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, C23.145(C22×S3), (C22×C6).232C23, 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), (C22×C3⋊D4)⋊16C2, (C2×C3⋊D4)⋊47C22, 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
Generators and relations for C2×C12⋊3D4
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 >
Subgroups: 1448 in 498 conjugacy classes, 143 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C42, C4⋊1D4, C22×D4, C22×D4, C4×Dic3, C2×D12, C2×D12, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, S3×C23, C23×C6, C2×C4⋊1D4, C2×C4×Dic3, C12⋊3D4, C22×D12, C22×C3⋊D4, D4×C2×C6, C2×C12⋊3D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C4⋊1D4, C22×D4, S3×D4, C2×C3⋊D4, S3×C23, C2×C4⋊1D4, C12⋊3D4, C2×S3×D4, C22×C3⋊D4, C2×C12⋊3D4
►On 96 points
(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 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 49)(25 47)(26 48)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 85)(84 86)
(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 17 96 36)(2 22 85 29)(3 15 86 34)(4 20 87 27)(5 13 88 32)(6 18 89 25)(7 23 90 30)(8 16 91 35)(9 21 92 28)(10 14 93 33)(11 19 94 26)(12 24 95 31)(37 64 57 73)(38 69 58 78)(39 62 59 83)(40 67 60 76)(41 72 49 81)(42 65 50 74)(43 70 51 79)(44 63 52 84)(45 68 53 77)(46 61 54 82)(47 66 55 75)(48 71 56 80)
(1 90)(2 89)(3 88)(4 87)(5 86)(6 85)(7 96)(8 95)(9 94)(10 93)(11 92)(12 91)(13 15)(16 24)(17 23)(18 22)(19 21)(25 29)(26 28)(30 36)(31 35)(32 34)(38 48)(39 47)(40 46)(41 45)(42 44)(49 53)(50 52)(54 60)(55 59)(56 58)(61 76)(62 75)(63 74)(64 73)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)
G:=sub<Sym(96)| (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,49)(25,47)(26,48)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,85)(84,86), (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), (1,17,96,36)(2,22,85,29)(3,15,86,34)(4,20,87,27)(5,13,88,32)(6,18,89,25)(7,23,90,30)(8,16,91,35)(9,21,92,28)(10,14,93,33)(11,19,94,26)(12,24,95,31)(37,64,57,73)(38,69,58,78)(39,62,59,83)(40,67,60,76)(41,72,49,81)(42,65,50,74)(43,70,51,79)(44,63,52,84)(45,68,53,77)(46,61,54,82)(47,66,55,75)(48,71,56,80), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(13,15)(16,24)(17,23)(18,22)(19,21)(25,29)(26,28)(30,36)(31,35)(32,34)(38,48)(39,47)(40,46)(41,45)(42,44)(49,53)(50,52)(54,60)(55,59)(56,58)(61,76)(62,75)(63,74)(64,73)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)>;
G:=Group( (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,49)(25,47)(26,48)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,85)(84,86), (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), (1,17,96,36)(2,22,85,29)(3,15,86,34)(4,20,87,27)(5,13,88,32)(6,18,89,25)(7,23,90,30)(8,16,91,35)(9,21,92,28)(10,14,93,33)(11,19,94,26)(12,24,95,31)(37,64,57,73)(38,69,58,78)(39,62,59,83)(40,67,60,76)(41,72,49,81)(42,65,50,74)(43,70,51,79)(44,63,52,84)(45,68,53,77)(46,61,54,82)(47,66,55,75)(48,71,56,80), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(13,15)(16,24)(17,23)(18,22)(19,21)(25,29)(26,28)(30,36)(31,35)(32,34)(38,48)(39,47)(40,46)(41,45)(42,44)(49,53)(50,52)(54,60)(55,59)(56,58)(61,76)(62,75)(63,74)(64,73)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77) );
G=PermutationGroup([[(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,49),(25,47),(26,48),(27,37),(28,38),(29,39),(30,40),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,85),(84,86)], [(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)], [(1,17,96,36),(2,22,85,29),(3,15,86,34),(4,20,87,27),(5,13,88,32),(6,18,89,25),(7,23,90,30),(8,16,91,35),(9,21,92,28),(10,14,93,33),(11,19,94,26),(12,24,95,31),(37,64,57,73),(38,69,58,78),(39,62,59,83),(40,67,60,76),(41,72,49,81),(42,65,50,74),(43,70,51,79),(44,63,52,84),(45,68,53,77),(46,61,54,82),(47,66,55,75),(48,71,56,80)], [(1,90),(2,89),(3,88),(4,87),(5,86),(6,85),(7,96),(8,95),(9,94),(10,93),(11,92),(12,91),(13,15),(16,24),(17,23),(18,22),(19,21),(25,29),(26,28),(30,36),(31,35),(32,34),(38,48),(39,47),(40,46),(41,45),(42,44),(49,53),(50,52),(54,60),(55,59),(56,58),(61,76),(62,75),(63,74),(64,73),(65,84),(66,83),(67,82),(68,81),(69,80),(70,79),(71,78),(72,77)]])
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 |
Matrix representation of C2×C12⋊3D4 ►in 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] >;
C2×C12⋊3D4 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