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), (D4×Dic3)⋊14C2, (C2×Dic3)⋊13D4, (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×C6).140C24, (C2×C12).34C23, 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, (C2×Dic3).223C23, (C2×Dic6).150C22, (C4×Dic3).254C22, 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
Generators and relations for C12⋊(C4○D4)
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 >
Subgroups: 784 in 310 conjugacy classes, 109 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C2×C42, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, D4⋊2S3, C22×Dic3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C22.26C24, 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, C12⋊(C4○D4)
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, S3×D4, D4⋊2S3, S3×C23, C22.26C24, C2×S3×D4, C2×D4⋊2S3, S3×C4○D4, C12⋊(C4○D4)
►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 16 48 76)(2 21 37 81)(3 14 38 74)(4 19 39 79)(5 24 40 84)(6 17 41 77)(7 22 42 82)(8 15 43 75)(9 20 44 80)(10 13 45 73)(11 18 46 78)(12 23 47 83)(25 64 96 60)(26 69 85 53)(27 62 86 58)(28 67 87 51)(29 72 88 56)(30 65 89 49)(31 70 90 54)(32 63 91 59)(33 68 92 52)(34 61 93 57)(35 66 94 50)(36 71 95 55)
(1 76 48 16)(2 83 37 23)(3 78 38 18)(4 73 39 13)(5 80 40 20)(6 75 41 15)(7 82 42 22)(8 77 43 17)(9 84 44 24)(10 79 45 19)(11 74 46 14)(12 81 47 21)(25 70 96 54)(26 65 85 49)(27 72 86 56)(28 67 87 51)(29 62 88 58)(30 69 89 53)(31 64 90 60)(32 71 91 55)(33 66 92 50)(34 61 93 57)(35 68 94 52)(36 63 95 59)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 85)(21 86)(22 87)(23 88)(24 89)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 58)(38 59)(39 60)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)
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), (1,16,48,76)(2,21,37,81)(3,14,38,74)(4,19,39,79)(5,24,40,84)(6,17,41,77)(7,22,42,82)(8,15,43,75)(9,20,44,80)(10,13,45,73)(11,18,46,78)(12,23,47,83)(25,64,96,60)(26,69,85,53)(27,62,86,58)(28,67,87,51)(29,72,88,56)(30,65,89,49)(31,70,90,54)(32,63,91,59)(33,68,92,52)(34,61,93,57)(35,66,94,50)(36,71,95,55), (1,76,48,16)(2,83,37,23)(3,78,38,18)(4,73,39,13)(5,80,40,20)(6,75,41,15)(7,82,42,22)(8,77,43,17)(9,84,44,24)(10,79,45,19)(11,74,46,14)(12,81,47,21)(25,70,96,54)(26,65,85,49)(27,72,86,56)(28,67,87,51)(29,62,88,58)(30,69,89,53)(31,64,90,60)(32,71,91,55)(33,66,92,50)(34,61,93,57)(35,68,94,52)(36,63,95,59), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,85)(21,86)(22,87)(23,88)(24,89)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,58)(38,59)(39,60)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57)>;
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), (1,16,48,76)(2,21,37,81)(3,14,38,74)(4,19,39,79)(5,24,40,84)(6,17,41,77)(7,22,42,82)(8,15,43,75)(9,20,44,80)(10,13,45,73)(11,18,46,78)(12,23,47,83)(25,64,96,60)(26,69,85,53)(27,62,86,58)(28,67,87,51)(29,72,88,56)(30,65,89,49)(31,70,90,54)(32,63,91,59)(33,68,92,52)(34,61,93,57)(35,66,94,50)(36,71,95,55), (1,76,48,16)(2,83,37,23)(3,78,38,18)(4,73,39,13)(5,80,40,20)(6,75,41,15)(7,82,42,22)(8,77,43,17)(9,84,44,24)(10,79,45,19)(11,74,46,14)(12,81,47,21)(25,70,96,54)(26,65,85,49)(27,72,86,56)(28,67,87,51)(29,62,88,58)(30,69,89,53)(31,64,90,60)(32,71,91,55)(33,66,92,50)(34,61,93,57)(35,68,94,52)(36,63,95,59), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,85)(21,86)(22,87)(23,88)(24,89)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,58)(38,59)(39,60)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57) );
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)], [(1,16,48,76),(2,21,37,81),(3,14,38,74),(4,19,39,79),(5,24,40,84),(6,17,41,77),(7,22,42,82),(8,15,43,75),(9,20,44,80),(10,13,45,73),(11,18,46,78),(12,23,47,83),(25,64,96,60),(26,69,85,53),(27,62,86,58),(28,67,87,51),(29,72,88,56),(30,65,89,49),(31,70,90,54),(32,63,91,59),(33,68,92,52),(34,61,93,57),(35,66,94,50),(36,71,95,55)], [(1,76,48,16),(2,83,37,23),(3,78,38,18),(4,73,39,13),(5,80,40,20),(6,75,41,15),(7,82,42,22),(8,77,43,17),(9,84,44,24),(10,79,45,19),(11,74,46,14),(12,81,47,21),(25,70,96,54),(26,65,85,49),(27,72,86,56),(28,67,87,51),(29,62,88,58),(30,69,89,53),(31,64,90,60),(32,71,91,55),(33,66,92,50),(34,61,93,57),(35,68,94,52),(36,63,95,59)], [(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,90),(14,91),(15,92),(16,93),(17,94),(18,95),(19,96),(20,85),(21,86),(22,87),(23,88),(24,89),(25,79),(26,80),(27,81),(28,82),(29,83),(30,84),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,58),(38,59),(39,60),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54),(46,55),(47,56),(48,57)]])
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 |
Matrix representation of C12⋊(C4○D4) ►in 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] >;
C12⋊(C4○D4) 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