metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊22D4, C6.192- (1+4), C4⋊C4.191D6, C3⋊4(Q8⋊5D4), C4.114(S3×D4), C22⋊Q8⋊10S3, D6⋊14(C4○D4), Dic3⋊D4⋊26C2, C12⋊D4⋊26C2, D6⋊Q8⋊20C2, C12.237(C2×D4), (C2×Q8).152D6, C22⋊C4.17D6, Dic3⋊5D4⋊27C2, C6.79(C22×D4), (C2×C12).56C23, (C2×C6).177C24, Dic3.25(C2×D4), (C22×C4).255D6, Dic6⋊C4⋊26C2, C12.23D4⋊13C2, D6⋊C4.128C22, (C6×Q8).109C22, C23.11D6⋊25C2, (C2×D12).265C22, Dic3⋊C4.29C22, (C22×C6).205C23, C23.130(C22×S3), C22.198(S3×C23), (C22×S3).199C23, (C22×C12).257C22, C2.20(Q8.15D6), (C2×Dic3).236C23, (C2×Dic6).295C22, (C4×Dic3).107C22, C6.D4.118C22, (C2×S3×Q8)⋊8C2, C2.52(C2×S3×D4), (C4×C3⋊D4)⋊24C2, C2.50(S3×C4○D4), (C2×C4○D12)⋊25C2, C6.162(C2×C4○D4), (S3×C2×C4).97C22, (C3×C22⋊Q8)⋊13C2, (C3×C4⋊C4).160C22, (C2×C4).592(C22×S3), (C2×C3⋊D4).125C22, (C3×C22⋊C4).32C22, SmallGroup(192,1192)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 752 in 290 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C3, C4 [×2], C4 [×12], C22, C22 [×13], S3 [×4], C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], C2×C4 [×17], D4 [×12], Q8 [×10], C23, C23 [×3], Dic3 [×4], Dic3 [×3], C12 [×2], C12 [×5], D6 [×2], D6 [×8], C2×C6, C2×C6 [×3], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4 [×6], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic6 [×4], Dic6 [×4], C4×S3 [×10], D12 [×6], 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, C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8, C22⋊Q8 [×2], C4.4D4 [×3], C22×Q8, C2×C4○D4, C4×Dic3, C4×Dic3 [×2], Dic3⋊C4, Dic3⋊C4 [×2], D6⋊C4, D6⋊C4 [×6], C6.D4, C3×C22⋊C4 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C2×Dic6, C2×Dic6 [×2], S3×C2×C4, S3×C2×C4 [×4], C2×D12, C2×D12 [×2], C4○D12 [×4], S3×Q8 [×4], C2×C3⋊D4, C2×C3⋊D4 [×2], C22×C12, C6×Q8, Q8⋊5D4, Dic3⋊D4 [×2], C23.11D6 [×2], Dic6⋊C4, Dic3⋊5D4 [×2], C12⋊D4, D6⋊Q8 [×2], C4×C3⋊D4, C12.23D4, C3×C22⋊Q8, C2×C4○D12, C2×S3×Q8, Dic6⋊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, Q8⋊5D4, C2×S3×D4, Q8.15D6, S3×C4○D4, Dic6⋊22D4
Generators and relations
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=dad=a5, cbc-1=a6b, bd=db, 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 80 7 74)(2 79 8 73)(3 78 9 84)(4 77 10 83)(5 76 11 82)(6 75 12 81)(13 61 19 67)(14 72 20 66)(15 71 21 65)(16 70 22 64)(17 69 23 63)(18 68 24 62)(25 60 31 54)(26 59 32 53)(27 58 33 52)(28 57 34 51)(29 56 35 50)(30 55 36 49)(37 85 43 91)(38 96 44 90)(39 95 45 89)(40 94 46 88)(41 93 47 87)(42 92 48 86)
(1 88 50 24)(2 93 51 17)(3 86 52 22)(4 91 53 15)(5 96 54 20)(6 89 55 13)(7 94 56 18)(8 87 57 23)(9 92 58 16)(10 85 59 21)(11 90 60 14)(12 95 49 19)(25 72 76 38)(26 65 77 43)(27 70 78 48)(28 63 79 41)(29 68 80 46)(30 61 81 39)(31 66 82 44)(32 71 83 37)(33 64 84 42)(34 69 73 47)(35 62 74 40)(36 67 75 45)
(1 68)(2 61)(3 66)(4 71)(5 64)(6 69)(7 62)(8 67)(9 72)(10 65)(11 70)(12 63)(13 73)(14 78)(15 83)(16 76)(17 81)(18 74)(19 79)(20 84)(21 77)(22 82)(23 75)(24 80)(25 92)(26 85)(27 90)(28 95)(29 88)(30 93)(31 86)(32 91)(33 96)(34 89)(35 94)(36 87)(37 53)(38 58)(39 51)(40 56)(41 49)(42 54)(43 59)(44 52)(45 57)(46 50)(47 55)(48 60)
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), (1,80,7,74)(2,79,8,73)(3,78,9,84)(4,77,10,83)(5,76,11,82)(6,75,12,81)(13,61,19,67)(14,72,20,66)(15,71,21,65)(16,70,22,64)(17,69,23,63)(18,68,24,62)(25,60,31,54)(26,59,32,53)(27,58,33,52)(28,57,34,51)(29,56,35,50)(30,55,36,49)(37,85,43,91)(38,96,44,90)(39,95,45,89)(40,94,46,88)(41,93,47,87)(42,92,48,86), (1,88,50,24)(2,93,51,17)(3,86,52,22)(4,91,53,15)(5,96,54,20)(6,89,55,13)(7,94,56,18)(8,87,57,23)(9,92,58,16)(10,85,59,21)(11,90,60,14)(12,95,49,19)(25,72,76,38)(26,65,77,43)(27,70,78,48)(28,63,79,41)(29,68,80,46)(30,61,81,39)(31,66,82,44)(32,71,83,37)(33,64,84,42)(34,69,73,47)(35,62,74,40)(36,67,75,45), (1,68)(2,61)(3,66)(4,71)(5,64)(6,69)(7,62)(8,67)(9,72)(10,65)(11,70)(12,63)(13,73)(14,78)(15,83)(16,76)(17,81)(18,74)(19,79)(20,84)(21,77)(22,82)(23,75)(24,80)(25,92)(26,85)(27,90)(28,95)(29,88)(30,93)(31,86)(32,91)(33,96)(34,89)(35,94)(36,87)(37,53)(38,58)(39,51)(40,56)(41,49)(42,54)(43,59)(44,52)(45,57)(46,50)(47,55)(48,60)>;
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), (1,80,7,74)(2,79,8,73)(3,78,9,84)(4,77,10,83)(5,76,11,82)(6,75,12,81)(13,61,19,67)(14,72,20,66)(15,71,21,65)(16,70,22,64)(17,69,23,63)(18,68,24,62)(25,60,31,54)(26,59,32,53)(27,58,33,52)(28,57,34,51)(29,56,35,50)(30,55,36,49)(37,85,43,91)(38,96,44,90)(39,95,45,89)(40,94,46,88)(41,93,47,87)(42,92,48,86), (1,88,50,24)(2,93,51,17)(3,86,52,22)(4,91,53,15)(5,96,54,20)(6,89,55,13)(7,94,56,18)(8,87,57,23)(9,92,58,16)(10,85,59,21)(11,90,60,14)(12,95,49,19)(25,72,76,38)(26,65,77,43)(27,70,78,48)(28,63,79,41)(29,68,80,46)(30,61,81,39)(31,66,82,44)(32,71,83,37)(33,64,84,42)(34,69,73,47)(35,62,74,40)(36,67,75,45), (1,68)(2,61)(3,66)(4,71)(5,64)(6,69)(7,62)(8,67)(9,72)(10,65)(11,70)(12,63)(13,73)(14,78)(15,83)(16,76)(17,81)(18,74)(19,79)(20,84)(21,77)(22,82)(23,75)(24,80)(25,92)(26,85)(27,90)(28,95)(29,88)(30,93)(31,86)(32,91)(33,96)(34,89)(35,94)(36,87)(37,53)(38,58)(39,51)(40,56)(41,49)(42,54)(43,59)(44,52)(45,57)(46,50)(47,55)(48,60) );
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)], [(1,80,7,74),(2,79,8,73),(3,78,9,84),(4,77,10,83),(5,76,11,82),(6,75,12,81),(13,61,19,67),(14,72,20,66),(15,71,21,65),(16,70,22,64),(17,69,23,63),(18,68,24,62),(25,60,31,54),(26,59,32,53),(27,58,33,52),(28,57,34,51),(29,56,35,50),(30,55,36,49),(37,85,43,91),(38,96,44,90),(39,95,45,89),(40,94,46,88),(41,93,47,87),(42,92,48,86)], [(1,88,50,24),(2,93,51,17),(3,86,52,22),(4,91,53,15),(5,96,54,20),(6,89,55,13),(7,94,56,18),(8,87,57,23),(9,92,58,16),(10,85,59,21),(11,90,60,14),(12,95,49,19),(25,72,76,38),(26,65,77,43),(27,70,78,48),(28,63,79,41),(29,68,80,46),(30,61,81,39),(31,66,82,44),(32,71,83,37),(33,64,84,42),(34,69,73,47),(35,62,74,40),(36,67,75,45)], [(1,68),(2,61),(3,66),(4,71),(5,64),(6,69),(7,62),(8,67),(9,72),(10,65),(11,70),(12,63),(13,73),(14,78),(15,83),(16,76),(17,81),(18,74),(19,79),(20,84),(21,77),(22,82),(23,75),(24,80),(25,92),(26,85),(27,90),(28,95),(29,88),(30,93),(31,86),(32,91),(33,96),(34,89),(35,94),(36,87),(37,53),(38,58),(39,51),(40,56),(41,49),(42,54),(43,59),(44,52),(45,57),(46,50),(47,55),(48,60)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [0,8,0,0,0,0,8,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 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 | Dic6⋊22D4 | Dic3⋊D4 | C23.11D6 | Dic6⋊C4 | Dic3⋊5D4 | C12⋊D4 | D6⋊Q8 | C4×C3⋊D4 | C12.23D4 | C3×C22⋊Q8 | C2×C4○D12 | C2×S3×Q8 | C22⋊Q8 | Dic6 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | D6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 1 | 1 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
Dic_6\rtimes_{22}D_4 % in TeX
G:=Group("Dic6:22D4"); // GroupNames label
G:=SmallGroup(192,1192);
// by ID
G=gap.SmallGroup(192,1192);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,100,1571,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=d*a*d=a^5,c*b*c^-1=a^6*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations