metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.64D6, (C2×C6).5Q16, (C2×C12).76D4, (C2×Q8).50D6, C6.37(C2×Q16), C22⋊Q8.3S3, C6.Q16⋊38C2, (C22×C6).90D4, Q8⋊2Dic3⋊13C2, (C22×C4).141D6, C12.188(C4○D4), (C6×Q8).44C22, C2.14(D4⋊D6), C4.94(D4⋊2S3), C6.115(C8⋊C22), (C2×C12).363C23, C12.55D4.7C2, C23.68(C3⋊D4), C3⋊5(C23.48D4), C22.3(C3⋊Q16), C4⋊Dic3.338C22, (C22×C12).167C22, C6.81(C22.D4), C2.15(C23.23D6), C2.8(C2×C3⋊Q16), (C2×C6).494(C2×D4), (C3×C22⋊Q8).2C2, (C2×C4).54(C3⋊D4), (C2×C3⋊C8).113C22, (C2×C4⋊Dic3).38C2, (C3×C4⋊C4).111C22, (C2×C4).463(C22×S3), C22.169(C2×C3⋊D4), SmallGroup(192,603)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C6).Q16
G = < a,b,c,d | a2=b6=c8=1, d2=b3c4, ab=ba, cac-1=dad-1=ab3, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 256 in 104 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×C6, C22⋊C8, Q8⋊C4, C2.D8, C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C6×Q8, C23.48D4, C6.Q16, C12.55D4, Q8⋊2Dic3, C2×C4⋊Dic3, C3×C22⋊Q8, (C2×C6).Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, C2×Q16, C8⋊C22, C3⋊Q16, D4⋊2S3, C2×C3⋊D4, C23.48D4, C23.23D6, C2×C3⋊Q16, D4⋊D6, (C2×C6).Q16
►On 96 points
Generators in S96
(1 5)(2 56)(3 7)(4 50)(6 52)(8 54)(9 78)(10 14)(11 80)(12 16)(13 74)(15 76)(17 60)(18 22)(19 62)(20 24)(21 64)(23 58)(25 83)(26 30)(27 85)(28 32)(29 87)(31 81)(33 91)(34 38)(35 93)(36 40)(37 95)(39 89)(41 45)(42 65)(43 47)(44 67)(46 69)(48 71)(49 53)(51 55)(57 61)(59 63)(66 70)(68 72)(73 77)(75 79)(82 86)(84 88)(90 94)(92 96)
(1 12 26 51 77 88)(2 81 78 52 27 13)(3 14 28 53 79 82)(4 83 80 54 29 15)(5 16 30 55 73 84)(6 85 74 56 31 9)(7 10 32 49 75 86)(8 87 76 50 25 11)(17 67 35 64 48 89)(18 90 41 57 36 68)(19 69 37 58 42 91)(20 92 43 59 38 70)(21 71 39 60 44 93)(22 94 45 61 40 72)(23 65 33 62 46 95)(24 96 47 63 34 66)
(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 23 55 58)(2 22 56 57)(3 21 49 64)(4 20 50 63)(5 19 51 62)(6 18 52 61)(7 17 53 60)(8 24 54 59)(9 68 78 45)(10 67 79 44)(11 66 80 43)(12 65 73 42)(13 72 74 41)(14 71 75 48)(15 70 76 47)(16 69 77 46)(25 34 83 92)(26 33 84 91)(27 40 85 90)(28 39 86 89)(29 38 87 96)(30 37 88 95)(31 36 81 94)(32 35 82 93)
G:=sub<Sym(96)| (1,5)(2,56)(3,7)(4,50)(6,52)(8,54)(9,78)(10,14)(11,80)(12,16)(13,74)(15,76)(17,60)(18,22)(19,62)(20,24)(21,64)(23,58)(25,83)(26,30)(27,85)(28,32)(29,87)(31,81)(33,91)(34,38)(35,93)(36,40)(37,95)(39,89)(41,45)(42,65)(43,47)(44,67)(46,69)(48,71)(49,53)(51,55)(57,61)(59,63)(66,70)(68,72)(73,77)(75,79)(82,86)(84,88)(90,94)(92,96), (1,12,26,51,77,88)(2,81,78,52,27,13)(3,14,28,53,79,82)(4,83,80,54,29,15)(5,16,30,55,73,84)(6,85,74,56,31,9)(7,10,32,49,75,86)(8,87,76,50,25,11)(17,67,35,64,48,89)(18,90,41,57,36,68)(19,69,37,58,42,91)(20,92,43,59,38,70)(21,71,39,60,44,93)(22,94,45,61,40,72)(23,65,33,62,46,95)(24,96,47,63,34,66), (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), (1,23,55,58)(2,22,56,57)(3,21,49,64)(4,20,50,63)(5,19,51,62)(6,18,52,61)(7,17,53,60)(8,24,54,59)(9,68,78,45)(10,67,79,44)(11,66,80,43)(12,65,73,42)(13,72,74,41)(14,71,75,48)(15,70,76,47)(16,69,77,46)(25,34,83,92)(26,33,84,91)(27,40,85,90)(28,39,86,89)(29,38,87,96)(30,37,88,95)(31,36,81,94)(32,35,82,93)>;
G:=Group( (1,5)(2,56)(3,7)(4,50)(6,52)(8,54)(9,78)(10,14)(11,80)(12,16)(13,74)(15,76)(17,60)(18,22)(19,62)(20,24)(21,64)(23,58)(25,83)(26,30)(27,85)(28,32)(29,87)(31,81)(33,91)(34,38)(35,93)(36,40)(37,95)(39,89)(41,45)(42,65)(43,47)(44,67)(46,69)(48,71)(49,53)(51,55)(57,61)(59,63)(66,70)(68,72)(73,77)(75,79)(82,86)(84,88)(90,94)(92,96), (1,12,26,51,77,88)(2,81,78,52,27,13)(3,14,28,53,79,82)(4,83,80,54,29,15)(5,16,30,55,73,84)(6,85,74,56,31,9)(7,10,32,49,75,86)(8,87,76,50,25,11)(17,67,35,64,48,89)(18,90,41,57,36,68)(19,69,37,58,42,91)(20,92,43,59,38,70)(21,71,39,60,44,93)(22,94,45,61,40,72)(23,65,33,62,46,95)(24,96,47,63,34,66), (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), (1,23,55,58)(2,22,56,57)(3,21,49,64)(4,20,50,63)(5,19,51,62)(6,18,52,61)(7,17,53,60)(8,24,54,59)(9,68,78,45)(10,67,79,44)(11,66,80,43)(12,65,73,42)(13,72,74,41)(14,71,75,48)(15,70,76,47)(16,69,77,46)(25,34,83,92)(26,33,84,91)(27,40,85,90)(28,39,86,89)(29,38,87,96)(30,37,88,95)(31,36,81,94)(32,35,82,93) );
G=PermutationGroup([[(1,5),(2,56),(3,7),(4,50),(6,52),(8,54),(9,78),(10,14),(11,80),(12,16),(13,74),(15,76),(17,60),(18,22),(19,62),(20,24),(21,64),(23,58),(25,83),(26,30),(27,85),(28,32),(29,87),(31,81),(33,91),(34,38),(35,93),(36,40),(37,95),(39,89),(41,45),(42,65),(43,47),(44,67),(46,69),(48,71),(49,53),(51,55),(57,61),(59,63),(66,70),(68,72),(73,77),(75,79),(82,86),(84,88),(90,94),(92,96)], [(1,12,26,51,77,88),(2,81,78,52,27,13),(3,14,28,53,79,82),(4,83,80,54,29,15),(5,16,30,55,73,84),(6,85,74,56,31,9),(7,10,32,49,75,86),(8,87,76,50,25,11),(17,67,35,64,48,89),(18,90,41,57,36,68),(19,69,37,58,42,91),(20,92,43,59,38,70),(21,71,39,60,44,93),(22,94,45,61,40,72),(23,65,33,62,46,95),(24,96,47,63,34,66)], [(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)], [(1,23,55,58),(2,22,56,57),(3,21,49,64),(4,20,50,63),(5,19,51,62),(6,18,52,61),(7,17,53,60),(8,24,54,59),(9,68,78,45),(10,67,79,44),(11,66,80,43),(12,65,73,42),(13,72,74,41),(14,71,75,48),(15,70,76,47),(16,69,77,46),(25,34,83,92),(26,33,84,91),(27,40,85,90),(28,39,86,89),(29,38,87,96),(30,37,88,95),(31,36,81,94),(32,35,82,93)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | Q16 | C3⋊D4 | C3⋊D4 | C8⋊C22 | D4⋊2S3 | C3⋊Q16 | D4⋊D6 |
| kernel | (C2×C6).Q16 | C6.Q16 | C12.55D4 | Q8⋊2Dic3 | C2×C4⋊Dic3 | C3×C22⋊Q8 | C22⋊Q8 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×Q8 | C12 | C2×C6 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 2 | 2 | 2 |
Matrix representation of (C2×C6).Q16 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 13 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 25 | 62 | 0 | 0 | 0 | 0 |
| 37 | 48 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 65 | 0 | 0 |
| 0 | 0 | 55 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 57 | 32 |
| 30 | 60 | 0 | 0 | 0 | 0 |
| 13 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 3 | 0 | 0 |
| 0 | 0 | 25 | 56 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 71 |
| 0 | 0 | 0 | 0 | 12 | 60 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,13,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,37,0,0,0,0,62,48,0,0,0,0,0,0,52,55,0,0,0,0,65,21,0,0,0,0,0,0,0,57,0,0,0,0,32,32],[30,13,0,0,0,0,60,43,0,0,0,0,0,0,17,25,0,0,0,0,3,56,0,0,0,0,0,0,13,12,0,0,0,0,71,60] >;
(C2×C6).Q16 in GAP, Magma, Sage, TeX
(C_2\times C_6).Q_{16} % in TeX
G:=Group("(C2xC6).Q16"); // GroupNames label
G:=SmallGroup(192,603);
// by ID
G=gap.SmallGroup(192,603);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,336,254,219,268,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^8=1,d^2=b^3*c^4,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^3,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations