metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.4C8, C16.18D6, Dic6.4C8, C24.73C23, C48.28C22, (C2×C16)⋊9S3, (S3×C16)⋊6C2, (C2×C48)⋊17C2, C3⋊1(D4○C16), C8.23(C4×S3), C4.10(S3×C8), D6.C8⋊7C2, D6.1(C2×C8), C3⋊D4.2C8, C8⋊S3.3C4, C24.66(C2×C4), C12.20(C2×C8), C8○D12.6C2, C4○D12.6C4, (C2×C8).325D6, C22.2(S3×C8), C3⋊C16.11C22, C6.14(C22×C8), C8.59(C22×S3), Dic3.3(C2×C8), C12.C8⋊15C2, C4.Dic3.7C4, (S3×C8).17C22, C12.130(C22×C4), (C2×C24).429C22, C2.15(S3×C2×C8), C3⋊C8.13(C2×C4), C4.104(S3×C2×C4), (C2×C6).16(C2×C8), (C4×S3).21(C2×C4), (C2×C4).105(C4×S3), (C2×C12).232(C2×C4), SmallGroup(192,460)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.4C8
G = < a,b,c | a12=b2=1, c8=a6, bab=a-1, ac=ca, bc=cb >
Subgroups: 152 in 84 conjugacy classes, 51 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C16, C16, C2×C8, C2×C8, M4(2), C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C16, C2×C16, M5(2), C8○D4, C3⋊C16, C48, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C4○D12, D4○C16, S3×C16, D6.C8, C12.C8, C2×C48, C8○D12, D12.4C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, C22×C4, C4×S3, C22×S3, C22×C8, S3×C8, S3×C2×C4, D4○C16, S3×C2×C8, D12.4C8
►On 96 points
(1 94 57 25 66 39 9 86 49 17 74 47)(2 95 58 26 67 40 10 87 50 18 75 48)(3 96 59 27 68 41 11 88 51 19 76 33)(4 81 60 28 69 42 12 89 52 20 77 34)(5 82 61 29 70 43 13 90 53 21 78 35)(6 83 62 30 71 44 14 91 54 22 79 36)(7 84 63 31 72 45 15 92 55 23 80 37)(8 85 64 32 73 46 16 93 56 24 65 38)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)
(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)
G:=sub<Sym(96)| (1,94,57,25,66,39,9,86,49,17,74,47)(2,95,58,26,67,40,10,87,50,18,75,48)(3,96,59,27,68,41,11,88,51,19,76,33)(4,81,60,28,69,42,12,89,52,20,77,34)(5,82,61,29,70,43,13,90,53,21,78,35)(6,83,62,30,71,44,14,91,54,22,79,36)(7,84,63,31,72,45,15,92,55,23,80,37)(8,85,64,32,73,46,16,93,56,24,65,38), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80), (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)>;
G:=Group( (1,94,57,25,66,39,9,86,49,17,74,47)(2,95,58,26,67,40,10,87,50,18,75,48)(3,96,59,27,68,41,11,88,51,19,76,33)(4,81,60,28,69,42,12,89,52,20,77,34)(5,82,61,29,70,43,13,90,53,21,78,35)(6,83,62,30,71,44,14,91,54,22,79,36)(7,84,63,31,72,45,15,92,55,23,80,37)(8,85,64,32,73,46,16,93,56,24,65,38), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80), (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) );
G=PermutationGroup([[(1,94,57,25,66,39,9,86,49,17,74,47),(2,95,58,26,67,40,10,87,50,18,75,48),(3,96,59,27,68,41,11,88,51,19,76,33),(4,81,60,28,69,42,12,89,52,20,77,34),(5,82,61,29,70,43,13,90,53,21,78,35),(6,83,62,30,71,44,14,91,54,22,79,36),(7,84,63,31,72,45,15,92,55,23,80,37),(8,85,64,32,73,46,16,93,56,24,65,38)], [(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(27,33),(28,34),(29,35),(30,36),(31,37),(32,38),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80)], [(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)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | 12B | 12C | 12D | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 16M | ··· | 16T | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 6 | 6 | 2 | 1 | 1 | 2 | 6 | 6 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | S3 | D6 | D6 | C4×S3 | C4×S3 | S3×C8 | S3×C8 | D4○C16 | D12.4C8 |
| kernel | D12.4C8 | S3×C16 | D6.C8 | C12.C8 | C2×C48 | C8○D12 | C8⋊S3 | C4.Dic3 | C4○D12 | Dic6 | D12 | C3⋊D4 | C2×C16 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of D12.4C8 ►in GL2(𝔽97) generated by
| 68 | 29 |
| 68 | 39 |
| 96 | 0 |
| 1 | 1 |
| 18 | 0 |
| 0 | 18 |
G:=sub<GL(2,GF(97))| [68,68,29,39],[96,1,0,1],[18,0,0,18] >;
D12.4C8 in GAP, Magma, Sage, TeX
D_{12}._4C_8 % in TeX
G:=Group("D12.4C8"); // GroupNames label
G:=SmallGroup(192,460);
// by ID
G=gap.SmallGroup(192,460);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,58,80,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^2=1,c^8=a^6,b*a*b=a^-1,a*c=c*a,b*c=c*b>;
// generators/relations