metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊22D4, C6.182- (1+4), C22⋊Q8⋊8S3, C3⋊5(D4⋊6D4), C4⋊C4.189D6, D6.21(C2×D4), C4.112(S3×D4), Dic3⋊D4⋊25C2, C4.D12⋊25C2, C12.235(C2×D4), (C2×Q8).150D6, C22⋊C4.16D6, Dic3⋊5D4⋊26C2, C6.77(C22×D4), Dic3⋊5(C4○D4), D6.D4⋊18C2, C23.9D6⋊25C2, (C2×C6).175C24, D6⋊C4.23C22, (C22×C4).253D6, Dic3⋊Q8⋊14C2, (C2×C12).503C23, (C6×Q8).107C22, (C2×D12).149C22, Dic3⋊C4.27C22, C4⋊Dic3.215C22, C23.129(C22×S3), (C22×C6).203C23, C22.196(S3×C23), (C22×S3).197C23, (C22×C12).255C22, C2.19(Q8.15D6), (C2×Dic3).234C23, (C4×Dic3).105C22, (C2×Dic6).294C22, C6.D4.116C22, (S3×C4⋊C4)⋊26C2, C2.50(C2×S3×D4), (C4×C3⋊D4)⋊23C2, C2.49(S3×C4○D4), (C2×C4○D12)⋊24C2, (C2×Q8⋊3S3)⋊8C2, C6.161(C2×C4○D4), (S3×C2×C4).95C22, (C3×C22⋊Q8)⋊11C2, (C2×C4).48(C22×S3), (C3×C4⋊C4).158C22, (C2×C3⋊D4).123C22, (C3×C22⋊C4).30C22, SmallGroup(192,1190)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 752 in 292 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×2], C4 [×11], C22, C22 [×14], S3 [×5], C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], C2×C4 [×21], D4 [×14], Q8 [×4], C23, C23 [×3], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×5], D6 [×4], D6 [×7], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4, C22×C4 [×7], C2×D4 [×6], C2×Q8, C2×Q8, C4○D4 [×8], Dic6 [×2], C4×S3 [×14], D12 [×4], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×2], C3⋊D4 [×6], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C3×Q8 [×2], C22×S3, C22×S3 [×2], C22×C6, C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], C4×Dic3, Dic3⋊C4, Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4, D6⋊C4 [×4], C6.D4, C3×C22⋊C4 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4, S3×C2×C4 [×6], C2×D12, C2×D12 [×2], C4○D12 [×4], Q8⋊3S3 [×4], C2×C3⋊D4, C2×C3⋊D4 [×2], C22×C12, C6×Q8, D4⋊6D4, C23.9D6 [×2], Dic3⋊D4 [×2], S3×C4⋊C4 [×2], Dic3⋊5D4, D6.D4 [×2], C4.D12, C4×C3⋊D4, Dic3⋊Q8, C3×C22⋊Q8, C2×C4○D12, C2×Q8⋊3S3, D12⋊22D4
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], S3×C23, D4⋊6D4, C2×S3×D4, Q8.15D6, S3×C4○D4, D12⋊22D4
Generators and relations
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=dad=a5, cbc-1=a4b, 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 61)(2 72)(3 71)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 64)(11 63)(12 62)(13 82)(14 81)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(21 74)(22 73)(23 84)(24 83)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 48)(36 47)(49 86)(50 85)(51 96)(52 95)(53 94)(54 93)(55 92)(56 91)(57 90)(58 89)(59 88)(60 87)
(1 48 88 22)(2 41 89 15)(3 46 90 20)(4 39 91 13)(5 44 92 18)(6 37 93 23)(7 42 94 16)(8 47 95 21)(9 40 96 14)(10 45 85 19)(11 38 86 24)(12 43 87 17)(25 53 75 67)(26 58 76 72)(27 51 77 65)(28 56 78 70)(29 49 79 63)(30 54 80 68)(31 59 81 61)(32 52 82 66)(33 57 83 71)(34 50 84 64)(35 55 73 69)(36 60 74 62)
(2 6)(3 11)(5 9)(8 12)(13 39)(14 44)(15 37)(16 42)(17 47)(18 40)(19 45)(20 38)(21 43)(22 48)(23 41)(24 46)(25 73)(26 78)(27 83)(28 76)(29 81)(30 74)(31 79)(32 84)(33 77)(34 82)(35 75)(36 80)(49 59)(50 52)(51 57)(53 55)(54 60)(56 58)(61 63)(62 68)(64 66)(65 71)(67 69)(70 72)(86 90)(87 95)(89 93)(92 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), (1,61)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,82)(14,81)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,74)(22,73)(23,84)(24,83)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,48)(36,47)(49,86)(50,85)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87), (1,48,88,22)(2,41,89,15)(3,46,90,20)(4,39,91,13)(5,44,92,18)(6,37,93,23)(7,42,94,16)(8,47,95,21)(9,40,96,14)(10,45,85,19)(11,38,86,24)(12,43,87,17)(25,53,75,67)(26,58,76,72)(27,51,77,65)(28,56,78,70)(29,49,79,63)(30,54,80,68)(31,59,81,61)(32,52,82,66)(33,57,83,71)(34,50,84,64)(35,55,73,69)(36,60,74,62), (2,6)(3,11)(5,9)(8,12)(13,39)(14,44)(15,37)(16,42)(17,47)(18,40)(19,45)(20,38)(21,43)(22,48)(23,41)(24,46)(25,73)(26,78)(27,83)(28,76)(29,81)(30,74)(31,79)(32,84)(33,77)(34,82)(35,75)(36,80)(49,59)(50,52)(51,57)(53,55)(54,60)(56,58)(61,63)(62,68)(64,66)(65,71)(67,69)(70,72)(86,90)(87,95)(89,93)(92,96)>;
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), (1,61)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,82)(14,81)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,74)(22,73)(23,84)(24,83)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,48)(36,47)(49,86)(50,85)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87), (1,48,88,22)(2,41,89,15)(3,46,90,20)(4,39,91,13)(5,44,92,18)(6,37,93,23)(7,42,94,16)(8,47,95,21)(9,40,96,14)(10,45,85,19)(11,38,86,24)(12,43,87,17)(25,53,75,67)(26,58,76,72)(27,51,77,65)(28,56,78,70)(29,49,79,63)(30,54,80,68)(31,59,81,61)(32,52,82,66)(33,57,83,71)(34,50,84,64)(35,55,73,69)(36,60,74,62), (2,6)(3,11)(5,9)(8,12)(13,39)(14,44)(15,37)(16,42)(17,47)(18,40)(19,45)(20,38)(21,43)(22,48)(23,41)(24,46)(25,73)(26,78)(27,83)(28,76)(29,81)(30,74)(31,79)(32,84)(33,77)(34,82)(35,75)(36,80)(49,59)(50,52)(51,57)(53,55)(54,60)(56,58)(61,63)(62,68)(64,66)(65,71)(67,69)(70,72)(86,90)(87,95)(89,93)(92,96) );
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)], [(1,61),(2,72),(3,71),(4,70),(5,69),(6,68),(7,67),(8,66),(9,65),(10,64),(11,63),(12,62),(13,82),(14,81),(15,80),(16,79),(17,78),(18,77),(19,76),(20,75),(21,74),(22,73),(23,84),(24,83),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,48),(36,47),(49,86),(50,85),(51,96),(52,95),(53,94),(54,93),(55,92),(56,91),(57,90),(58,89),(59,88),(60,87)], [(1,48,88,22),(2,41,89,15),(3,46,90,20),(4,39,91,13),(5,44,92,18),(6,37,93,23),(7,42,94,16),(8,47,95,21),(9,40,96,14),(10,45,85,19),(11,38,86,24),(12,43,87,17),(25,53,75,67),(26,58,76,72),(27,51,77,65),(28,56,78,70),(29,49,79,63),(30,54,80,68),(31,59,81,61),(32,52,82,66),(33,57,83,71),(34,50,84,64),(35,55,73,69),(36,60,74,62)], [(2,6),(3,11),(5,9),(8,12),(13,39),(14,44),(15,37),(16,42),(17,47),(18,40),(19,45),(20,38),(21,43),(22,48),(23,41),(24,46),(25,73),(26,78),(27,83),(28,76),(29,81),(30,74),(31,79),(32,84),(33,77),(34,82),(35,75),(36,80),(49,59),(50,52),(51,57),(53,55),(54,60),(56,58),(61,63),(62,68),(64,66),(65,71),(67,69),(70,72),(86,90),(87,95),(89,93),(92,96)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 3 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 12 | 10 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 8 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,8,5,0,0,0,0,0,5],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,8,5,0,0,0,0,3,5],[12,5,0,0,0,0,10,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,8,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 6 | 6 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4○D4 | 2- (1+4) | S3×D4 | Q8.15D6 | S3×C4○D4 |
| kernel | D12⋊22D4 | C23.9D6 | Dic3⋊D4 | S3×C4⋊C4 | Dic3⋊5D4 | D6.D4 | C4.D12 | C4×C3⋊D4 | Dic3⋊Q8 | C3×C22⋊Q8 | C2×C4○D12 | C2×Q8⋊3S3 | C22⋊Q8 | D12 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | Dic3 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 1 | 1 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_{12}\rtimes_{22}D_4 % in TeX
G:=Group("D12:22D4"); // GroupNames label
G:=SmallGroup(192,1190);
// by ID
G=gap.SmallGroup(192,1190);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,1571,570,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^5,c*b*c^-1=a^4*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations