metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.17D4, D6⋊C8⋊19C2, (C2×Q16)⋊4S3, (C2×C8).39D6, C4.68(S3×D4), D6⋊3Q8⋊7C2, (C6×Q16)⋊14C2, C12.53(C2×D4), (C2×Q8).87D6, (C3×Q8).10D4, C6.62C22≀C2, C6.80(C4○D8), C2.D24⋊18C2, C3⋊6(D4.7D4), Q8⋊2Dic3⋊34C2, Q8.16(C3⋊D4), (C22×S3).40D4, C22.278(S3×D4), (C6×Q8).90C22, C2.30(C23⋊2D6), (C2×C24).181C22, (C2×C12).461C23, (C2×Dic3).188D4, C2.29(Q16⋊S3), C6.79(C8.C22), C2.17(D24⋊C2), (C2×D12).125C22, C4⋊Dic3.184C22, C4.49(C2×C3⋊D4), (C2×C6).372(C2×D4), (S3×C2×C4).53C22, (C2×Q8⋊2S3)⋊20C2, (C2×C3⋊C8).166C22, (C2×Q8⋊3S3).5C2, (C2×C4).549(C22×S3), SmallGroup(192,746)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.17D4
G = < a,b,c,d | a12=b2=c4=1, d2=a6, bab=cac-1=a-1, dad-1=a7, cbc-1=a7b, dbd-1=a9b, dcd-1=a6c-1 >
Subgroups: 472 in 152 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C22×S3, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, D6⋊C4, Q8⋊2S3, C2×C24, C3×Q16, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, Q8⋊3S3, C6×Q8, D4.7D4, D6⋊C8, C2.D24, Q8⋊2Dic3, C2×Q8⋊2S3, D6⋊3Q8, C6×Q16, C2×Q8⋊3S3, 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, D4.7D4, Q16⋊S3, D24⋊C2, C23⋊2D6, 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 22)(14 21)(15 20)(16 19)(17 18)(23 24)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 45)(38 44)(39 43)(40 42)(46 48)(49 51)(52 60)(53 59)(54 58)(55 57)(61 71)(62 70)(63 69)(64 68)(65 67)(73 78)(74 77)(75 76)(79 84)(80 83)(81 82)(86 96)(87 95)(88 94)(89 93)(90 92)
(1 62 31 37)(2 61 32 48)(3 72 33 47)(4 71 34 46)(5 70 35 45)(6 69 36 44)(7 68 25 43)(8 67 26 42)(9 66 27 41)(10 65 28 40)(11 64 29 39)(12 63 30 38)(13 86 83 57)(14 85 84 56)(15 96 73 55)(16 95 74 54)(17 94 75 53)(18 93 76 52)(19 92 77 51)(20 91 78 50)(21 90 79 49)(22 89 80 60)(23 88 81 59)(24 87 82 58)
(1 55 7 49)(2 50 8 56)(3 57 9 51)(4 52 10 58)(5 59 11 53)(6 54 12 60)(13 47 19 41)(14 42 20 48)(15 37 21 43)(16 44 22 38)(17 39 23 45)(18 46 24 40)(25 90 31 96)(26 85 32 91)(27 92 33 86)(28 87 34 93)(29 94 35 88)(30 89 36 95)(61 84 67 78)(62 79 68 73)(63 74 69 80)(64 81 70 75)(65 76 71 82)(66 83 72 77)
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)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,45)(38,44)(39,43)(40,42)(46,48)(49,51)(52,60)(53,59)(54,58)(55,57)(61,71)(62,70)(63,69)(64,68)(65,67)(73,78)(74,77)(75,76)(79,84)(80,83)(81,82)(86,96)(87,95)(88,94)(89,93)(90,92), (1,62,31,37)(2,61,32,48)(3,72,33,47)(4,71,34,46)(5,70,35,45)(6,69,36,44)(7,68,25,43)(8,67,26,42)(9,66,27,41)(10,65,28,40)(11,64,29,39)(12,63,30,38)(13,86,83,57)(14,85,84,56)(15,96,73,55)(16,95,74,54)(17,94,75,53)(18,93,76,52)(19,92,77,51)(20,91,78,50)(21,90,79,49)(22,89,80,60)(23,88,81,59)(24,87,82,58), (1,55,7,49)(2,50,8,56)(3,57,9,51)(4,52,10,58)(5,59,11,53)(6,54,12,60)(13,47,19,41)(14,42,20,48)(15,37,21,43)(16,44,22,38)(17,39,23,45)(18,46,24,40)(25,90,31,96)(26,85,32,91)(27,92,33,86)(28,87,34,93)(29,94,35,88)(30,89,36,95)(61,84,67,78)(62,79,68,73)(63,74,69,80)(64,81,70,75)(65,76,71,82)(66,83,72,77)>;
