metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊12D4, C42.172D6, C6.352- (1+4), C4⋊Q8⋊10S3, C4.73(S3×D4), (C4×D12)⋊51C2, C12⋊7(C4○D4), C3⋊7(D4⋊6D4), C4⋊C4.123D6, D6.25(C2×D4), C12.71(C2×D4), C12⋊D4⋊40C2, D6⋊3Q8⋊35C2, C4⋊2(Q8⋊3S3), (C2×Q8).169D6, D6.D4⋊46C2, (C2×C6).270C24, C6.100(C22×D4), (C4×C12).211C22, (C2×C12).103C23, D6⋊C4.151C22, (C6×Q8).137C22, (C2×D12).271C22, Dic3⋊C4.60C22, C4⋊Dic3.384C22, C22.291(S3×C23), (C22×S3).231C23, C2.36(Q8.15D6), (C2×Dic3).141C23, (S3×C4⋊C4)⋊44C2, C2.73(C2×S3×D4), (C3×C4⋊Q8)⋊12C2, C6.121(C2×C4○D4), (C2×Q8⋊3S3)⋊13C2, (S3×C2×C4).144C22, (C2×C4).93(C22×S3), C2.28(C2×Q8⋊3S3), (C3×C4⋊C4).213C22, SmallGroup(192,1285)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 768 in 292 conjugacy classes, 107 normal (27 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×4], C4 [×9], C22, C22 [×14], S3 [×6], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×20], D4 [×14], Q8 [×4], C23 [×4], Dic3 [×4], C12 [×4], C12 [×5], D6 [×4], D6 [×10], C2×C6, C42, C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4 [×8], C2×D4 [×6], C2×Q8 [×2], C4○D4 [×8], C4×S3 [×16], D12 [×4], D12 [×10], C2×Dic3 [×4], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×4], C22×S3 [×4], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×8], C4×C12, C3×C4⋊C4 [×4], S3×C2×C4 [×8], C2×D12 [×2], C2×D12 [×4], Q8⋊3S3 [×8], C6×Q8 [×2], D4⋊6D4, C4×D12 [×2], S3×C4⋊C4 [×2], D6.D4 [×4], C12⋊D4 [×2], D6⋊3Q8 [×2], C3×C4⋊Q8, C2×Q8⋊3S3 [×2], D12⋊12D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2- (1+4), S3×D4 [×2], Q8⋊3S3 [×2], S3×C23, D4⋊6D4, C2×S3×D4, C2×Q8⋊3S3, Q8.15D6, D12⋊12D4
Generators and relations
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, dbd=a10b, dcd=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 25)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 48)(22 47)(23 46)(24 45)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 72)(58 71)(59 70)(60 69)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(81 96)(82 95)(83 94)(84 93)
(1 67 45 83)(2 68 46 84)(3 69 47 73)(4 70 48 74)(5 71 37 75)(6 72 38 76)(7 61 39 77)(8 62 40 78)(9 63 41 79)(10 64 42 80)(11 65 43 81)(12 66 44 82)(13 95 26 51)(14 96 27 52)(15 85 28 53)(16 86 29 54)(17 87 30 55)(18 88 31 56)(19 89 32 57)(20 90 33 58)(21 91 34 59)(22 92 35 60)(23 93 36 49)(24 94 25 50)
(1 45)(2 38)(3 43)(4 48)(5 41)(6 46)(7 39)(8 44)(9 37)(10 42)(11 47)(12 40)(13 32)(14 25)(15 30)(16 35)(17 28)(18 33)(19 26)(20 31)(21 36)(22 29)(23 34)(24 27)(49 59)(50 52)(51 57)(53 55)(54 60)(56 58)(62 66)(63 71)(65 69)(68 72)(73 81)(75 79)(76 84)(78 82)(85 87)(86 92)(88 90)(89 95)(91 93)(94 96)
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), (1,25)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,48)(22,47)(23,46)(24,45)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,72)(58,71)(59,70)(60,69)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,96)(82,95)(83,94)(84,93), (1,67,45,83)(2,68,46,84)(3,69,47,73)(4,70,48,74)(5,71,37,75)(6,72,38,76)(7,61,39,77)(8,62,40,78)(9,63,41,79)(10,64,42,80)(11,65,43,81)(12,66,44,82)(13,95,26,51)(14,96,27,52)(15,85,28,53)(16,86,29,54)(17,87,30,55)(18,88,31,56)(19,89,32,57)(20,90,33,58)(21,91,34,59)(22,92,35,60)(23,93,36,49)(24,94,25,50), (1,45)(2,38)(3,43)(4,48)(5,41)(6,46)(7,39)(8,44)(9,37)(10,42)(11,47)(12,40)(13,32)(14,25)(15,30)(16,35)(17,28)(18,33)(19,26)(20,31)(21,36)(22,29)(23,34)(24,27)(49,59)(50,52)(51,57)(53,55)(54,60)(56,58)(62,66)(63,71)(65,69)(68,72)(73,81)(75,79)(76,84)(78,82)(85,87)(86,92)(88,90)(89,95)(91,93)(94,96)>;
