metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊8Q8, Dic6⋊12D4, C42.173D6, C6.362- (1+4), C3⋊4(D4×Q8), C4⋊1(S3×Q8), C4⋊Q8⋊11S3, D6⋊7(C2×Q8), C12⋊3(C2×Q8), C12⋊Q8⋊44C2, C4.74(S3×D4), C4⋊C4.218D6, C12.72(C2×D4), D6⋊Q8⋊48C2, D6⋊3Q8⋊36C2, (C4×Dic6)⋊52C2, (C4×D12).26C2, (C2×Q8).170D6, C6.47(C22×Q8), (C2×C6).271C24, Dic3.30(C2×D4), C6.101(C22×D4), Dic3⋊5D4.13C2, (C4×C12).212C22, (C2×C12).104C23, D6⋊C4.152C22, (C6×Q8).138C22, (C2×D12).272C22, C4⋊Dic3.385C22, C22.292(S3×C23), Dic3⋊C4.166C22, (C22×S3).232C23, C2.37(Q8.15D6), (C2×Dic3).142C23, (C2×Dic6).189C22, (C4×Dic3).160C22, (C2×S3×Q8)⋊13C2, C2.74(C2×S3×D4), C2.30(C2×S3×Q8), (C3×C4⋊Q8)⋊13C2, (S3×C2×C4).145C22, (C3×C4⋊C4).214C22, (C2×C4).218(C22×S3), SmallGroup(192,1286)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 672 in 280 conjugacy classes, 115 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×13], C22, C22 [×8], S3 [×4], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×18], D4 [×4], Q8 [×16], C23 [×2], Dic3 [×4], Dic3 [×4], C12 [×4], C12 [×5], D6 [×4], D6 [×4], C2×C6, C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×6], C2×D4, C2×Q8 [×2], C2×Q8 [×13], Dic6 [×4], Dic6 [×8], C4×S3 [×12], D12 [×4], C2×Dic3 [×6], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×4], C22×S3 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C4⋊Q8, C4⋊Q8 [×2], C22×Q8 [×2], C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3 [×2], D6⋊C4 [×6], C4×C12, C3×C4⋊C4 [×4], C2×Dic6, C2×Dic6 [×4], S3×C2×C4 [×6], C2×D12, S3×Q8 [×8], C6×Q8 [×2], D4×Q8, C4×Dic6, C4×D12, C12⋊Q8 [×2], Dic3⋊5D4 [×2], D6⋊Q8 [×4], D6⋊3Q8 [×2], C3×C4⋊Q8, C2×S3×Q8 [×2], D12⋊8Q8
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], Q8 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C2×Q8 [×6], C24, C22×S3 [×7], C22×D4, C22×Q8, 2- (1+4), S3×D4 [×2], S3×Q8 [×2], S3×C23, D4×Q8, C2×S3×D4, C2×S3×Q8, Q8.15D6, D12⋊8Q8
Generators and relations
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=c-1 >
►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 3)(4 12)(5 11)(6 10)(7 9)(13 17)(14 16)(18 24)(19 23)(20 22)(25 27)(28 36)(29 35)(30 34)(31 33)(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 83)(74 82)(75 81)(76 80)(77 79)(85 89)(86 88)(90 96)(91 95)(92 94)
(1 86 46 25)(2 87 47 26)(3 88 48 27)(4 89 37 28)(5 90 38 29)(6 91 39 30)(7 92 40 31)(8 93 41 32)(9 94 42 33)(10 95 43 34)(11 96 44 35)(12 85 45 36)(13 60 82 64)(14 49 83 65)(15 50 84 66)(16 51 73 67)(17 52 74 68)(18 53 75 69)(19 54 76 70)(20 55 77 71)(21 56 78 72)(22 57 79 61)(23 58 80 62)(24 59 81 63)
(1 62 46 58)(2 69 47 53)(3 64 48 60)(4 71 37 55)(5 66 38 50)(6 61 39 57)(7 68 40 52)(8 63 41 59)(9 70 42 54)(10 65 43 49)(11 72 44 56)(12 67 45 51)(13 88 82 27)(14 95 83 34)(15 90 84 29)(16 85 73 36)(17 92 74 31)(18 87 75 26)(19 94 76 33)(20 89 77 28)(21 96 78 35)(22 91 79 30)(23 86 80 25)(24 93 81 32)
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,17)(14,16)(18,24)(19,23)(20,22)(25,27)(28,36)(29,35)(30,34)(31,33)(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)(73,83)(74,82)(75,81)(76,80)(77,79)(85,89)(86,88)(90,96)(91,95)(92,94), (1,86,46,25)(2,87,47,26)(3,88,48,27)(4,89,37,28)(5,90,38,29)(6,91,39,30)(7,92,40,31)(8,93,41,32)(9,94,42,33)(10,95,43,34)(11,96,44,35)(12,85,45,36)(13,60,82,64)(14,49,83,65)(15,50,84,66)(16,51,73,67)(17,52,74,68)(18,53,75,69)(19,54,76,70)(20,55,77,71)(21,56,78,72)(22,57,79,61)(23,58,80,62)(24,59,81,63), (1,62,46,58)(2,69,47,53)(3,64,48,60)(4,71,37,55)(5,66,38,50)(6,61,39,57)(7,68,40,52)(8,63,41,59)(9,70,42,54)(10,65,43,49)(11,72,44,56)(12,67,45,51)(13,88,82,27)(14,95,83,34)(15,90,84,29)(16,85,73,36)(17,92,74,31)(18,87,75,26)(19,94,76,33)(20,89,77,28)(21,96,78,35)(22,91,79,30)(23,86,80,25)(24,93,81,32)>;
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,17)(14,16)(18,24)(19,23)(20,22)(25,27)(28,36)(29,35)(30,34)(31,33)(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)(73,83)(74,82)(75,81)(76,80)(77,79)(85,89)(86,88)(90,96)(91,95)(92,94), (1,86,46,25)(2,87,47,26)(3,88,48,27)(4,89,37,28)(5,90,38,29)(6,91,39,30)(7,92,40,31)(8,93,41,32)(9,94,42,33)(10,95,43,34)(11,96,44,35)(12,85,45,36)(13,60,82,64)(14,49,83,65)(15,50,84,66)(16,51,73,67)(17,52,74,68)(18,53,75,69)(19,54,76,70)(20,55,77,71)(21,56,78,72)(22,57,79,61)(23,58,80,62)(24,59,81,63), (1,62,46,58)(2,69,47,53)(3,64,48,60)(4,71,37,55)(5,66,38,50)(6,61,39,57)(7,68,40,52)(8,63,41,59)(9,70,42,54)(10,65,43,49)(11,72,44,56)(12,67,45,51)(13,88,82,27)(14,95,83,34)(15,90,84,29)(16,85,73,36)(17,92,74,31)(18,87,75,26)(19,94,76,33)(20,89,77,28)(21,96,78,35)(22,91,79,30)(23,86,80,25)(24,93,81,32) );
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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,17),(14,16),(18,24),(19,23),(20,22),(25,27),(28,36),(29,35),(30,34),(31,33),(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),(73,83),(74,82),(75,81),(76,80),(77,79),(85,89),(86,88),(90,96),(91,95),(92,94)], [(1,86,46,25),(2,87,47,26),(3,88,48,27),(4,89,37,28),(5,90,38,29),(6,91,39,30),(7,92,40,31),(8,93,41,32),(9,94,42,33),(10,95,43,34),(11,96,44,35),(12,85,45,36),(13,60,82,64),(14,49,83,65),(15,50,84,66),(16,51,73,67),(17,52,74,68),(18,53,75,69),(19,54,76,70),(20,55,77,71),(21,56,78,72),(22,57,79,61),(23,58,80,62),(24,59,81,63)], [(1,62,46,58),(2,69,47,53),(3,64,48,60),(4,71,37,55),(5,66,38,50),(6,61,39,57),(7,68,40,52),(8,63,41,59),(9,70,42,54),(10,65,43,49),(11,72,44,56),(12,67,45,51),(13,88,82,27),(14,95,83,34),(15,90,84,29),(16,85,73,36),(17,92,74,31),(18,87,75,26),(19,94,76,33),(20,89,77,28),(21,96,78,35),(22,91,79,30),(23,86,80,25),(24,93,81,32)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,12,12],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | D6 | 2- (1+4) | S3×D4 | S3×Q8 | Q8.15D6 |
| kernel | D12⋊8Q8 | C4×Dic6 | C4×D12 | C12⋊Q8 | Dic3⋊5D4 | D6⋊Q8 | D6⋊3Q8 | C3×C4⋊Q8 | C2×S3×Q8 | C4⋊Q8 | Dic6 | D12 | C42 | C4⋊C4 | C2×Q8 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 1 | 4 | 4 | 1 | 4 | 2 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_{12}\rtimes_8Q_8 % in TeX
G:=Group("D12:8Q8"); // GroupNames label
G:=SmallGroup(192,1286);
// by ID
G=gap.SmallGroup(192,1286);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,100,1571,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations