metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊8D12, C24⋊13D4, D6⋊3SD16, C4.Q8⋊8S3, C3⋊3(C8⋊8D4), C4⋊C4.38D6, C4.D12⋊6C2, C4.50(C2×D12), (C2×C8).260D6, C12⋊D4.5C2, C6.55(C4○D8), C6.D8⋊15C2, C12.130(C2×D4), C6.40(C2×SD16), C2.24(S3×SD16), C12.29(C4○D4), C6.SD16⋊16C2, C6.43(C4⋊D4), C4.3(Q8⋊3S3), (C2×Dic3).99D4, (C22×S3).53D4, C22.216(S3×D4), C2.16(C12⋊D4), (C2×C12).280C23, (C2×C24).161C22, (C2×D12).74C22, C2.22(Q8.7D6), (C2×Dic6).83C22, (S3×C2×C8)⋊7C2, (C3×C4.Q8)⋊9C2, (C2×C24⋊C2)⋊28C2, (C2×C6).285(C2×D4), (C3×C4⋊C4).73C22, (C2×C3⋊C8).228C22, (S3×C2×C4).233C22, (C2×C4).383(C22×S3), SmallGroup(192,423)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊8D12
G = < a,b,c | a8=b12=c2=1, bab-1=cac=a3, cbc=b-1 >
Subgroups: 416 in 124 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, S3×C8, C24⋊C2, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C2×D12, C8⋊8D4, C6.D8, C6.SD16, C3×C4.Q8, C12⋊D4, C4.D12, S3×C2×C8, C2×C24⋊C2, C8⋊8D12
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×SD16, C4○D8, C2×D12, S3×D4, Q8⋊3S3, C8⋊8D4, C12⋊D4, S3×SD16, Q8.7D6, C8⋊8D12
►On 96 points
(1 17 68 33 77 94 50 39)(2 34 51 18 78 40 69 95)(3 19 70 35 79 96 52 41)(4 36 53 20 80 42 71 85)(5 21 72 25 81 86 54 43)(6 26 55 22 82 44 61 87)(7 23 62 27 83 88 56 45)(8 28 57 24 84 46 63 89)(9 13 64 29 73 90 58 47)(10 30 59 14 74 48 65 91)(11 15 66 31 75 92 60 37)(12 32 49 16 76 38 67 93)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(37 86)(38 85)(39 96)(40 95)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(60 72)(73 83)(74 82)(75 81)(76 80)(77 79)
G:=sub<Sym(96)| (1,17,68,33,77,94,50,39)(2,34,51,18,78,40,69,95)(3,19,70,35,79,96,52,41)(4,36,53,20,80,42,71,85)(5,21,72,25,81,86,54,43)(6,26,55,22,82,44,61,87)(7,23,62,27,83,88,56,45)(8,28,57,24,84,46,63,89)(9,13,64,29,73,90,58,47)(10,30,59,14,74,48,65,91)(11,15,66,31,75,92,60,37)(12,32,49,16,76,38,67,93), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(37,86)(38,85)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,72)(73,83)(74,82)(75,81)(76,80)(77,79)>;
G:=Group( (1,17,68,33,77,94,50,39)(2,34,51,18,78,40,69,95)(3,19,70,35,79,96,52,41)(4,36,53,20,80,42,71,85)(5,21,72,25,81,86,54,43)(6,26,55,22,82,44,61,87)(7,23,62,27,83,88,56,45)(8,28,57,24,84,46,63,89)(9,13,64,29,73,90,58,47)(10,30,59,14,74,48,65,91)(11,15,66,31,75,92,60,37)(12,32,49,16,76,38,67,93), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(37,86)(38,85)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,72)(73,83)(74,82)(75,81)(76,80)(77,79) );
G=PermutationGroup([[(1,17,68,33,77,94,50,39),(2,34,51,18,78,40,69,95),(3,19,70,35,79,96,52,41),(4,36,53,20,80,42,71,85),(5,21,72,25,81,86,54,43),(6,26,55,22,82,44,61,87),(7,23,62,27,83,88,56,45),(8,28,57,24,84,46,63,89),(9,13,64,29,73,90,58,47),(10,30,59,14,74,48,65,91),(11,15,66,31,75,92,60,37),(12,32,49,16,76,38,67,93)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,27),(14,26),(15,25),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,28),(37,86),(38,85),(39,96),(40,95),(41,94),(42,93),(43,92),(44,91),(45,90),(46,89),(47,88),(48,87),(49,71),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61),(60,72),(73,83),(74,82),(75,81),(76,80),(77,79)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 24 | 2 | 2 | 2 | 6 | 6 | 8 | 8 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | D6 | D6 | C4○D4 | SD16 | D12 | C4○D8 | Q8⋊3S3 | S3×D4 | S3×SD16 | Q8.7D6 |
| kernel | C8⋊8D12 | C6.D8 | C6.SD16 | C3×C4.Q8 | C12⋊D4 | C4.D12 | S3×C2×C8 | C2×C24⋊C2 | C4.Q8 | C24 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C12 | D6 | C8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C8⋊8D12 ►in GL4(𝔽73) generated by
| 6 | 67 | 0 | 0 |
| 6 | 6 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 66 | 7 |
| 0 | 0 | 66 | 59 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [6,6,0,0,67,6,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,1,0,0,0,0,66,66,0,0,7,59],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,1,72] >;
C8⋊8D12 in GAP, Magma, Sage, TeX
C_8\rtimes_8D_{12} % in TeX
G:=Group("C8:8D12"); // GroupNames label
G:=SmallGroup(192,423);
// by ID
G=gap.SmallGroup(192,423);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,555,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^8=b^12=c^2=1,b*a*b^-1=c*a*c=a^3,c*b*c=b^-1>;
// generators/relations