metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊17D4, Dic6⋊16D4, C4⋊D4⋊2S3, C4⋊C4.57D6, C4.99(S3×D4), C3⋊4(D4⋊D4), (C2×D4).37D6, C6.45C22≀C2, C6.96(C4○D8), C12.146(C2×D4), C6.D8⋊34C2, (C2×C12).262D4, (C22×C6).83D4, C6.SD16⋊33C2, C6.90(C8⋊C22), (C6×D4).53C22, (C22×C4).136D6, C12.55D4⋊11C2, C2.13(C23⋊2D6), (C2×C12).356C23, C23.30(C3⋊D4), C2.15(Q8.13D6), C2.11(D12⋊6C22), (C2×D12).241C22, (C22×C12).160C22, (C2×Dic6).268C22, (C2×D4⋊S3)⋊9C2, (C3×C4⋊D4)⋊2C2, (C2×D4.S3)⋊8C2, (C2×C4○D12)⋊15C2, (C2×C6).487(C2×D4), (C2×C3⋊C8).108C22, (C2×C4).171(C3⋊D4), (C3×C4⋊C4).104C22, (C2×C4).456(C22×S3), C22.162(C2×C3⋊D4), SmallGroup(192,596)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊17D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=a7, ad=da, cbc-1=a9b, dbd=a6b, dcd=c-1 >
Subgroups: 496 in 162 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊C8, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C2×C3⋊C8, D4⋊S3, D4.S3, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, D4⋊D4, C6.D8, C6.SD16, C12.55D4, C2×D4⋊S3, C2×D4.S3, C3×C4⋊D4, C2×C4○D12, D12⋊17D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, C4○D8, C8⋊C22, S3×D4, C2×C3⋊D4, D4⋊D4, D12⋊6C22, C23⋊2D6, Q8.13D6, D12⋊17D4
►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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 35)(26 34)(27 33)(28 32)(29 31)(37 45)(38 44)(39 43)(40 42)(46 48)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 75)(76 84)(77 83)(78 82)(79 81)(85 89)(86 88)(90 96)(91 95)(92 94)
(1 73 53 86)(2 80 54 93)(3 75 55 88)(4 82 56 95)(5 77 57 90)(6 84 58 85)(7 79 59 92)(8 74 60 87)(9 81 49 94)(10 76 50 89)(11 83 51 96)(12 78 52 91)(13 37 70 32)(14 44 71 27)(15 39 72 34)(16 46 61 29)(17 41 62 36)(18 48 63 31)(19 43 64 26)(20 38 65 33)(21 45 66 28)(22 40 67 35)(23 47 68 30)(24 42 69 25)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 84)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 81)(35 82)(36 83)(37 92)(38 93)(39 94)(40 95)(41 96)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 72)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,35)(26,34)(27,33)(28,32)(29,31)(37,45)(38,44)(39,43)(40,42)(46,48)(49,56)(50,55)(51,54)(52,53)(57,60)(58,59)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,75)(76,84)(77,83)(78,82)(79,81)(85,89)(86,88)(90,96)(91,95)(92,94), (1,73,53,86)(2,80,54,93)(3,75,55,88)(4,82,56,95)(5,77,57,90)(6,84,58,85)(7,79,59,92)(8,74,60,87)(9,81,49,94)(10,76,50,89)(11,83,51,96)(12,78,52,91)(13,37,70,32)(14,44,71,27)(15,39,72,34)(16,46,61,29)(17,41,62,36)(18,48,63,31)(19,43,64,26)(20,38,65,33)(21,45,66,28)(22,40,67,35)(23,47,68,30)(24,42,69,25), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,84)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,81)(35,82)(36,83)(37,92)(38,93)(39,94)(40,95)(41,96)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,72)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)>;
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,35)(26,34)(27,33)(28,32)(29,31)(37,45)(38,44)(39,43)(40,42)(46,48)(49,56)(50,55)(51,54)(52,53)(57,60)(58,59)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,75)(76,84)(77,83)(78,82)(79,81)(85,89)(86,88)(90,96)(91,95)(92,94), (1,73,53,86)(2,80,54,93)(3,75,55,88)(4,82,56,95)(5,77,57,90)(6,84,58,85)(7,79,59,92)(8,74,60,87)(9,81,49,94)(10,76,50,89)(11,83,51,96)(12,78,52,91)(13,37,70,32)(14,44,71,27)(15,39,72,34)(16,46,61,29)(17,41,62,36)(18,48,63,31)(19,43,64,26)(20,38,65,33)(21,45,66,28)(22,40,67,35)(23,47,68,30)(24,42,69,25), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,84)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,81)(35,82)(36,83)(37,92)(38,93)(39,94)(40,95)(41,96)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,72)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71) );
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22),(25,35),(26,34),(27,33),(28,32),(29,31),(37,45),(38,44),(39,43),(40,42),(46,48),(49,56),(50,55),(51,54),(52,53),(57,60),(58,59),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(73,75),(76,84),(77,83),(78,82),(79,81),(85,89),(86,88),(90,96),(91,95),(92,94)], [(1,73,53,86),(2,80,54,93),(3,75,55,88),(4,82,56,95),(5,77,57,90),(6,84,58,85),(7,79,59,92),(8,74,60,87),(9,81,49,94),(10,76,50,89),(11,83,51,96),(12,78,52,91),(13,37,70,32),(14,44,71,27),(15,39,72,34),(16,46,61,29),(17,41,62,36),(18,48,63,31),(19,43,64,26),(20,38,65,33),(21,45,66,28),(22,40,67,35),(23,47,68,30),(24,42,69,25)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,84),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(33,80),(34,81),(35,82),(36,83),(37,92),(38,93),(39,94),(40,95),(41,96),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,72),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 8 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | D6 | C3⋊D4 | C3⋊D4 | C4○D8 | C8⋊C22 | S3×D4 | D12⋊6C22 | Q8.13D6 |
| kernel | D12⋊17D4 | C6.D8 | C6.SD16 | C12.55D4 | C2×D4⋊S3 | C2×D4.S3 | C3×C4⋊D4 | C2×C4○D12 | C4⋊D4 | Dic6 | D12 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×D4 | C2×C4 | C23 | C6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of D12⋊17D4 ►in GL6(𝔽73)
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 48 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 48 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 4 | 0 | 0 |
| 0 | 0 | 55 | 61 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 17 |
| 0 | 0 | 0 | 0 | 56 | 19 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 55 | 0 | 0 |
| 0 | 0 | 8 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 56 | 19 |
| 0 | 0 | 0 | 0 | 54 | 17 |
G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,3,0,0,0,0,48,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,48,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,12,55,0,0,0,0,4,61,0,0,0,0,0,0,54,56,0,0,0,0,17,19],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,8,0,0,0,0,55,27,0,0,0,0,0,0,56,54,0,0,0,0,19,17] >;
D12⋊17D4 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{17}D_4 % in TeX
G:=Group("D12:17D4"); // GroupNames label
G:=SmallGroup(192,596);
// by ID
G=gap.SmallGroup(192,596);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^9*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations