metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.13D12, Q8.18D12, C24.12C23, C12.63C24, D24.14C22, D12.26C23, M4(2).28D6, Dic6.26C23, Dic12.10C22, C8○D4⋊9S3, C3⋊1(Q8○D8), Q8○D12⋊4C2, C4○D24⋊13C2, C4○D4.56D6, (C3×D4).25D4, C4.29(C2×D12), (C2×C8).102D6, C12.75(C2×D4), (C3×Q8).25D4, C8.D6⋊12C2, C4.60(S3×C23), C8.54(C22×S3), C6.30(C22×D4), C22.5(C2×D12), (C2×Dic12)⋊15C2, C24⋊C2.2C22, (C2×C24).70C22, C2.32(C22×D12), (C2×C12).517C23, C4○D12.27C22, (C2×Dic6).194C22, (C3×M4(2)).30C22, (C3×C8○D4)⋊5C2, (C2×C6).10(C2×D4), (C2×C4).228(C22×S3), (C3×C4○D4).47C22, SmallGroup(192,1312)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.13D12
G = < a,b,c,d | a4=b2=d2=1, c12=a2, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=a2c11 >
Subgroups: 624 in 248 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, D6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C4○D4, C4○D4, C24, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, C24⋊C2, D24, Dic12, C2×C24, C3×M4(2), C2×Dic6, C4○D12, D4⋊2S3, S3×Q8, C3×C4○D4, Q8○D8, C4○D24, C2×Dic12, C8.D6, C3×C8○D4, Q8○D12, D4.13D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22×D4, C2×D12, S3×C23, Q8○D8, C22×D12, D4.13D12
►On 96 points
Generators in S96
(1 28 13 40)(2 29 14 41)(3 30 15 42)(4 31 16 43)(5 32 17 44)(6 33 18 45)(7 34 19 46)(8 35 20 47)(9 36 21 48)(10 37 22 25)(11 38 23 26)(12 39 24 27)(49 76 61 88)(50 77 62 89)(51 78 63 90)(52 79 64 91)(53 80 65 92)(54 81 66 93)(55 82 67 94)(56 83 68 95)(57 84 69 96)(58 85 70 73)(59 86 71 74)(60 87 72 75)
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 85)(23 86)(24 87)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 30)(26 29)(27 28)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(81 82)(91 96)(92 95)(93 94)
G:=sub<Sym(96)| (1,28,13,40)(2,29,14,41)(3,30,15,42)(4,31,16,43)(5,32,17,44)(6,33,18,45)(7,34,19,46)(8,35,20,47)(9,36,21,48)(10,37,22,25)(11,38,23,26)(12,39,24,27)(49,76,61,88)(50,77,62,89)(51,78,63,90)(52,79,64,91)(53,80,65,92)(54,81,66,93)(55,82,67,94)(56,83,68,95)(57,84,69,96)(58,85,70,73)(59,86,71,74)(60,87,72,75), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,30)(26,29)(27,28)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)(91,96)(92,95)(93,94)>;
G:=Group( (1,28,13,40)(2,29,14,41)(3,30,15,42)(4,31,16,43)(5,32,17,44)(6,33,18,45)(7,34,19,46)(8,35,20,47)(9,36,21,48)(10,37,22,25)(11,38,23,26)(12,39,24,27)(49,76,61,88)(50,77,62,89)(51,78,63,90)(52,79,64,91)(53,80,65,92)(54,81,66,93)(55,82,67,94)(56,83,68,95)(57,84,69,96)(58,85,70,73)(59,86,71,74)(60,87,72,75), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,30)(26,29)(27,28)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)(91,96)(92,95)(93,94) );
G=PermutationGroup([[(1,28,13,40),(2,29,14,41),(3,30,15,42),(4,31,16,43),(5,32,17,44),(6,33,18,45),(7,34,19,46),(8,35,20,47),(9,36,21,48),(10,37,22,25),(11,38,23,26),(12,39,24,27),(49,76,61,88),(50,77,62,89),(51,78,63,90),(52,79,64,91),(53,80,65,92),(54,81,66,93),(55,82,67,94),(56,83,68,95),(57,84,69,96),(58,85,70,73),(59,86,71,74),(60,87,72,75)], [(1,88),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,85),(23,86),(24,87),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54),(46,55),(47,56),(48,57)], [(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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13),(25,30),(26,29),(27,28),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85),(79,84),(80,83),(81,82),(91,96),(92,95),(93,94)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D | 24E | ··· | 24J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D12 | D12 | Q8○D8 | D4.13D12 |
| kernel | D4.13D12 | C4○D24 | C2×Dic12 | C8.D6 | C3×C8○D4 | Q8○D12 | C8○D4 | C3×D4 | C3×Q8 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 1 | 3 | 1 | 3 | 3 | 1 | 6 | 2 | 2 | 4 |
Matrix representation of D4.13D12 ►in GL4(𝔽73) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 42 | 11 | 42 | 11 |
| 62 | 31 | 62 | 31 |
| 42 | 11 | 31 | 62 |
| 62 | 31 | 11 | 42 |
| 50 | 55 | 0 | 0 |
| 18 | 68 | 0 | 0 |
| 0 | 0 | 50 | 55 |
| 0 | 0 | 18 | 68 |
| 23 | 18 | 0 | 0 |
| 68 | 50 | 0 | 0 |
| 0 | 0 | 23 | 18 |
| 0 | 0 | 68 | 50 |
G:=sub<GL(4,GF(73))| [0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0],[42,62,42,62,11,31,11,31,42,62,31,11,11,31,62,42],[50,18,0,0,55,68,0,0,0,0,50,18,0,0,55,68],[23,68,0,0,18,50,0,0,0,0,23,68,0,0,18,50] >;
D4.13D12 in GAP, Magma, Sage, TeX
D_4._{13}D_{12} % in TeX
G:=Group("D4.13D12"); // GroupNames label
G:=SmallGroup(192,1312);
// by ID
G=gap.SmallGroup(192,1312);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,675,192,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^12=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=a^2*c^11>;
// generators/relations