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)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,45)(38,44)(39,43)(40,42)(46,48)(49,51)(52,60)(53,59)(54,58)(55,57)(61,71)(62,70)(63,69)(64,68)(65,67)(73,78)(74,77)(75,76)(79,84)(80,83)(81,82)(86,96)(87,95)(88,94)(89,93)(90,92), (1,62,31,37)(2,61,32,48)(3,72,33,47)(4,71,34,46)(5,70,35,45)(6,69,36,44)(7,68,25,43)(8,67,26,42)(9,66,27,41)(10,65,28,40)(11,64,29,39)(12,63,30,38)(13,86,83,57)(14,85,84,56)(15,96,73,55)(16,95,74,54)(17,94,75,53)(18,93,76,52)(19,92,77,51)(20,91,78,50)(21,90,79,49)(22,89,80,60)(23,88,81,59)(24,87,82,58), (1,55,7,49)(2,50,8,56)(3,57,9,51)(4,52,10,58)(5,59,11,53)(6,54,12,60)(13,47,19,41)(14,42,20,48)(15,37,21,43)(16,44,22,38)(17,39,23,45)(18,46,24,40)(25,90,31,96)(26,85,32,91)(27,92,33,86)(28,87,34,93)(29,94,35,88)(30,89,36,95)(61,84,67,78)(62,79,68,73)(63,74,69,80)(64,81,70,75)(65,76,71,82)(66,83,72,77) );
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),(13,22),(14,21),(15,20),(16,19),(17,18),(23,24),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(37,45),(38,44),(39,43),(40,42),(46,48),(49,51),(52,60),(53,59),(54,58),(55,57),(61,71),(62,70),(63,69),(64,68),(65,67),(73,78),(74,77),(75,76),(79,84),(80,83),(81,82),(86,96),(87,95),(88,94),(89,93),(90,92)], [(1,62,31,37),(2,61,32,48),(3,72,33,47),(4,71,34,46),(5,70,35,45),(6,69,36,44),(7,68,25,43),(8,67,26,42),(9,66,27,41),(10,65,28,40),(11,64,29,39),(12,63,30,38),(13,86,83,57),(14,85,84,56),(15,96,73,55),(16,95,74,54),(17,94,75,53),(18,93,76,52),(19,92,77,51),(20,91,78,50),(21,90,79,49),(22,89,80,60),(23,88,81,59),(24,87,82,58)], [(1,55,7,49),(2,50,8,56),(3,57,9,51),(4,52,10,58),(5,59,11,53),(6,54,12,60),(13,47,19,41),(14,42,20,48),(15,37,21,43),(16,44,22,38),(17,39,23,45),(18,46,24,40),(25,90,31,96),(26,85,32,91),(27,92,33,86),(28,87,34,93),(29,94,35,88),(30,89,36,95),(61,84,67,78),(62,79,68,73),(63,74,69,80),(64,81,70,75),(65,76,71,82),(66,83,72,77)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 8 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | C3⋊D4 | C4○D8 | C8.C22 | S3×D4 | S3×D4 | Q16⋊S3 | D24⋊C2 |
| kernel | D12.17D4 | D6⋊C8 | C2.D24 | Q8⋊2Dic3 | C2×Q8⋊2S3 | D6⋊3Q8 | C6×Q16 | C2×Q8⋊3S3 | C2×Q16 | D12 | C2×Dic3 | C3×Q8 | C22×S3 | C2×C8 | C2×Q8 | Q8 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D12.17D4 ►in GL6(𝔽73)
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 52 | 0 | 0 |
| 0 | 0 | 2 | 48 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 57 |
| 0 | 0 | 0 | 0 | 57 | 57 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 48 | 20 | 0 | 0 |
| 0 | 0 | 71 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 0 | 6 | 67 |
G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[72,0,0,0,0,0,72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,25,2,0,0,0,0,52,48,0,0,0,0,0,0,16,57,0,0,0,0,57,57],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,48,71,0,0,0,0,20,25,0,0,0,0,0,0,6,6,0,0,0,0,6,67] >;
D12.17D4 in GAP, Magma, Sage, TeX
D_{12}._{17}D_4 % in TeX
G:=Group("D12.17D4"); // GroupNames label
G:=SmallGroup(192,746);
// by ID
G=gap.SmallGroup(192,746);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,758,135,184,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^6,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^7,c*b*c^-1=a^7*b,d*b*d^-1=a^9*b,d*c*d^-1=a^6*c^-1>;
// generators/relations