metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.30D4, Q16.11D6, C12.13C24, C24.42C23, Dic6.30D4, D12.8C23, D24.13C22, Dic6.8C23, Dic12.15C22, C3⋊2(Q8○D8), C4○D24⋊4C2, C8○D12⋊3C2, (C6×Q16)⋊3C2, (S3×Q16)⋊6C2, C4.77(S3×D4), C3⋊C8.5C23, (C2×Q16)⋊12S3, D6.28(C2×D4), C12.88(C2×D4), (C2×C8).105D6, Q16⋊S3⋊5C2, C3⋊D4.10D4, D24⋊C2⋊6C2, (C4×S3).7C23, (S3×C8).7C22, C4.13(S3×C23), C8.14(C22×S3), (C2×Q8).115D6, C22.22(S3×D4), Q8⋊2S3.C22, (S3×Q8).1C22, (C3×Q8).7C23, Q8.11D6⋊8C2, C24⋊C2.3C22, C8⋊S3.3C22, (C2×C24).35C22, Q8.15D6⋊4C2, Dic3.33(C2×D4), C6.114(C22×D4), Q8.17(C22×S3), C3⋊Q16.1C22, (C2×C12).530C23, C4○D12.53C22, (C6×Q8).152C22, (C3×Q16).11C22, Q8⋊3S3.1C22, C4.Dic3.48C22, C2.87(C2×S3×D4), (C2×C6).403(C2×D4), (C2×C4).231(C22×S3), SmallGroup(192,1325)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 616 in 248 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×8], C22, C22 [×4], S3 [×4], C6, C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×14], D4 [×11], Q8 [×4], Q8 [×9], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×4], D6 [×2], D6 [×2], C2×C6, C2×C8, C2×C8 [×2], M4(2) [×3], D8, SD16 [×6], Q16 [×4], Q16 [×5], C2×Q8 [×2], C2×Q8 [×6], C4○D4 [×13], C3⋊C8 [×2], C24 [×2], Dic6, Dic6 [×2], Dic6 [×4], C4×S3 [×2], C4×S3 [×10], D12, D12 [×2], D12 [×4], C3⋊D4 [×2], C3⋊D4 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×4], C3×Q8 [×2], C8○D4, C2×Q16, C2×Q16 [×2], C4○D8 [×3], C8.C22 [×6], 2- (1+4) [×2], S3×C8 [×2], C8⋊S3 [×2], C24⋊C2 [×2], D24, Dic12, C4.Dic3, Q8⋊2S3 [×4], C3⋊Q16 [×4], C2×C24, C3×Q16 [×4], C4○D12, C4○D12 [×2], C4○D12 [×4], S3×Q8 [×4], S3×Q8 [×2], Q8⋊3S3 [×4], Q8⋊3S3 [×2], C6×Q8 [×2], Q8○D8, C8○D12, C4○D24, S3×Q16 [×2], Q16⋊S3 [×4], D24⋊C2 [×2], Q8.11D6 [×2], C6×Q16, Q8.15D6 [×2], D12.30D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, S3×D4 [×2], S3×C23, Q8○D8, C2×S3×D4, D12.30D4
Generators and relations
G = < a,b,c,d | a12=b2=1, c4=d2=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=a6c3 >
►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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(25 86)(26 85)(27 96)(28 95)(29 94)(30 93)(31 92)(32 91)(33 90)(34 89)(35 88)(36 87)(37 70)(38 69)(39 68)(40 67)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 72)(48 71)(49 76)(50 75)(51 74)(52 73)(53 84)(54 83)(55 82)(56 81)(57 80)(58 79)(59 78)(60 77)
(1 47 50 94 7 41 56 88)(2 48 51 95 8 42 57 89)(3 37 52 96 9 43 58 90)(4 38 53 85 10 44 59 91)(5 39 54 86 11 45 60 92)(6 40 55 87 12 46 49 93)(13 67 82 36 19 61 76 30)(14 68 83 25 20 62 77 31)(15 69 84 26 21 63 78 32)(16 70 73 27 22 64 79 33)(17 71 74 28 23 65 80 34)(18 72 75 29 24 66 81 35)
(1 79 7 73)(2 74 8 80)(3 81 9 75)(4 76 10 82)(5 83 11 77)(6 78 12 84)(13 53 19 59)(14 60 20 54)(15 55 21 49)(16 50 22 56)(17 57 23 51)(18 52 24 58)(25 86 31 92)(26 93 32 87)(27 88 33 94)(28 95 34 89)(29 90 35 96)(30 85 36 91)(37 66 43 72)(38 61 44 67)(39 68 45 62)(40 63 46 69)(41 70 47 64)(42 65 48 71)
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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(25,86)(26,85)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,72)(48,71)(49,76)(50,75)(51,74)(52,73)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77), (1,47,50,94,7,41,56,88)(2,48,51,95,8,42,57,89)(3,37,52,96,9,43,58,90)(4,38,53,85,10,44,59,91)(5,39,54,86,11,45,60,92)(6,40,55,87,12,46,49,93)(13,67,82,36,19,61,76,30)(14,68,83,25,20,62,77,31)(15,69,84,26,21,63,78,32)(16,70,73,27,22,64,79,33)(17,71,74,28,23,65,80,34)(18,72,75,29,24,66,81,35), (1,79,7,73)(2,74,8,80)(3,81,9,75)(4,76,10,82)(5,83,11,77)(6,78,12,84)(13,53,19,59)(14,60,20,54)(15,55,21,49)(16,50,22,56)(17,57,23,51)(18,52,24,58)(25,86,31,92)(26,93,32,87)(27,88,33,94)(28,95,34,89)(29,90,35,96)(30,85,36,91)(37,66,43,72)(38,61,44,67)(39,68,45,62)(40,63,46,69)(41,70,47,64)(42,65,48,71)>;