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), (1,25)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,48)(22,47)(23,46)(24,45)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,72)(58,71)(59,70)(60,69)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,96)(82,95)(83,94)(84,93), (1,67,45,83)(2,68,46,84)(3,69,47,73)(4,70,48,74)(5,71,37,75)(6,72,38,76)(7,61,39,77)(8,62,40,78)(9,63,41,79)(10,64,42,80)(11,65,43,81)(12,66,44,82)(13,95,26,51)(14,96,27,52)(15,85,28,53)(16,86,29,54)(17,87,30,55)(18,88,31,56)(19,89,32,57)(20,90,33,58)(21,91,34,59)(22,92,35,60)(23,93,36,49)(24,94,25,50), (1,45)(2,38)(3,43)(4,48)(5,41)(6,46)(7,39)(8,44)(9,37)(10,42)(11,47)(12,40)(13,32)(14,25)(15,30)(16,35)(17,28)(18,33)(19,26)(20,31)(21,36)(22,29)(23,34)(24,27)(49,59)(50,52)(51,57)(53,55)(54,60)(56,58)(62,66)(63,71)(65,69)(68,72)(73,81)(75,79)(76,84)(78,82)(85,87)(86,92)(88,90)(89,95)(91,93)(94,96) );
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)], [(1,25),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,48),(22,47),(23,46),(24,45),(49,68),(50,67),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,72),(58,71),(59,70),(60,69),(73,92),(74,91),(75,90),(76,89),(77,88),(78,87),(79,86),(80,85),(81,96),(82,95),(83,94),(84,93)], [(1,67,45,83),(2,68,46,84),(3,69,47,73),(4,70,48,74),(5,71,37,75),(6,72,38,76),(7,61,39,77),(8,62,40,78),(9,63,41,79),(10,64,42,80),(11,65,43,81),(12,66,44,82),(13,95,26,51),(14,96,27,52),(15,85,28,53),(16,86,29,54),(17,87,30,55),(18,88,31,56),(19,89,32,57),(20,90,33,58),(21,91,34,59),(22,92,35,60),(23,93,36,49),(24,94,25,50)], [(1,45),(2,38),(3,43),(4,48),(5,41),(6,46),(7,39),(8,44),(9,37),(10,42),(11,47),(12,40),(13,32),(14,25),(15,30),(16,35),(17,28),(18,33),(19,26),(20,31),(21,36),(22,29),(23,34),(24,27),(49,59),(50,52),(51,57),(53,55),(54,60),(56,58),(62,66),(63,71),(65,69),(68,72),(73,81),(75,79),(76,84),(78,82),(85,87),(86,92),(88,90),(89,95),(91,93),(94,96)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 2 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 5 | 1 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,2,8],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,5,0,0,0,0,0,1] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 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 | S3 | D4 | D6 | D6 | D6 | C4○D4 | 2- (1+4) | S3×D4 | Q8⋊3S3 | Q8.15D6 |
| kernel | D12⋊12D4 | C4×D12 | S3×C4⋊C4 | D6.D4 | C12⋊D4 | D6⋊3Q8 | C3×C4⋊Q8 | C2×Q8⋊3S3 | C4⋊Q8 | D12 | C42 | C4⋊C4 | C2×Q8 | C12 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 4 | 1 | 4 | 2 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_{12}\rtimes_{12}D_4 % in TeX
G:=Group("D12:12D4"); // GroupNames label
G:=SmallGroup(192,1285);
// by ID
G=gap.SmallGroup(192,1285);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,1571,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations