metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊10Q8, C42.130D6, C6.1112+ (1+4), (C4×Q8)⋊16S3, D6.8(C2×Q8), C4.50(S3×Q8), C4⋊C4.326D6, D6⋊3Q8⋊8C2, (Q8×C12)⋊14C2, C3⋊4(D4⋊3Q8), D6⋊Q8⋊11C2, C12⋊2Q8⋊28C2, (C4×D12).21C2, (C2×Q8).202D6, C12.108(C2×Q8), C4.67(C4○D12), C2.23(D4○D12), C6.31(C22×Q8), (C2×C6).123C24, C12.6Q8⋊18C2, C12.118(C4○D4), (C2×C12).590C23, (C4×C12).175C22, D6⋊C4.103C22, (C6×Q8).223C22, (C2×D12).289C22, Dic3⋊C4.69C22, C4⋊Dic3.202C22, C22.144(S3×C23), (C2×Dic6).31C22, (C2×Dic3).55C23, (C22×S3).180C23, (S3×C4⋊C4)⋊18C2, C2.14(C2×S3×Q8), C6.55(C2×C4○D4), C2.62(C2×C4○D12), (S3×C2×C4).74C22, (C3×C4⋊C4).351C22, (C2×C4).169(C22×S3), SmallGroup(192,1138)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 552 in 228 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×11], C22, C22 [×8], S3 [×4], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×4], Q8 [×4], C23 [×2], Dic3 [×6], C12 [×4], C12 [×5], D6 [×4], D6 [×4], C2×C6, C42, C42 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×13], C22×C4 [×6], C2×D4, C2×Q8, C2×Q8 [×2], Dic6 [×2], C4×S3 [×8], D12 [×4], C2×Dic3 [×6], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×2], C22×S3 [×2], C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, Dic3⋊C4 [×8], C4⋊Dic3, C4⋊Dic3 [×4], D6⋊C4 [×6], C4×C12, C4×C12 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C2×Dic6 [×2], S3×C2×C4 [×6], C2×D12, C6×Q8, D4⋊3Q8, C12⋊2Q8, C12.6Q8 [×2], C4×D12, C4×D12 [×2], S3×C4⋊C4 [×2], D6⋊Q8 [×4], D6⋊3Q8 [×2], Q8×C12, D12⋊10Q8
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), C4○D12 [×2], S3×Q8 [×2], S3×C23, D4⋊3Q8, C2×C4○D12, C2×S3×Q8, D4○D12, D12⋊10Q8
Generators and relations
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a6b, bd=db, 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)(38 48)(39 47)(40 46)(41 45)(42 44)(49 51)(52 60)(53 59)(54 58)(55 57)(61 71)(62 70)(63 69)(64 68)(65 67)(73 77)(74 76)(78 84)(79 83)(80 82)(85 87)(88 96)(89 95)(90 94)(91 93)
(1 58 74 39)(2 59 75 40)(3 60 76 41)(4 49 77 42)(5 50 78 43)(6 51 79 44)(7 52 80 45)(8 53 81 46)(9 54 82 47)(10 55 83 48)(11 56 84 37)(12 57 73 38)(13 27 70 87)(14 28 71 88)(15 29 72 89)(16 30 61 90)(17 31 62 91)(18 32 63 92)(19 33 64 93)(20 34 65 94)(21 35 66 95)(22 36 67 96)(23 25 68 85)(24 26 69 86)
(1 14 74 71)(2 15 75 72)(3 16 76 61)(4 17 77 62)(5 18 78 63)(6 19 79 64)(7 20 80 65)(8 21 81 66)(9 22 82 67)(10 23 83 68)(11 24 84 69)(12 13 73 70)(25 55 85 48)(26 56 86 37)(27 57 87 38)(28 58 88 39)(29 59 89 40)(30 60 90 41)(31 49 91 42)(32 50 92 43)(33 51 93 44)(34 52 94 45)(35 53 95 46)(36 54 96 47)
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)(38,48)(39,47)(40,46)(41,45)(42,44)(49,51)(52,60)(53,59)(54,58)(55,57)(61,71)(62,70)(63,69)(64,68)(65,67)(73,77)(74,76)(78,84)(79,83)(80,82)(85,87)(88,96)(89,95)(90,94)(91,93), (1,58,74,39)(2,59,75,40)(3,60,76,41)(4,49,77,42)(5,50,78,43)(6,51,79,44)(7,52,80,45)(8,53,81,46)(9,54,82,47)(10,55,83,48)(11,56,84,37)(12,57,73,38)(13,27,70,87)(14,28,71,88)(15,29,72,89)(16,30,61,90)(17,31,62,91)(18,32,63,92)(19,33,64,93)(20,34,65,94)(21,35,66,95)(22,36,67,96)(23,25,68,85)(24,26,69,86), (1,14,74,71)(2,15,75,72)(3,16,76,61)(4,17,77,62)(5,18,78,63)(6,19,79,64)(7,20,80,65)(8,21,81,66)(9,22,82,67)(10,23,83,68)(11,24,84,69)(12,13,73,70)(25,55,85,48)(26,56,86,37)(27,57,87,38)(28,58,88,39)(29,59,89,40)(30,60,90,41)(31,49,91,42)(32,50,92,43)(33,51,93,44)(34,52,94,45)(35,53,95,46)(36,54,96,47)>;
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)(38,48)(39,47)(40,46)(41,45)(42,44)(49,51)(52,60)(53,59)(54,58)(55,57)(61,71)(62,70)(63,69)(64,68)(65,67)(73,77)(74,76)(78,84)(79,83)(80,82)(85,87)(88,96)(89,95)(90,94)(91,93), (1,58,74,39)(2,59,75,40)(3,60,76,41)(4,49,77,42)(5,50,78,43)(6,51,79,44)(7,52,80,45)(8,53,81,46)(9,54,82,47)(10,55,83,48)(11,56,84,37)(12,57,73,38)(13,27,70,87)(14,28,71,88)(15,29,72,89)(16,30,61,90)(17,31,62,91)(18,32,63,92)(19,33,64,93)(20,34,65,94)(21,35,66,95)(22,36,67,96)(23,25,68,85)(24,26,69,86), (1,14,74,71)(2,15,75,72)(3,16,76,61)(4,17,77,62)(5,18,78,63)(6,19,79,64)(7,20,80,65)(8,21,81,66)(9,22,82,67)(10,23,83,68)(11,24,84,69)(12,13,73,70)(25,55,85,48)(26,56,86,37)(27,57,87,38)(28,58,88,39)(29,59,89,40)(30,60,90,41)(31,49,91,42)(32,50,92,43)(33,51,93,44)(34,52,94,45)(35,53,95,46)(36,54,96,47) );
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),(38,48),(39,47),(40,46),(41,45),(42,44),(49,51),(52,60),(53,59),(54,58),(55,57),(61,71),(62,70),(63,69),(64,68),(65,67),(73,77),(74,76),(78,84),(79,83),(80,82),(85,87),(88,96),(89,95),(90,94),(91,93)], [(1,58,74,39),(2,59,75,40),(3,60,76,41),(4,49,77,42),(5,50,78,43),(6,51,79,44),(7,52,80,45),(8,53,81,46),(9,54,82,47),(10,55,83,48),(11,56,84,37),(12,57,73,38),(13,27,70,87),(14,28,71,88),(15,29,72,89),(16,30,61,90),(17,31,62,91),(18,32,63,92),(19,33,64,93),(20,34,65,94),(21,35,66,95),(22,36,67,96),(23,25,68,85),(24,26,69,86)], [(1,14,74,71),(2,15,75,72),(3,16,76,61),(4,17,77,62),(5,18,78,63),(6,19,79,64),(7,20,80,65),(8,21,81,66),(9,22,82,67),(10,23,83,68),(11,24,84,69),(12,13,73,70),(25,55,85,48),(26,56,86,37),(27,57,87,38),(28,58,88,39),(29,59,89,40),(30,60,90,41),(31,49,91,42),(32,50,92,43),(33,51,93,44),(34,52,94,45),(35,53,95,46),(36,54,96,47)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 7 | 3 | 0 | 0 |
| 10 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 4 | 0 | 0 |
| 9 | 2 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 9 |
| 0 | 0 | 9 | 10 |
G:=sub<GL(4,GF(13))| [7,10,0,0,3,10,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[11,9,0,0,4,2,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,3,9,0,0,9,10] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ (1+4) | S3×Q8 | D4○D12 |
| kernel | D12⋊10Q8 | C12⋊2Q8 | C12.6Q8 | C4×D12 | S3×C4⋊C4 | D6⋊Q8 | D6⋊3Q8 | Q8×C12 | C4×Q8 | D12 | C42 | C4⋊C4 | C2×Q8 | C12 | C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 3 | 2 | 4 | 2 | 1 | 1 | 4 | 3 | 3 | 1 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_{12}\rtimes_{10}Q_8 % in TeX
G:=Group("D12:10Q8"); // GroupNames label
G:=SmallGroup(192,1138);
// by ID
G=gap.SmallGroup(192,1138);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,100,185,192,6278]);
// 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,a*d=d*a,c*b*c^-1=a^6*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations