direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12⋊3D4, C24.51D6, C12⋊9(C2×D4), (C2×D4)⋊39D6, (C2×C12)⋊13D4, C6⋊2(C4⋊1D4), Dic3⋊2(C2×D4), (C2×Dic3)⋊14D4, (C6×D4)⋊44C22, (C22×D4)⋊12S3, (C22×D12)⋊19C2, (C2×D12)⋊56C22, (C2×C6).298C24, C22.149(S3×D4), C6.145(C22×D4), (C22×C4).396D6, (C2×C12).544C23, (C4×Dic3)⋊68C22, (C23×C6).78C22, (S3×C23).77C22, (C22×C6).232C23, C23.145(C22×S3), C22.311(S3×C23), (C22×S3).129C23, (C22×C12).276C22, (C2×Dic3).285C23, (C22×Dic3).232C22, (D4×C2×C6)⋊6C2, C4⋊1(C2×C3⋊D4), C3⋊3(C2×C4⋊1D4), C2.105(C2×S3×D4), (C2×C4×Dic3)⋊12C2, (C2×C4)⋊10(C3⋊D4), (C2×C6).581(C2×D4), (C2×C3⋊D4)⋊47C22, (C22×C3⋊D4)⋊16C2, C2.18(C22×C3⋊D4), (C2×C4).627(C22×S3), C22.111(C2×C3⋊D4), SmallGroup(192,1362)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1448 in 498 conjugacy classes, 143 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C4 [×8], C22, C22 [×6], C22 [×40], S3 [×4], C6, C6 [×6], C6 [×4], C2×C4 [×6], C2×C4 [×12], D4 [×48], C23, C23 [×4], C23 [×28], Dic3 [×8], C12 [×4], D6 [×20], C2×C6, C2×C6 [×6], C2×C6 [×20], C42 [×4], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×44], C24 [×2], C24 [×2], D12 [×8], C2×Dic3 [×12], C3⋊D4 [×32], C2×C12 [×6], C3×D4 [×8], C22×S3 [×4], C22×S3 [×12], C22×C6, C22×C6 [×4], C22×C6 [×12], C2×C42, C4⋊1D4 [×8], C22×D4, C22×D4 [×5], C4×Dic3 [×4], C2×D12 [×4], C2×D12 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×16], C2×C3⋊D4 [×16], C22×C12, C6×D4 [×4], C6×D4 [×4], S3×C23 [×2], C23×C6 [×2], C2×C4⋊1D4, C2×C4×Dic3, C12⋊3D4 [×8], C22×D12, C22×C3⋊D4 [×4], D4×C2×C6, C2×C12⋊3D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×12], C23 [×15], D6 [×7], C2×D4 [×18], C24, C3⋊D4 [×4], C22×S3 [×7], C4⋊1D4 [×4], C22×D4 [×3], S3×D4 [×4], C2×C3⋊D4 [×6], S3×C23, C2×C4⋊1D4, C12⋊3D4 [×4], C2×S3×D4 [×2], C22×C3⋊D4, C2×C12⋊3D4
Generators and relations
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >
►On 96 points
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 85)(10 86)(11 87)(12 88)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 25)(23 26)(24 27)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 73)(58 74)(59 75)(60 76)
(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 18 46 78)(2 23 47 83)(3 16 48 76)(4 21 37 81)(5 14 38 74)(6 19 39 79)(7 24 40 84)(8 17 41 77)(9 22 42 82)(10 15 43 75)(11 20 44 80)(12 13 45 73)(25 72 54 85)(26 65 55 90)(27 70 56 95)(28 63 57 88)(29 68 58 93)(30 61 59 86)(31 66 60 91)(32 71 49 96)(33 64 50 89)(34 69 51 94)(35 62 52 87)(36 67 53 92)
(1 40)(2 39)(3 38)(4 37)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 17)(14 16)(18 24)(19 23)(20 22)(25 35)(26 34)(27 33)(28 32)(29 31)(49 57)(50 56)(51 55)(52 54)(58 60)(61 86)(62 85)(63 96)(64 95)(65 94)(66 93)(67 92)(68 91)(69 90)(70 89)(71 88)(72 87)(73 77)(74 76)(78 84)(79 83)(80 82)
G:=sub<Sym(96)| (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,85)(10,86)(11,87)(12,88)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,25)(23,26)(24,27)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(57,73)(58,74)(59,75)(60,76), (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), (1,18,46,78)(2,23,47,83)(3,16,48,76)(4,21,37,81)(5,14,38,74)(6,19,39,79)(7,24,40,84)(8,17,41,77)(9,22,42,82)(10,15,43,75)(11,20,44,80)(12,13,45,73)(25,72,54,85)(26,65,55,90)(27,70,56,95)(28,63,57,88)(29,68,58,93)(30,61,59,86)(31,66,60,91)(32,71,49,96)(33,64,50,89)(34,69,51,94)(35,62,52,87)(36,67,53,92), (1,40)(2,39)(3,38)(4,37)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,17)(14,16)(18,24)(19,23)(20,22)(25,35)(26,34)(27,33)(28,32)(29,31)(49,57)(50,56)(51,55)(52,54)(58,60)(61,86)(62,85)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,77)(74,76)(78,84)(79,83)(80,82)>;
G:=Group( (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,85)(10,86)(11,87)(12,88)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,25)(23,26)(24,27)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(57,73)(58,74)(59,75)(60,76), (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), (1,18,46,78)(2,23,47,83)(3,16,48,76)(4,21,37,81)(5,14,38,74)(6,19,39,79)(7,24,40,84)(8,17,41,77)(9,22,42,82)(10,15,43,75)(11,20,44,80)(12,13,45,73)(25,72,54,85)(26,65,55,90)(27,70,56,95)(28,63,57,88)(29,68,58,93)(30,61,59,86)(31,66,60,91)(32,71,49,96)(33,64,50,89)(34,69,51,94)(35,62,52,87)(36,67,53,92), (1,40)(2,39)(3,38)(4,37)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,17)(14,16)(18,24)(19,23)(20,22)(25,35)(26,34)(27,33)(28,32)(29,31)(49,57)(50,56)(51,55)(52,54)(58,60)(61,86)(62,85)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,77)(74,76)(78,84)(79,83)(80,82) );
G=PermutationGroup([(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,85),(10,86),(11,87),(12,88),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(21,36),(22,25),(23,26),(24,27),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,77),(50,78),(51,79),(52,80),(53,81),(54,82),(55,83),(56,84),(57,73),(58,74),(59,75),(60,76)], [(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)], [(1,18,46,78),(2,23,47,83),(3,16,48,76),(4,21,37,81),(5,14,38,74),(6,19,39,79),(7,24,40,84),(8,17,41,77),(9,22,42,82),(10,15,43,75),(11,20,44,80),(12,13,45,73),(25,72,54,85),(26,65,55,90),(27,70,56,95),(28,63,57,88),(29,68,58,93),(30,61,59,86),(31,66,60,91),(32,71,49,96),(33,64,50,89),(34,69,51,94),(35,62,52,87),(36,67,53,92)], [(1,40),(2,39),(3,38),(4,37),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,17),(14,16),(18,24),(19,23),(20,22),(25,35),(26,34),(27,33),(28,32),(29,31),(49,57),(50,56),(51,55),(52,54),(58,60),(61,86),(62,85),(63,96),(64,95),(65,94),(66,93),(67,92),(68,91),(69,90),(70,89),(71,88),(72,87),(73,77),(74,76),(78,84),(79,83),(80,82)])
Matrix representation ►G ⊆ GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 2 |
| 0 | 0 | 0 | 12 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 2 | 4 | 0 | 0 |
| 0 | 2 | 11 | 0 | 0 |
| 0 | 0 | 0 | 12 | 2 |
| 0 | 0 | 0 | 12 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 12 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,12,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,2,1],[12,0,0,0,0,0,2,2,0,0,0,4,11,0,0,0,0,0,12,12,0,0,0,2,1],[12,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,12] >;
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6G | 6H | ··· | 6O | 12A | 12B | 12C | 12D |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C3⋊D4 | S3×D4 |
| kernel | C2×C12⋊3D4 | C2×C4×Dic3 | C12⋊3D4 | C22×D12 | C22×C3⋊D4 | D4×C2×C6 | C22×D4 | C2×Dic3 | C2×C12 | C22×C4 | C2×D4 | C24 | C2×C4 | C22 |
| # reps | 1 | 1 | 8 | 1 | 4 | 1 | 1 | 8 | 4 | 1 | 4 | 2 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_2\times C_{12}\rtimes_3D_4 % in TeX
G:=Group("C2xC12:3D4"); // GroupNames label
G:=SmallGroup(192,1362);
// by ID
G=gap.SmallGroup(192,1362);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations