metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊23D4, C42.110D6, C6.602- (1+4), (C4×D4)⋊14S3, (D4×C12)⋊16C2, C12⋊7D4⋊8C2, C4⋊C4.283D6, D6⋊Q8⋊7C2, C3⋊1(Q8⋊5D4), C4.141(S3×D4), (C4×Dic6)⋊31C2, (C2×D4).215D6, C12.347(C2×D4), (C2×C6).96C24, C6.51(C22×D4), C42⋊7S3⋊17C2, C22⋊2(C4○D12), D6⋊C4.66C22, C2.17(Q8○D12), C22⋊C4.111D6, Dic3.17(C2×D4), (C22×Dic6)⋊9C2, (C22×C4).225D6, Dic3⋊4D4⋊47C2, C23.14D6⋊26C2, C12.48D4⋊21C2, (C2×C12).784C23, (C4×C12).153C22, C23.11D6⋊6C2, (C6×D4).306C22, (C2×D12).211C22, (C22×S3).31C23, C4⋊Dic3.298C22, (C22×C6).166C23, C22.121(S3×C23), C23.106(C22×S3), Dic3⋊C4.154C22, (C22×C12).108C22, (C4×Dic3).204C22, (C2×Dic3).204C23, (C2×Dic6).240C22, C6.D4.13C22, (C22×Dic3).96C22, C2.24(C2×S3×D4), (C2×C4○D12)⋊9C2, (C2×C6)⋊3(C4○D4), C6.43(C2×C4○D4), C2.47(C2×C4○D12), (S3×C2×C4).199C22, (C3×C4⋊C4).327C22, (C2×C4).159(C22×S3), (C2×C3⋊D4).114C22, (C3×C22⋊C4).123C22, SmallGroup(192,1111)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 728 in 290 conjugacy classes, 107 normal (51 characteristic)
C1, C2 [×3], C2 [×5], C3, C4 [×2], C4 [×12], C22, C22 [×2], C22 [×11], S3 [×2], C6 [×3], C6 [×3], C2×C4 [×5], C2×C4 [×18], D4 [×12], Q8 [×10], C23 [×2], C23 [×2], Dic3 [×4], Dic3 [×4], C12 [×2], C12 [×4], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×5], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8 [×8], C4○D4 [×4], Dic6 [×4], Dic6 [×6], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C2×Dic3 [×4], C3⋊D4 [×8], C2×C12 [×5], C2×C12 [×4], C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3, D6⋊C4 [×6], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6 [×2], C2×Dic6 [×2], C2×Dic6 [×4], S3×C2×C4 [×2], C2×D12, C4○D12 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, Q8⋊5D4, C4×Dic6, C42⋊7S3, Dic3⋊4D4 [×2], C23.11D6 [×2], D6⋊Q8 [×2], C12.48D4, C12⋊7D4, C23.14D6 [×2], D4×C12, C22×Dic6, C2×C4○D12, Dic6⋊23D4
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), C4○D12 [×2], S3×D4 [×2], S3×C23, Q8⋊5D4, C2×C4○D12, C2×S3×D4, Q8○D12, Dic6⋊23D4
Generators and relations
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, ac=ca, ad=da, 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 67 7 61)(2 66 8 72)(3 65 9 71)(4 64 10 70)(5 63 11 69)(6 62 12 68)(13 77 19 83)(14 76 20 82)(15 75 21 81)(16 74 22 80)(17 73 23 79)(18 84 24 78)(25 91 31 85)(26 90 32 96)(27 89 33 95)(28 88 34 94)(29 87 35 93)(30 86 36 92)(37 59 43 53)(38 58 44 52)(39 57 45 51)(40 56 46 50)(41 55 47 49)(42 54 48 60)
(1 40 96 24)(2 41 85 13)(3 42 86 14)(4 43 87 15)(5 44 88 16)(6 45 89 17)(7 46 90 18)(8 47 91 19)(9 48 92 20)(10 37 93 21)(11 38 94 22)(12 39 95 23)(25 83 66 49)(26 84 67 50)(27 73 68 51)(28 74 69 52)(29 75 70 53)(30 76 71 54)(31 77 72 55)(32 78 61 56)(33 79 62 57)(34 80 63 58)(35 81 64 59)(36 82 65 60)
(1 84)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 68)(14 69)(15 70)(16 71)(17 72)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 37)(36 38)(49 95)(50 96)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)
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), (1,67,7,61)(2,66,8,72)(3,65,9,71)(4,64,10,70)(5,63,11,69)(6,62,12,68)(13,77,19,83)(14,76,20,82)(15,75,21,81)(16,74,22,80)(17,73,23,79)(18,84,24,78)(25,91,31,85)(26,90,32,96)(27,89,33,95)(28,88,34,94)(29,87,35,93)(30,86,36,92)(37,59,43,53)(38,58,44,52)(39,57,45,51)(40,56,46,50)(41,55,47,49)(42,54,48,60), (1,40,96,24)(2,41,85,13)(3,42,86,14)(4,43,87,15)(5,44,88,16)(6,45,89,17)(7,46,90,18)(8,47,91,19)(9,48,92,20)(10,37,93,21)(11,38,94,22)(12,39,95,23)(25,83,66,49)(26,84,67,50)(27,73,68,51)(28,74,69,52)(29,75,70,53)(30,76,71,54)(31,77,72,55)(32,78,61,56)(33,79,62,57)(34,80,63,58)(35,81,64,59)(36,82,65,60), (1,84)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,68)(14,69)(15,70)(16,71)(17,72)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,37)(36,38)(49,95)(50,96)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)>;
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), (1,67,7,61)(2,66,8,72)(3,65,9,71)(4,64,10,70)(5,63,11,69)(6,62,12,68)(13,77,19,83)(14,76,20,82)(15,75,21,81)(16,74,22,80)(17,73,23,79)(18,84,24,78)(25,91,31,85)(26,90,32,96)(27,89,33,95)(28,88,34,94)(29,87,35,93)(30,86,36,92)(37,59,43,53)(38,58,44,52)(39,57,45,51)(40,56,46,50)(41,55,47,49)(42,54,48,60), (1,40,96,24)(2,41,85,13)(3,42,86,14)(4,43,87,15)(5,44,88,16)(6,45,89,17)(7,46,90,18)(8,47,91,19)(9,48,92,20)(10,37,93,21)(11,38,94,22)(12,39,95,23)(25,83,66,49)(26,84,67,50)(27,73,68,51)(28,74,69,52)(29,75,70,53)(30,76,71,54)(31,77,72,55)(32,78,61,56)(33,79,62,57)(34,80,63,58)(35,81,64,59)(36,82,65,60), (1,84)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,68)(14,69)(15,70)(16,71)(17,72)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,37)(36,38)(49,95)(50,96)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94) );
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)], [(1,67,7,61),(2,66,8,72),(3,65,9,71),(4,64,10,70),(5,63,11,69),(6,62,12,68),(13,77,19,83),(14,76,20,82),(15,75,21,81),(16,74,22,80),(17,73,23,79),(18,84,24,78),(25,91,31,85),(26,90,32,96),(27,89,33,95),(28,88,34,94),(29,87,35,93),(30,86,36,92),(37,59,43,53),(38,58,44,52),(39,57,45,51),(40,56,46,50),(41,55,47,49),(42,54,48,60)], [(1,40,96,24),(2,41,85,13),(3,42,86,14),(4,43,87,15),(5,44,88,16),(6,45,89,17),(7,46,90,18),(8,47,91,19),(9,48,92,20),(10,37,93,21),(11,38,94,22),(12,39,95,23),(25,83,66,49),(26,84,67,50),(27,73,68,51),(28,74,69,52),(29,75,70,53),(30,76,71,54),(31,77,72,55),(32,78,61,56),(33,79,62,57),(34,80,63,58),(35,81,64,59),(36,82,65,60)], [(1,84),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,81),(11,82),(12,83),(13,68),(14,69),(15,70),(16,71),(17,72),(18,61),(19,62),(20,63),(21,64),(22,65),(23,66),(24,67),(25,39),(26,40),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,37),(36,38),(49,95),(50,96),(51,85),(52,86),(53,87),(54,88),(55,89),(56,90),(57,91),(58,92),(59,93),(60,94)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 10 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 2 |
| 0 | 0 | 4 | 11 |
| 6 | 3 | 0 | 0 |
| 5 | 7 | 0 | 0 |
| 0 | 0 | 2 | 9 |
| 0 | 0 | 4 | 11 |
| 6 | 3 | 0 | 0 |
| 10 | 7 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,3,10,0,0,3,6],[1,0,0,0,0,1,0,0,0,0,2,4,0,0,2,11],[6,5,0,0,3,7,0,0,0,0,2,4,0,0,9,11],[6,10,0,0,3,7,0,0,0,0,12,0,0,0,0,12] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| 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 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2- (1+4) | S3×D4 | Q8○D12 |
| kernel | Dic6⋊23D4 | C4×Dic6 | C42⋊7S3 | Dic3⋊4D4 | C23.11D6 | D6⋊Q8 | C12.48D4 | C12⋊7D4 | C23.14D6 | D4×C12 | C22×Dic6 | C2×C4○D12 | C4×D4 | Dic6 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C2×C6 | C22 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
Dic_6\rtimes_{23}D_4 % in TeX
G:=Group("Dic6:23D4"); // GroupNames label
G:=SmallGroup(192,1111);
// by ID
G=gap.SmallGroup(192,1111);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,387,100,675,570,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^6*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations