91)(51,94,85,60)(52,63,86,89)(53,92,87,58)(54,61,88,95)(55,90,81,64)(56,59,82,93)>;
G:=Group( (1,75)(2,24,94,76,35,60)(3,77)(4,62,37,78,96,18)(5,79)(6,20,90,80,39,64)(7,73)(8,58,33,74,92,22)(9,81,71,48,55,29)(10,41)(11,31,49,42,65,83)(12,43)(13,85,67,44,51,25)(14,45)(15,27,53,46,69,87)(16,47)(17,36)(19,38)(21,40)(23,34)(26,68)(28,70)(30,72)(32,66)(50,84)(52,86)(54,88)(56,82)(57,91)(59,93)(61,95)(63,89), (1,38,93,5,34,89)(2,6)(3,91,36,7,95,40)(4,8)(9,13)(10,68,56,14,72,52)(11,15)(12,54,66,16,50,70)(17,73,61,21,77,57)(18,22)(19,59,79,23,63,75)(20,24)(25,29)(26,82,45,30,86,41)(27,31)(28,43,88,32,47,84)(33,37)(35,39)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64)(65,69)(67,71)(74,78)(76,80)(81,85)(83,87)(90,94)(92,96), (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), (1,10,75,41)(2,44,76,13)(3,16,77,47)(4,42,78,11)(5,14,79,45)(6,48,80,9)(7,12,73,43)(8,46,74,15)(17,28,36,70)(18,65,37,31)(19,26,38,68)(20,71,39,29)(21,32,40,66)(22,69,33,27)(23,30,34,72)(24,67,35,25)(49,96,83,62)(50,57,84,91)(51,94,85,60)(52,63,86,89)(53,92,87,58)(54,61,88,95)(55,90,81,64)(56,59,82,93) );
G=PermutationGroup([[(1,75),(2,24,94,76,35,60),(3,77),(4,62,37,78,96,18),(5,79),(6,20,90,80,39,64),(7,73),(8,58,33,74,92,22),(9,81,71,48,55,29),(10,41),(11,31,49,42,65,83),(12,43),(13,85,67,44,51,25),(14,45),(15,27,53,46,69,87),(16,47),(17,36),(19,38),(21,40),(23,34),(26,68),(28,70),(30,72),(32,66),(50,84),(52,86),(54,88),(56,82),(57,91),(59,93),(61,95),(63,89)], [(1,38,93,5,34,89),(2,6),(3,91,36,7,95,40),(4,8),(9,13),(10,68,56,14,72,52),(11,15),(12,54,66,16,50,70),(17,73,61,21,77,57),(18,22),(19,59,79,23,63,75),(20,24),(25,29),(26,82,45,30,86,41),(27,31),(28,43,88,32,47,84),(33,37),(35,39),(42,46),(44,48),(49,53),(51,55),(58,62),(60,64),(65,69),(67,71),(74,78),(76,80),(81,85),(83,87),(90,94),(92,96)], [(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)], [(1,10,75,41),(2,44,76,13),(3,16,77,47),(4,42,78,11),(5,14,79,45),(6,48,80,9),(7,12,73,43),(8,46,74,15),(17,28,36,70),(18,65,37,31),(19,26,38,68),(20,71,39,29),(21,32,40,66),(22,69,33,27),(23,30,34,72),(24,67,35,25),(49,96,83,62),(50,57,84,91),(51,94,85,60),(52,63,86,89),(53,92,87,58),(54,61,88,95),(55,90,81,64),(56,59,82,93)]])
Matrix representation of C62.4D4 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 40 | 18 | 0 | 0 | 0 | 0 |
| 67 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 7 | 14 | 0 | 0 |
| 0 | 0 | 59 | 66 | 0 | 0 |
| 56 | 55 | 0 | 0 | 0 | 0 |
| 16 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 59 | 0 | 0 |
| 0 | 0 | 14 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 27 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[40,67,0,0,0,0,18,21,0,0,0,0,0,0,0,0,7,59,0,0,0,0,14,66,0,0,0,27,0,0,0,0,27,0,0,0],[56,16,0,0,0,0,55,17,0,0,0,0,0,0,66,14,0,0,0,0,59,7,0,0,0,0,0,0,0,27,0,0,0,0,27,0] >;
C62.4D4 in GAP, Magma, Sage, TeX
C_6^2._4D_4
% in TeX
G:=Group("C6^2.4D4"); // GroupNames label
G:=SmallGroup(288,388);
// by ID
G=gap.SmallGroup(288,388);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,85,120,422,219,100,2693,2028,691,797,2372]);
// 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=a^3*c^3>;
// generators/relations
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