metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4:C4.197D6, (D4xDic3):27C2, (C2xD4).159D6, C22:C4.66D6, Dic3.Q8:24C2, (C2xC12).67C23, (C2xC6).193C24, D6:C4.31C22, (C22xC4).337D6, Dic3:4D4:17C2, Dic6:C4:30C2, Dic3.8(C4oD4), C23.14D6.1C2, (C6xD4).131C22, C22.D4:17S3, C23.8D6:27C2, (C22xC6).29C23, C23.34(C22xS3), C23.11D6:30C2, Dic3.D4:28C2, C23.16D6:11C2, C23.28D6:19C2, Dic3:C4.38C22, (C22xS3).84C23, C4:Dic3.224C22, C22.214(S3xC23), (C2xDic3).98C23, C22.11(D4:2S3), (C22xC12).367C22, C3:8(C23.36C23), (C4xDic3).120C22, (C2xDic6).166C22, C6.D4.39C22, (C22xDic3).226C22, C4:C4:S3:26C2, C4:C4:7S3:31C2, (C2xC4xDic3):36C2, C2.57(S3xC4oD4), C6.169(C2xC4oD4), (C2xC6).45(C4oD4), C2.51(C2xD4:2S3), (S3xC2xC4).109C22, (C2xC4).58(C22xS3), (C3xC4:C4).173C22, (C3xC22.D4):3C2, (C2xC3:D4).45C22, (C3xC22:C4).48C22, SmallGroup(192,1208)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4:C4.197D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, bab-1=a-1, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 528 in 234 conjugacy classes, 99 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, Dic6, C4xS3, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C22xS3, C22xC6, C2xC42, C42:C2, C4xD4, C4xQ8, C4:D4, C22:Q8, C22.D4, C22.D4, C4.4D4, C42.C2, C42:2C2, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C3xC22:C4, C3xC4:C4, C2xDic6, S3xC2xC4, C22xDic3, C2xC3:D4, C22xC12, C6xD4, C23.36C23, C23.16D6, Dic3.D4, C23.8D6, Dic3:4D4, C23.11D6, Dic6:C4, Dic3.Q8, C4:C4:7S3, C4:C4:S3, C2xC4xDic3, C23.28D6, D4xDic3, C23.14D6, C3xC22.D4, C4:C4.197D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C24, C22xS3, C2xC4oD4, D4:2S3, S3xC23, C23.36C23, C2xD4:2S3, S3xC4oD4, C4:C4.197D6
►On 96 points
(1 10 73 54)(2 29 74 95)(3 12 75 50)(4 25 76 91)(5 8 77 52)(6 27 78 93)(7 32 51 60)(9 34 53 56)(11 36 49 58)(13 72 79 44)(14 23 80 89)(15 68 81 46)(16 19 82 85)(17 70 83 48)(18 21 84 87)(20 39 86 61)(22 41 88 63)(24 37 90 65)(26 33 92 55)(28 35 94 57)(30 31 96 59)(38 47 66 69)(40 43 62 71)(42 45 64 67)
(1 69 57 19)(2 70 58 20)(3 71 59 21)(4 72 60 22)(5 67 55 23)(6 68 56 24)(7 63 25 13)(8 64 26 14)(9 65 27 15)(10 66 28 16)(11 61 29 17)(12 62 30 18)(31 87 75 43)(32 88 76 44)(33 89 77 45)(34 90 78 46)(35 85 73 47)(36 86 74 48)(37 93 81 53)(38 94 82 54)(39 95 83 49)(40 96 84 50)(41 91 79 51)(42 92 80 52)
(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 27 73 93)(2 26 74 92)(3 25 75 91)(4 30 76 96)(5 29 77 95)(6 28 78 94)(7 31 51 59)(8 36 52 58)(9 35 53 57)(10 34 54 56)(11 33 49 55)(12 32 50 60)(13 43 79 71)(14 48 80 70)(15 47 81 69)(16 46 82 68)(17 45 83 67)(18 44 84 72)(19 65 85 37)(20 64 86 42)(21 63 87 41)(22 62 88 40)(23 61 89 39)(24 66 90 38)
G:=sub<Sym(96)| (1,10,73,54)(2,29,74,95)(3,12,75,50)(4,25,76,91)(5,8,77,52)(6,27,78,93)(7,32,51,60)(9,34,53,56)(11,36,49,58)(13,72,79,44)(14,23,80,89)(15,68,81,46)(16,19,82,85)(17,70,83,48)(18,21,84,87)(20,39,86,61)(22,41,88,63)(24,37,90,65)(26,33,92,55)(28,35,94,57)(30,31,96,59)(38,47,66,69)(40,43,62,71)(42,45,64,67), (1,69,57,19)(2,70,58,20)(3,71,59,21)(4,72,60,22)(5,67,55,23)(6,68,56,24)(7,63,25,13)(8,64,26,14)(9,65,27,15)(10,66,28,16)(11,61,29,17)(12,62,30,18)(31,87,75,43)(32,88,76,44)(33,89,77,45)(34,90,78,46)(35,85,73,47)(36,86,74,48)(37,93,81,53)(38,94,82,54)(39,95,83,49)(40,96,84,50)(41,91,79,51)(42,92,80,52), (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), (1,27,73,93)(2,26,74,92)(3,25,75,91)(4,30,76,96)(5,29,77,95)(6,28,78,94)(7,31,51,59)(8,36,52,58)(9,35,53,57)(10,34,54,56)(11,33,49,55)(12,32,50,60)(13,43,79,71)(14,48,80,70)(15,47,81,69)(16,46,82,68)(17,45,83,67)(18,44,84,72)(19,65,85,37)(20,64,86,42)(21,63,87,41)(22,62,88,40)(23,61,89,39)(24,66,90,38)>;
G:=Group( (1,10,73,54)(2,29,74,95)(3,12,75,50)(4,25,76,91)(5,8,77,52)(6,27,78,93)(7,32,51,60)(9,34,53,56)(11,36,49,58)(13,72,79,44)(14,23,80,89)(15,68,81,46)(16,19,82,85)(17,70,83,48)(18,21,84,87)(20,39,86,61)(22,41,88,63)(24,37,90,65)(26,33,92,55)(28,35,94,57)(30,31,96,59)(38,47,66,69)(40,43,62,71)(42,45,64,67), (1,69,57,19)(2,70,58,20)(3,71,59,21)(4,72,60,22)(5,67,55,23)(6,68,56,24)(7,63,25,13)(8,64,26,14)(9,65,27,15)(10,66,28,16)(11,61,29,17)(12,62,30,18)(31,87,75,43)(32,88,76,44)(33,89,77,45)(34,90,78,46)(35,85,73,47)(36,86,74,48)(37,93,81,53)(38,94,82,54)(39,95,83,49)(40,96,84,50)(41,91,79,51)(42,92,80,52), (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), (1,27,73,93)(2,26,74,92)(3,25,75,91)(4,30,76,96)(5,29,77,95)(6,28,78,94)(7,31,51,59)(8,36,52,58)(9,35,53,57)(10,34,54,56)(11,33,49,55)(12,32,50,60)(13,43,79,71)(14,48,80,70)(15,47,81,69)(16,46,82,68)(17,45,83,67)(18,44,84,72)(19,65,85,37)(20,64,86,42)(21,63,87,41)(22,62,88,40)(23,61,89,39)(24,66,90,38) );
G=PermutationGroup([[(1,10,73,54),(2,29,74,95),(3,12,75,50),(4,25,76,91),(5,8,77,52),(6,27,78,93),(7,32,51,60),(9,34,53,56),(11,36,49,58),(13,72,79,44),(14,23,80,89),(15,68,81,46),(16,19,82,85),(17,70,83,48),(18,21,84,87),(20,39,86,61),(22,41,88,63),(24,37,90,65),(26,33,92,55),(28,35,94,57),(30,31,96,59),(38,47,66,69),(40,43,62,71),(42,45,64,67)], [(1,69,57,19),(2,70,58,20),(3,71,59,21),(4,72,60,22),(5,67,55,23),(6,68,56,24),(7,63,25,13),(8,64,26,14),(9,65,27,15),(10,66,28,16),(11,61,29,17),(12,62,30,18),(31,87,75,43),(32,88,76,44),(33,89,77,45),(34,90,78,46),(35,85,73,47),(36,86,74,48),(37,93,81,53),(38,94,82,54),(39,95,83,49),(40,96,84,50),(41,91,79,51),(42,92,80,52)], [(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)], [(1,27,73,93),(2,26,74,92),(3,25,75,91),(4,30,76,96),(5,29,77,95),(6,28,78,94),(7,31,51,59),(8,36,52,58),(9,35,53,57),(10,34,54,56),(11,33,49,55),(12,32,50,60),(13,43,79,71),(14,48,80,70),(15,47,81,69),(16,46,82,68),(17,45,83,67),(18,44,84,72),(19,65,85,37),(20,64,86,42),(21,63,87,41),(22,62,88,40),(23,61,89,39),(24,66,90,38)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 4R | 4S | 4T | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 12 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4oD4 | C4oD4 | D4:2S3 | S3xC4oD4 |
| kernel | C4:C4.197D6 | C23.16D6 | Dic3.D4 | C23.8D6 | Dic3:4D4 | C23.11D6 | Dic6:C4 | Dic3.Q8 | C4:C4:7S3 | C4:C4:S3 | C2xC4xDic3 | C23.28D6 | D4xDic3 | C23.14D6 | C3xC22.D4 | C22.D4 | C22:C4 | C4:C4 | C22xC4 | C2xD4 | Dic3 | C2xC6 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 8 | 4 | 2 | 4 |
Matrix representation of C4:C4.197D6 ►in GL6(F13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 3 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 9 | 4 | 0 | 0 | 0 | 0 |
| 12 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 5 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 9 | 4 | 0 | 0 | 0 | 0 |
| 12 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [8,3,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[9,12,0,0,0,0,4,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,5,0],[7,5,0,0,0,0,6,6,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[9,12,0,0,0,0,4,4,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;
C4:C4.197D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{197}D_6 % in TeX
G:=Group("C4:C4.197D6"); // GroupNames label
G:=SmallGroup(192,1208);
// by ID
G=gap.SmallGroup(192,1208);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,100,346,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations