metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.19D8, C12.19SD16, C42.224D6, C4⋊Q8⋊6S3, C4⋊C4.84D6, C6.61(C2×D8), C4.7(D4⋊S3), C3⋊4(C4.4D8), C6.D8⋊42C2, (C2×C12).158D4, C4⋊D12.9C2, C6.78(C2×SD16), C12.83(C4○D4), C4.5(Q8⋊2S3), (C2×C12).406C23, (C4×C12).135C22, C4.16(Q8⋊3S3), C6.58(C4.4D4), (C2×D12).109C22, C2.11(C12.23D4), (C4×C3⋊C8)⋊18C2, (C3×C4⋊Q8)⋊6C2, C2.16(C2×D4⋊S3), (C2×C6).537(C2×D4), (C2×C3⋊C8).263C22, C2.16(C2×Q8⋊2S3), (C2×C4).137(C3⋊D4), (C3×C4⋊C4).131C22, (C2×C4).503(C22×S3), C22.209(C2×C3⋊D4), SmallGroup(192,647)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.D8
G = < a,b,c | a12=b8=c2=1, bab-1=a5, cac=a-1, cbc=a6b-1 >
Subgroups: 432 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C12, C12, D6, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, C3⋊C8, D12, C2×C12, C2×C12, C3×Q8, C22×S3, C4×C8, D4⋊C4, C4⋊1D4, C4⋊Q8, C2×C3⋊C8, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×D12, C2×D12, C6×Q8, C4.4D8, C4×C3⋊C8, C6.D8, C4⋊D12, C3×C4⋊Q8, C12.D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, SD16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, C2×D8, C2×SD16, D4⋊S3, Q8⋊2S3, Q8⋊3S3, C2×C3⋊D4, C4.4D8, C2×D4⋊S3, C2×Q8⋊2S3, C12.23D4, C12.D8
►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 33 46 51 81 66 96 20)(2 26 47 56 82 71 85 13)(3 31 48 49 83 64 86 18)(4 36 37 54 84 69 87 23)(5 29 38 59 73 62 88 16)(6 34 39 52 74 67 89 21)(7 27 40 57 75 72 90 14)(8 32 41 50 76 65 91 19)(9 25 42 55 77 70 92 24)(10 30 43 60 78 63 93 17)(11 35 44 53 79 68 94 22)(12 28 45 58 80 61 95 15)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 36)(24 35)(37 93)(38 92)(39 91)(40 90)(41 89)(42 88)(43 87)(44 86)(45 85)(46 96)(47 95)(48 94)(49 62)(50 61)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(73 77)(74 76)(78 84)(79 83)(80 82)
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), (1,33,46,51,81,66,96,20)(2,26,47,56,82,71,85,13)(3,31,48,49,83,64,86,18)(4,36,37,54,84,69,87,23)(5,29,38,59,73,62,88,16)(6,34,39,52,74,67,89,21)(7,27,40,57,75,72,90,14)(8,32,41,50,76,65,91,19)(9,25,42,55,77,70,92,24)(10,30,43,60,78,63,93,17)(11,35,44,53,79,68,94,22)(12,28,45,58,80,61,95,15), (2,12)(3,11)(4,10)(5,9)(6,8)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,96)(47,95)(48,94)(49,62)(50,61)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(73,77)(74,76)(78,84)(79,83)(80,82)>;
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), (1,33,46,51,81,66,96,20)(2,26,47,56,82,71,85,13)(3,31,48,49,83,64,86,18)(4,36,37,54,84,69,87,23)(5,29,38,59,73,62,88,16)(6,34,39,52,74,67,89,21)(7,27,40,57,75,72,90,14)(8,32,41,50,76,65,91,19)(9,25,42,55,77,70,92,24)(10,30,43,60,78,63,93,17)(11,35,44,53,79,68,94,22)(12,28,45,58,80,61,95,15), (2,12)(3,11)(4,10)(5,9)(6,8)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,96)(47,95)(48,94)(49,62)(50,61)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(73,77)(74,76)(78,84)(79,83)(80,82) );
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)], [(1,33,46,51,81,66,96,20),(2,26,47,56,82,71,85,13),(3,31,48,49,83,64,86,18),(4,36,37,54,84,69,87,23),(5,29,38,59,73,62,88,16),(6,34,39,52,74,67,89,21),(7,27,40,57,75,72,90,14),(8,32,41,50,76,65,91,19),(9,25,42,55,77,70,92,24),(10,30,43,60,78,63,93,17),(11,35,44,53,79,68,94,22),(12,28,45,58,80,61,95,15)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,36),(24,35),(37,93),(38,92),(39,91),(40,90),(41,89),(42,88),(43,87),(44,86),(45,85),(46,96),(47,95),(48,94),(49,62),(50,61),(51,72),(52,71),(53,70),(54,69),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63),(73,77),(74,76),(78,84),(79,83),(80,82)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 24 | 24 | 2 | 2 | ··· | 2 | 8 | 8 | 2 | 2 | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D8 | SD16 | C4○D4 | C3⋊D4 | D4⋊S3 | Q8⋊2S3 | Q8⋊3S3 |
| kernel | C12.D8 | C4×C3⋊C8 | C6.D8 | C4⋊D12 | C3×C4⋊Q8 | C4⋊Q8 | C2×C12 | C42 | C4⋊C4 | C12 | C12 | C12 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 |
Matrix representation of C12.D8 ►in GL6(𝔽73)
| 0 | 27 | 0 | 0 | 0 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 57 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [0,27,0,0,0,0,27,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,57,16],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;
C12.D8 in GAP, Magma, Sage, TeX
C_{12}.D_8 % in TeX
G:=Group("C12.D8"); // GroupNames label
G:=SmallGroup(192,647);
// by ID
G=gap.SmallGroup(192,647);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,219,100,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=a^6*b^-1>;
// generators/relations