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, C23.130(C22×S3), C22.198(S3×C23), (C22×C6).205C23, (C22×S3).199C23, (C22×C12).257C22, C2.20(Q8.15D6), (C2×Dic6).295C22, (C4×Dic3).107C22, (C2×Dic3).236C23, 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
Generators and relations for Dic6⋊22D4
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 >
Subgroups: 752 in 290 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C22×C6, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C4○D12, S3×Q8, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×Q8, Q8⋊5D4, 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, Dic6⋊22D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, 2- 1+4, S3×D4, S3×C23, Q8⋊5D4, C2×S3×D4, Q8.15D6, S3×C4○D4, Dic6⋊22D4
►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 69 7 63)(2 68 8 62)(3 67 9 61)(4 66 10 72)(5 65 11 71)(6 64 12 70)(13 35 19 29)(14 34 20 28)(15 33 21 27)(16 32 22 26)(17 31 23 25)(18 30 24 36)(37 74 43 80)(38 73 44 79)(39 84 45 78)(40 83 46 77)(41 82 47 76)(42 81 48 75)(49 96 55 90)(50 95 56 89)(51 94 57 88)(52 93 58 87)(53 92 59 86)(54 91 60 85)
(1 95 77 26)(2 88 78 31)(3 93 79 36)(4 86 80 29)(5 91 81 34)(6 96 82 27)(7 89 83 32)(8 94 84 25)(9 87 73 30)(10 92 74 35)(11 85 75 28)(12 90 76 33)(13 72 53 43)(14 65 54 48)(15 70 55 41)(16 63 56 46)(17 68 57 39)(18 61 58 44)(19 66 59 37)(20 71 60 42)(21 64 49 47)(22 69 50 40)(23 62 51 45)(24 67 52 38)
(1 22)(2 15)(3 20)(4 13)(5 18)(6 23)(7 16)(8 21)(9 14)(10 19)(11 24)(12 17)(25 64)(26 69)(27 62)(28 67)(29 72)(30 65)(31 70)(32 63)(33 68)(34 61)(35 66)(36 71)(37 92)(38 85)(39 90)(40 95)(41 88)(42 93)(43 86)(44 91)(45 96)(46 89)(47 94)(48 87)(49 84)(50 77)(51 82)(52 75)(53 80)(54 73)(55 78)(56 83)(57 76)(58 81)(59 74)(60 79)
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), (1,69,7,63)(2,68,8,62)(3,67,9,61)(4,66,10,72)(5,65,11,71)(6,64,12,70)(13,35,19,29)(14,34,20,28)(15,33,21,27)(16,32,22,26)(17,31,23,25)(18,30,24,36)(37,74,43,80)(38,73,44,79)(39,84,45,78)(40,83,46,77)(41,82,47,76)(42,81,48,75)(49,96,55,90)(50,95,56,89)(51,94,57,88)(52,93,58,87)(53,92,59,86)(54,91,60,85), (1,95,77,26)(2,88,78,31)(3,93,79,36)(4,86,80,29)(5,91,81,34)(6,96,82,27)(7,89,83,32)(8,94,84,25)(9,87,73,30)(10,92,74,35)(11,85,75,28)(12,90,76,33)(13,72,53,43)(14,65,54,48)(15,70,55,41)(16,63,56,46)(17,68,57,39)(18,61,58,44)(19,66,59,37)(20,71,60,42)(21,64,49,47)(22,69,50,40)(23,62,51,45)(24,67,52,38), (1,22)(2,15)(3,20)(4,13)(5,18)(6,23)(7,16)(8,21)(9,14)(10,19)(11,24)(12,17)(25,64)(26,69)(27,62)(28,67)(29,72)(30,65)(31,70)(32,63)(33,68)(34,61)(35,66)(36,71)(37,92)(38,85)(39,90)(40,95)(41,88)(42,93)(43,86)(44,91)(45,96)(46,89)(47,94)(48,87)(49,84)(50,77)(51,82)(52,75)(53,80)(54,73)(55,78)(56,83)(57,76)(58,81)(59,74)(60,79)>;
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), (1,69,7,63)(2,68,8,62)(3,67,9,61)(4,66,10,72)(5,65,11,71)(6,64,12,70)(13,35,19,29)(14,34,20,28)(15,33,21,27)(16,32,22,26)(17,31,23,25)(18,30,24,36)(37,74,43,80)(38,73,44,79)(39,84,45,78)(40,83,46,77)(41,82,47,76)(42,81,48,75)(49,96,55,90)(50,95,56,89)(51,94,57,88)(52,93,58,87)(53,92,59,86)(54,91,60,85), (1,95,77,26)(2,88,78,31)(3,93,79,36)(4,86,80,29)(5,91,81,34)(6,96,82,27)(7,89,83,32)(8,94,84,25)(9,87,73,30)(10,92,74,35)(11,85,75,28)(12,90,76,33)(13,72,53,43)(14,65,54,48)(15,70,55,41)(16,63,56,46)(17,68,57,39)(18,61,58,44)(19,66,59,37)(20,71,60,42)(21,64,49,47)(22,69,50,40)(23,62,51,45)(24,67,52,38), (1,22)(2,15)(3,20)(4,13)(5,18)(6,23)(7,16)(8,21)(9,14)(10,19)(11,24)(12,17)(25,64)(26,69)(27,62)(28,67)(29,72)(30,65)(31,70)(32,63)(33,68)(34,61)(35,66)(36,71)(37,92)(38,85)(39,90)(40,95)(41,88)(42,93)(43,86)(44,91)(45,96)(46,89)(47,94)(48,87)(49,84)(50,77)(51,82)(52,75)(53,80)(54,73)(55,78)(56,83)(57,76)(58,81)(59,74)(60,79) );
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)], [(1,69,7,63),(2,68,8,62),(3,67,9,61),(4,66,10,72),(5,65,11,71),(6,64,12,70),(13,35,19,29),(14,34,20,28),(15,33,21,27),(16,32,22,26),(17,31,23,25),(18,30,24,36),(37,74,43,80),(38,73,44,79),(39,84,45,78),(40,83,46,77),(41,82,47,76),(42,81,48,75),(49,96,55,90),(50,95,56,89),(51,94,57,88),(52,93,58,87),(53,92,59,86),(54,91,60,85)], [(1,95,77,26),(2,88,78,31),(3,93,79,36),(4,86,80,29),(5,91,81,34),(6,96,82,27),(7,89,83,32),(8,94,84,25),(9,87,73,30),(10,92,74,35),(11,85,75,28),(12,90,76,33),(13,72,53,43),(14,65,54,48),(15,70,55,41),(16,63,56,46),(17,68,57,39),(18,61,58,44),(19,66,59,37),(20,71,60,42),(21,64,49,47),(22,69,50,40),(23,62,51,45),(24,67,52,38)], [(1,22),(2,15),(3,20),(4,13),(5,18),(6,23),(7,16),(8,21),(9,14),(10,19),(11,24),(12,17),(25,64),(26,69),(27,62),(28,67),(29,72),(30,65),(31,70),(32,63),(33,68),(34,61),(35,66),(36,71),(37,92),(38,85),(39,90),(40,95),(41,88),(42,93),(43,86),(44,91),(45,96),(46,89),(47,94),(48,87),(49,84),(50,77),(51,82),(52,75),(53,80),(54,73),(55,78),(56,83),(57,76),(58,81),(59,74),(60,79)]])
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 |
Matrix representation of Dic6⋊22D4 ►in 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] >;
Dic6⋊22D4 in GAP, Magma, Sage, TeX
{\rm 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