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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(25,86)(26,85)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,72)(48,71)(49,76)(50,75)(51,74)(52,73)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77), (1,47,50,94,7,41,56,88)(2,48,51,95,8,42,57,89)(3,37,52,96,9,43,58,90)(4,38,53,85,10,44,59,91)(5,39,54,86,11,45,60,92)(6,40,55,87,12,46,49,93)(13,67,82,36,19,61,76,30)(14,68,83,25,20,62,77,31)(15,69,84,26,21,63,78,32)(16,70,73,27,22,64,79,33)(17,71,74,28,23,65,80,34)(18,72,75,29,24,66,81,35), (1,79,7,73)(2,74,8,80)(3,81,9,75)(4,76,10,82)(5,83,11,77)(6,78,12,84)(13,53,19,59)(14,60,20,54)(15,55,21,49)(16,50,22,56)(17,57,23,51)(18,52,24,58)(25,86,31,92)(26,93,32,87)(27,88,33,94)(28,95,34,89)(29,90,35,96)(30,85,36,91)(37,66,43,72)(38,61,44,67)(39,68,45,62)(40,63,46,69)(41,70,47,64)(42,65,48,71) );
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)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(25,86),(26,85),(27,96),(28,95),(29,94),(30,93),(31,92),(32,91),(33,90),(34,89),(35,88),(36,87),(37,70),(38,69),(39,68),(40,67),(41,66),(42,65),(43,64),(44,63),(45,62),(46,61),(47,72),(48,71),(49,76),(50,75),(51,74),(52,73),(53,84),(54,83),(55,82),(56,81),(57,80),(58,79),(59,78),(60,77)], [(1,47,50,94,7,41,56,88),(2,48,51,95,8,42,57,89),(3,37,52,96,9,43,58,90),(4,38,53,85,10,44,59,91),(5,39,54,86,11,45,60,92),(6,40,55,87,12,46,49,93),(13,67,82,36,19,61,76,30),(14,68,83,25,20,62,77,31),(15,69,84,26,21,63,78,32),(16,70,73,27,22,64,79,33),(17,71,74,28,23,65,80,34),(18,72,75,29,24,66,81,35)], [(1,79,7,73),(2,74,8,80),(3,81,9,75),(4,76,10,82),(5,83,11,77),(6,78,12,84),(13,53,19,59),(14,60,20,54),(15,55,21,49),(16,50,22,56),(17,57,23,51),(18,52,24,58),(25,86,31,92),(26,93,32,87),(27,88,33,94),(28,95,34,89),(29,90,35,96),(30,85,36,91),(37,66,43,72),(38,61,44,67),(39,68,45,62),(40,63,46,69),(41,70,47,64),(42,65,48,71)])
Matrix representation ►G ⊆ GL4(𝔽7) generated by
| 0 | 6 | 6 | 1 |
| 5 | 2 | 6 | 3 |
| 6 | 3 | 1 | 6 |
| 5 | 5 | 5 | 4 |
| 1 | 0 | 2 | 4 |
| 2 | 2 | 5 | 5 |
| 6 | 3 | 1 | 6 |
| 4 | 2 | 6 | 3 |
| 3 | 5 | 2 | 2 |
| 4 | 5 | 1 | 2 |
| 6 | 6 | 5 | 2 |
| 1 | 6 | 3 | 0 |
| 1 | 5 | 0 | 2 |
| 0 | 0 | 4 | 3 |
| 6 | 6 | 4 | 2 |
| 6 | 1 | 4 | 2 |
G:=sub<GL(4,GF(7))| [0,5,6,5,6,2,3,5,6,6,1,5,1,3,6,4],[1,2,6,4,0,2,3,2,2,5,1,6,4,5,6,3],[3,4,6,1,5,5,6,6,2,1,5,3,2,2,2,0],[1,0,6,6,5,0,6,1,0,4,4,4,2,3,2,2] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 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 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | S3×D4 | S3×D4 | Q8○D8 | D12.30D4 |
| kernel | D12.30D4 | C8○D12 | C4○D24 | S3×Q16 | Q16⋊S3 | D24⋊C2 | Q8.11D6 | C6×Q16 | Q8.15D6 | C2×Q16 | Dic6 | D12 | C3⋊D4 | C2×C8 | Q16 | C2×Q8 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 1 | 1 | 2 | 4 |
In GAP, Magma, Sage, TeX
D_{12}._{30}D_4 % in TeX
G:=Group("D12.30D4"); // GroupNames label
G:=SmallGroup(192,1325);
// by ID
G=gap.SmallGroup(192,1325);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,184,185,136,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^4=d^2=a^6,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=a^6*c^3>;
// generators/relations