metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊Q8⋊18C2, C12⋊9(C4○D4), C4⋊D4⋊25S3, C4⋊C4.175D6, C12⋊3D4⋊14C2, C12⋊7D4⋊31C2, C4⋊3(D4⋊2S3), C22.1(S3×D4), (C2×Dic3)⋊13D4, (D4×Dic3)⋊14C2, (C2×D4).150D6, C22⋊C4.45D6, Dic3.5(C2×D4), Dic3⋊5D4⋊19C2, C6.59(C22×D4), Dic3⋊2(C4○D4), Dic3⋊4D4⋊5C2, C23.14D6⋊7C2, (C2×C12).34C23, (C2×C6).140C24, D6⋊C4.57C22, (C22×C4).382D6, (C6×D4).114C22, (C22×C6).11C23, C23.11D6⋊16C2, (C2×D12).141C22, Dic3⋊C4.12C22, (C22×S3).59C23, C4⋊Dic3.203C22, C23.188(C22×S3), C22.161(S3×C23), C3⋊3(C22.26C24), (C22×C12).235C22, (C4×Dic3).254C22, (C2×Dic6).150C22, (C2×Dic3).223C23, C6.D4.18C22, (C22×Dic3).221C22, C2.32(C2×S3×D4), (C2×C4×Dic3)⋊7C2, (C2×C6).3(C2×D4), (C3×C4⋊D4)⋊5C2, C6.79(C2×C4○D4), C2.33(S3×C4○D4), (C2×D4⋊2S3)⋊8C2, (S3×C2×C4).79C22, C2.30(C2×D4⋊2S3), (C2×C4).34(C22×S3), (C3×C4⋊C4).136C22, (C2×C3⋊D4).23C22, (C3×C22⋊C4).5C22, SmallGroup(192,1155)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 784 in 310 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×2], C4 [×12], C22, C22 [×2], C22 [×14], S3 [×2], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×22], D4 [×20], Q8 [×4], C23, C23 [×2], C23 [×2], Dic3 [×6], Dic3 [×3], C12 [×2], C12 [×3], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×8], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8 [×2], C4○D4 [×8], Dic6 [×4], C4×S3 [×4], D12 [×2], C2×Dic3 [×4], C2×Dic3 [×6], C2×Dic3 [×6], C3⋊D4 [×12], C2×C12 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×6], C22×S3 [×2], C22×C6, C22×C6 [×2], C2×C42, C4×D4 [×4], C4⋊D4, C4⋊D4 [×3], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C2×C4○D4 [×2], C4×Dic3 [×4], Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×4], C6.D4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6 [×2], S3×C2×C4 [×2], C2×D12, D4⋊2S3 [×8], C22×Dic3 [×2], C22×Dic3 [×2], C2×C3⋊D4 [×6], C22×C12, C6×D4, C6×D4 [×2], C22.26C24, Dic3⋊4D4 [×2], C23.11D6 [×2], C12⋊Q8, Dic3⋊5D4, C2×C4×Dic3, C12⋊7D4, D4×Dic3, C23.14D6 [×2], C12⋊3D4, C3×C4⋊D4, C2×D4⋊2S3 [×2], C12⋊(C4○D4)
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C22×S3 [×7], C22×D4, C2×C4○D4 [×2], S3×D4 [×2], D4⋊2S3 [×2], S3×C23, C22.26C24, C2×S3×D4, C2×D4⋊2S3, S3×C4○D4, C12⋊(C4○D4)
Generators and relations
G = < a,b,c,d | a12=b4=d2=1, c2=b2, bab-1=a5, cac-1=a7, ad=da, bc=cb, bd=db, dcd=b2c >
►On 96 points
(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 61 90 58)(2 66 91 51)(3 71 92 56)(4 64 93 49)(5 69 94 54)(6 62 95 59)(7 67 96 52)(8 72 85 57)(9 65 86 50)(10 70 87 55)(11 63 88 60)(12 68 89 53)(13 82 25 40)(14 75 26 45)(15 80 27 38)(16 73 28 43)(17 78 29 48)(18 83 30 41)(19 76 31 46)(20 81 32 39)(21 74 33 44)(22 79 34 37)(23 84 35 42)(24 77 36 47)
(1 58 90 61)(2 53 91 68)(3 60 92 63)(4 55 93 70)(5 50 94 65)(6 57 95 72)(7 52 96 67)(8 59 85 62)(9 54 86 69)(10 49 87 64)(11 56 88 71)(12 51 89 66)(13 80 25 38)(14 75 26 45)(15 82 27 40)(16 77 28 47)(17 84 29 42)(18 79 30 37)(19 74 31 44)(20 81 32 39)(21 76 33 46)(22 83 34 41)(23 78 35 48)(24 73 36 43)
(1 81)(2 82)(3 83)(4 84)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 61)(33 62)(34 63)(35 64)(36 65)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 85)(47 86)(48 87)
G:=sub<Sym(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), (1,61,90,58)(2,66,91,51)(3,71,92,56)(4,64,93,49)(5,69,94,54)(6,62,95,59)(7,67,96,52)(8,72,85,57)(9,65,86,50)(10,70,87,55)(11,63,88,60)(12,68,89,53)(13,82,25,40)(14,75,26,45)(15,80,27,38)(16,73,28,43)(17,78,29,48)(18,83,30,41)(19,76,31,46)(20,81,32,39)(21,74,33,44)(22,79,34,37)(23,84,35,42)(24,77,36,47), (1,58,90,61)(2,53,91,68)(3,60,92,63)(4,55,93,70)(5,50,94,65)(6,57,95,72)(7,52,96,67)(8,59,85,62)(9,54,86,69)(10,49,87,64)(11,56,88,71)(12,51,89,66)(13,80,25,38)(14,75,26,45)(15,82,27,40)(16,77,28,47)(17,84,29,42)(18,79,30,37)(19,74,31,44)(20,81,32,39)(21,76,33,46)(22,83,34,41)(23,78,35,48)(24,73,36,43), (1,81)(2,82)(3,83)(4,84)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,61)(33,62)(34,63)(35,64)(36,65)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,85)(47,86)(48,87)>;
G:=Group( (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), (1,61,90,58)(2,66,91,51)(3,71,92,56)(4,64,93,49)(5,69,94,54)(6,62,95,59)(7,67,96,52)(8,72,85,57)(9,65,86,50)(10,70,87,55)(11,63,88,60)(12,68,89,53)(13,82,25,40)(14,75,26,45)(15,80,27,38)(16,73,28,43)(17,78,29,48)(18,83,30,41)(19,76,31,46)(20,81,32,39)(21,74,33,44)(22,79,34,37)(23,84,35,42)(24,77,36,47), (1,58,90,61)(2,53,91,68)(3,60,92,63)(4,55,93,70)(5,50,94,65)(6,57,95,72)(7,52,96,67)(8,59,85,62)(9,54,86,69)(10,49,87,64)(11,56,88,71)(12,51,89,66)(13,80,25,38)(14,75,26,45)(15,82,27,40)(16,77,28,47)(17,84,29,42)(18,79,30,37)(19,74,31,44)(20,81,32,39)(21,76,33,46)(22,83,34,41)(23,78,35,48)(24,73,36,43), (1,81)(2,82)(3,83)(4,84)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,61)(33,62)(34,63)(35,64)(36,65)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,85)(47,86)(48,87) );
G=PermutationGroup([(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)], [(1,61,90,58),(2,66,91,51),(3,71,92,56),(4,64,93,49),(5,69,94,54),(6,62,95,59),(7,67,96,52),(8,72,85,57),(9,65,86,50),(10,70,87,55),(11,63,88,60),(12,68,89,53),(13,82,25,40),(14,75,26,45),(15,80,27,38),(16,73,28,43),(17,78,29,48),(18,83,30,41),(19,76,31,46),(20,81,32,39),(21,74,33,44),(22,79,34,37),(23,84,35,42),(24,77,36,47)], [(1,58,90,61),(2,53,91,68),(3,60,92,63),(4,55,93,70),(5,50,94,65),(6,57,95,72),(7,52,96,67),(8,59,85,62),(9,54,86,69),(10,49,87,64),(11,56,88,71),(12,51,89,66),(13,80,25,38),(14,75,26,45),(15,82,27,40),(16,77,28,47),(17,84,29,42),(18,79,30,37),(19,74,31,44),(20,81,32,39),(21,76,33,46),(22,83,34,41),(23,78,35,48),(24,73,36,43)], [(1,81),(2,82),(3,83),(4,84),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,79),(12,80),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,49),(24,50),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,61),(33,62),(34,63),(35,64),(36,65),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,94),(44,95),(45,96),(46,85),(47,86),(48,87)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 2 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,12,0,0,0,0,0,0,12,12,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,8,0] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 4Q | 4R | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | D4⋊2S3 | S3×D4 | S3×C4○D4 |
| kernel | C12⋊(C4○D4) | Dic3⋊4D4 | C23.11D6 | C12⋊Q8 | Dic3⋊5D4 | C2×C4×Dic3 | C12⋊7D4 | D4×Dic3 | C23.14D6 | C12⋊3D4 | C3×C4⋊D4 | C2×D4⋊2S3 | C4⋊D4 | C2×Dic3 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | Dic3 | C12 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 2 | 1 | 1 | 3 | 4 | 4 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_{12}\rtimes (C_4\circ D_4) % in TeX
G:=Group("C12:(C4oD4)"); // GroupNames label
G:=SmallGroup(192,1155);
// by ID
G=gap.SmallGroup(192,1155);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,387,570,185,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^4=d^2=1,c^2=b^2,b*a*b^-1=a^5,c*a*c^-1=a^7,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations