direct product, metabelian, supersoluble, monomial
Aliases: C3xC4:C4:7S3, C62.183C23, (S3xC12):5C4, (C4xS3):2C12, D6:C4.4C6, C4.14(S3xC12), C4:Dic3:12C6, D6.4(C2xC12), C12.105(C4xS3), C12.11(C2xC12), (C4xDic3):13C6, (C2xC12).270D6, (Dic3xC12):29C2, C6.10(C22xC12), Dic3.9(C2xC12), (C6xC12).248C22, C6.57(Q8:3S3), C6.119(D4:2S3), C32:15(C42:C2), (C6xDic3).126C22, (C3xC4:C4):3C6, C4:C4:7(C3xS3), (S3xC2xC4).1C6, (C3xC4:C4):16S3, C6.109(S3xC2xC4), C2.12(S3xC2xC12), (C32xC4:C4):4C2, (S3xC2xC12).10C2, (C2xC4).31(S3xC6), C6.26(C3xC4oD4), C22.17(S3xC2xC6), (S3xC6).22(C2xC4), (C3xC12).66(C2xC4), (C2xC12).57(C2xC6), (C3xD6:C4).12C2, (C3xC4:Dic3):21C2, C3:3(C3xC42:C2), C2.4(C3xD4:2S3), (S3xC2xC6).92C22, C2.1(C3xQ8:3S3), (C2xC6).38(C22xC6), (C3xC6).81(C22xC4), (C3xC6).133(C4oD4), (C22xS3).19(C2xC6), (C2xC6).316(C22xS3), (C3xDic3).30(C2xC4), (C2xDic3).47(C2xC6), SmallGroup(288,663)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xC4:C4:7S3
G = < a,b,c,d,e | a3=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 338 in 165 conjugacy classes, 82 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, C2xC4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C4:C4, C22xC4, C3xS3, C3xC6, C4xS3, C2xDic3, C2xDic3, C2xC12, C2xC12, C2xC12, C22xS3, C22xC6, C42:C2, C3xDic3, C3xDic3, C3xC12, C3xC12, S3xC6, S3xC6, C62, C4xDic3, C4:Dic3, D6:C4, C4xC12, C3xC22:C4, C3xC4:C4, C3xC4:C4, S3xC2xC4, C22xC12, S3xC12, C6xDic3, C6xDic3, C6xC12, C6xC12, S3xC2xC6, C4:C4:7S3, C3xC42:C2, Dic3xC12, C3xC4:Dic3, C3xD6:C4, C32xC4:C4, S3xC2xC12, C3xC4:C4:7S3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, C23, C12, D6, C2xC6, C22xC4, C4oD4, C3xS3, C4xS3, C2xC12, C22xS3, C22xC6, C42:C2, S3xC6, S3xC2xC4, D4:2S3, Q8:3S3, C22xC12, C3xC4oD4, S3xC12, S3xC2xC6, C4:C4:7S3, C3xC42:C2, S3xC2xC12, C3xD4:2S3, C3xQ8:3S3, C3xC4:C4:7S3
►On 96 points
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 60 76)(6 57 73)(7 58 74)(8 59 75)(9 25 42)(10 26 43)(11 27 44)(12 28 41)(21 37 46)(22 38 47)(23 39 48)(24 40 45)(29 34 83)(30 35 84)(31 36 81)(32 33 82)(49 54 89)(50 55 90)(51 56 91)(52 53 92)(61 66 79)(62 67 80)(63 68 77)(64 65 78)(69 85 94)(70 86 95)(71 87 96)(72 88 93)
(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 43 8)(2 46 44 7)(3 45 41 6)(4 48 42 5)(9 60 15 23)(10 59 16 22)(11 58 13 21)(12 57 14 24)(17 38 26 75)(18 37 27 74)(19 40 28 73)(20 39 25 76)(29 61 71 52)(30 64 72 51)(31 63 69 50)(32 62 70 49)(33 67 86 54)(34 66 87 53)(35 65 88 56)(36 68 85 55)(77 94 90 81)(78 93 91 84)(79 96 92 83)(80 95 89 82)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 76 60)(6 73 57)(7 74 58)(8 75 59)(9 42 25)(10 43 26)(11 44 27)(12 41 28)(21 46 37)(22 47 38)(23 48 39)(24 45 40)(29 34 83)(30 35 84)(31 36 81)(32 33 82)(49 54 89)(50 55 90)(51 56 91)(52 53 92)(61 66 79)(62 67 80)(63 68 77)(64 65 78)(69 85 94)(70 86 95)(71 87 96)(72 88 93)
(1 78)(2 79)(3 80)(4 77)(5 83)(6 84)(7 81)(8 82)(9 50)(10 51)(11 52)(12 49)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 55)(26 56)(27 53)(28 54)(29 60)(30 57)(31 58)(32 59)(33 75)(34 76)(35 73)(36 74)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
G:=sub<Sym(96)| (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,60,76)(6,57,73)(7,58,74)(8,59,75)(9,25,42)(10,26,43)(11,27,44)(12,28,41)(21,37,46)(22,38,47)(23,39,48)(24,40,45)(29,34,83)(30,35,84)(31,36,81)(32,33,82)(49,54,89)(50,55,90)(51,56,91)(52,53,92)(61,66,79)(62,67,80)(63,68,77)(64,65,78)(69,85,94)(70,86,95)(71,87,96)(72,88,93), (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), (1,47,43,8)(2,46,44,7)(3,45,41,6)(4,48,42,5)(9,60,15,23)(10,59,16,22)(11,58,13,21)(12,57,14,24)(17,38,26,75)(18,37,27,74)(19,40,28,73)(20,39,25,76)(29,61,71,52)(30,64,72,51)(31,63,69,50)(32,62,70,49)(33,67,86,54)(34,66,87,53)(35,65,88,56)(36,68,85,55)(77,94,90,81)(78,93,91,84)(79,96,92,83)(80,95,89,82), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,76,60)(6,73,57)(7,74,58)(8,75,59)(9,42,25)(10,43,26)(11,44,27)(12,41,28)(21,46,37)(22,47,38)(23,48,39)(24,45,40)(29,34,83)(30,35,84)(31,36,81)(32,33,82)(49,54,89)(50,55,90)(51,56,91)(52,53,92)(61,66,79)(62,67,80)(63,68,77)(64,65,78)(69,85,94)(70,86,95)(71,87,96)(72,88,93), (1,78)(2,79)(3,80)(4,77)(5,83)(6,84)(7,81)(8,82)(9,50)(10,51)(11,52)(12,49)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,55)(26,56)(27,53)(28,54)(29,60)(30,57)(31,58)(32,59)(33,75)(34,76)(35,73)(36,74)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)>;
G:=Group( (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,60,76)(6,57,73)(7,58,74)(8,59,75)(9,25,42)(10,26,43)(11,27,44)(12,28,41)(21,37,46)(22,38,47)(23,39,48)(24,40,45)(29,34,83)(30,35,84)(31,36,81)(32,33,82)(49,54,89)(50,55,90)(51,56,91)(52,53,92)(61,66,79)(62,67,80)(63,68,77)(64,65,78)(69,85,94)(70,86,95)(71,87,96)(72,88,93), (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), (1,47,43,8)(2,46,44,7)(3,45,41,6)(4,48,42,5)(9,60,15,23)(10,59,16,22)(11,58,13,21)(12,57,14,24)(17,38,26,75)(18,37,27,74)(19,40,28,73)(20,39,25,76)(29,61,71,52)(30,64,72,51)(31,63,69,50)(32,62,70,49)(33,67,86,54)(34,66,87,53)(35,65,88,56)(36,68,85,55)(77,94,90,81)(78,93,91,84)(79,96,92,83)(80,95,89,82), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,76,60)(6,73,57)(7,74,58)(8,75,59)(9,42,25)(10,43,26)(11,44,27)(12,41,28)(21,46,37)(22,47,38)(23,48,39)(24,45,40)(29,34,83)(30,35,84)(31,36,81)(32,33,82)(49,54,89)(50,55,90)(51,56,91)(52,53,92)(61,66,79)(62,67,80)(63,68,77)(64,65,78)(69,85,94)(70,86,95)(71,87,96)(72,88,93), (1,78)(2,79)(3,80)(4,77)(5,83)(6,84)(7,81)(8,82)(9,50)(10,51)(11,52)(12,49)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,55)(26,56)(27,53)(28,54)(29,60)(30,57)(31,58)(32,59)(33,75)(34,76)(35,73)(36,74)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96) );
G=PermutationGroup([[(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,60,76),(6,57,73),(7,58,74),(8,59,75),(9,25,42),(10,26,43),(11,27,44),(12,28,41),(21,37,46),(22,38,47),(23,39,48),(24,40,45),(29,34,83),(30,35,84),(31,36,81),(32,33,82),(49,54,89),(50,55,90),(51,56,91),(52,53,92),(61,66,79),(62,67,80),(63,68,77),(64,65,78),(69,85,94),(70,86,95),(71,87,96),(72,88,93)], [(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)], [(1,47,43,8),(2,46,44,7),(3,45,41,6),(4,48,42,5),(9,60,15,23),(10,59,16,22),(11,58,13,21),(12,57,14,24),(17,38,26,75),(18,37,27,74),(19,40,28,73),(20,39,25,76),(29,61,71,52),(30,64,72,51),(31,63,69,50),(32,62,70,49),(33,67,86,54),(34,66,87,53),(35,65,88,56),(36,68,85,55),(77,94,90,81),(78,93,91,84),(79,96,92,83),(80,95,89,82)], [(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,76,60),(6,73,57),(7,74,58),(8,75,59),(9,42,25),(10,43,26),(11,44,27),(12,41,28),(21,46,37),(22,47,38),(23,48,39),(24,45,40),(29,34,83),(30,35,84),(31,36,81),(32,33,82),(49,54,89),(50,55,90),(51,56,91),(52,53,92),(61,66,79),(62,67,80),(63,68,77),(64,65,78),(69,85,94),(70,86,95),(71,87,96),(72,88,93)], [(1,78),(2,79),(3,80),(4,77),(5,83),(6,84),(7,81),(8,82),(9,50),(10,51),(11,52),(12,49),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,55),(26,56),(27,53),(28,54),(29,60),(30,57),(31,58),(32,59),(33,75),(34,76),(35,73),(36,74),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | ··· | 6F | 6G | ··· | 6O | 6P | 6Q | 6R | 6S | 12A | ··· | 12L | 12M | ··· | 12T | 12U | ··· | 12AL | 12AM | ··· | 12AT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | S3 | D6 | C4oD4 | C3xS3 | C4xS3 | S3xC6 | C3xC4oD4 | S3xC12 | D4:2S3 | Q8:3S3 | C3xD4:2S3 | C3xQ8:3S3 |
| kernel | C3xC4:C4:7S3 | Dic3xC12 | C3xC4:Dic3 | C3xD6:C4 | C32xC4:C4 | S3xC2xC12 | C4:C4:7S3 | S3xC12 | C4xDic3 | C4:Dic3 | D6:C4 | C3xC4:C4 | S3xC2xC4 | C4xS3 | C3xC4:C4 | C2xC12 | C3xC6 | C4:C4 | C12 | C2xC4 | C6 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 8 | 4 | 2 | 4 | 2 | 2 | 16 | 1 | 3 | 4 | 2 | 4 | 6 | 8 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3xC4:C4:7S3 ►in GL4(F13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 12 | 4 | 0 | 0 |
| 6 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 12 | 0 | 0 |
| 2 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 2 | 9 |
| 8 | 7 | 0 | 0 |
| 4 | 5 | 0 | 0 |
| 0 | 0 | 5 | 2 |
| 0 | 0 | 1 | 8 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[12,6,0,0,4,1,0,0,0,0,1,0,0,0,0,1],[1,2,0,0,12,12,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,3,2,0,0,0,9],[8,4,0,0,7,5,0,0,0,0,5,1,0,0,2,8] >;
C3xC4:C4:7S3 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes C_4\rtimes_7S_3
% in TeX
G:=Group("C3xC4:C4:7S3"); // GroupNames label
G:=SmallGroup(288,663);
// by ID
G=gap.SmallGroup(288,663);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,1094,555,142,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations