metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊19D4, C6.692- (1+4), C4⋊D4⋊6S3, C4⋊C4.176D6, C3⋊3(Q8⋊5D4), C4.108(S3×D4), C22⋊C4.5D6, C4.D12⋊19C2, (D4×Dic3)⋊15C2, (C2×D4).151D6, C12.224(C2×D4), C6.61(C22×D4), C23.14D6⋊9C2, (C2×C6).142C24, D6⋊C4.11C22, C2.27(Q8○D12), Dic3.20(C2×D4), (C22×C4).234D6, Dic6⋊C4⋊19C2, C23.12D6⋊14C2, C22⋊2(D4⋊2S3), (C2×C12).500C23, (C22×Dic6)⋊16C2, (C6×D4).116C22, C23.19(C22×S3), (C22×C6).13C23, Dic3.D4⋊16C2, C23.11D6⋊17C2, Dic3⋊C4.13C22, (C22×S3).61C23, C4⋊Dic3.204C22, C22.163(S3×C23), (C4×Dic3).89C22, (C2×Dic3).65C23, (C22×C12).236C22, (C2×Dic6).245C22, C6.D4.20C22, (C22×Dic3).103C22, C2.34(C2×S3×D4), (C2×C6)⋊4(C4○D4), (C3×C4⋊D4)⋊7C2, (C4×C3⋊D4)⋊14C2, C6.80(C2×C4○D4), (S3×C2×C4).81C22, (C2×D4⋊2S3)⋊10C2, C2.31(C2×D4⋊2S3), (C3×C4⋊C4).138C22, (C2×C4).173(C22×S3), (C3×C22⋊C4).7C22, (C2×C3⋊D4).118C22, SmallGroup(192,1157)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 704 in 290 conjugacy classes, 107 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C3, C4 [×2], C4 [×12], C22, C22 [×2], C22 [×11], S3, C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×19], D4 [×12], Q8 [×10], C23, C23 [×2], C23, Dic3 [×4], Dic3 [×5], C12 [×2], C12 [×3], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×8], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×5], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×3], C2×Q8 [×8], C4○D4 [×4], Dic6 [×4], Dic6 [×6], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×4], C2×Dic3 [×8], C3⋊D4 [×6], C2×C12 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×6], C22×S3, C22×C6, C22×C6 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C4⋊D4 [×2], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, C4×Dic3, C4×Dic3 [×2], Dic3⋊C4, Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4, D6⋊C4 [×2], C6.D4, C6.D4 [×4], C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6 [×2], C2×Dic6 [×2], C2×Dic6 [×4], S3×C2×C4, D4⋊2S3 [×4], C22×Dic3 [×4], C2×C3⋊D4, C2×C3⋊D4 [×2], C22×C12, C6×D4, C6×D4 [×2], Q8⋊5D4, Dic3.D4 [×2], C23.11D6 [×2], Dic6⋊C4, C4.D12, C4×C3⋊D4, D4×Dic3 [×2], C23.12D6, C23.14D6 [×2], C3×C4⋊D4, C22×Dic6, C2×D4⋊2S3, Dic6⋊19D4
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], D4⋊2S3 [×2], S3×C23, Q8⋊5D4, C2×S3×D4, C2×D4⋊2S3, Q8○D12, Dic6⋊19D4
Generators and relations
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=a7, ad=da, bc=cb, 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 56 7 50)(2 55 8 49)(3 54 9 60)(4 53 10 59)(5 52 11 58)(6 51 12 57)(13 32 19 26)(14 31 20 25)(15 30 21 36)(16 29 22 35)(17 28 23 34)(18 27 24 33)(37 72 43 66)(38 71 44 65)(39 70 45 64)(40 69 46 63)(41 68 47 62)(42 67 48 61)(73 87 79 93)(74 86 80 92)(75 85 81 91)(76 96 82 90)(77 95 83 89)(78 94 84 88)
(1 30 96 45)(2 25 85 40)(3 32 86 47)(4 27 87 42)(5 34 88 37)(6 29 89 44)(7 36 90 39)(8 31 91 46)(9 26 92 41)(10 33 93 48)(11 28 94 43)(12 35 95 38)(13 74 68 60)(14 81 69 55)(15 76 70 50)(16 83 71 57)(17 78 72 52)(18 73 61 59)(19 80 62 54)(20 75 63 49)(21 82 64 56)(22 77 65 51)(23 84 66 58)(24 79 67 53)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 73)(37 57)(38 58)(39 59)(40 60)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)
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), (1,56,7,50)(2,55,8,49)(3,54,9,60)(4,53,10,59)(5,52,11,58)(6,51,12,57)(13,32,19,26)(14,31,20,25)(15,30,21,36)(16,29,22,35)(17,28,23,34)(18,27,24,33)(37,72,43,66)(38,71,44,65)(39,70,45,64)(40,69,46,63)(41,68,47,62)(42,67,48,61)(73,87,79,93)(74,86,80,92)(75,85,81,91)(76,96,82,90)(77,95,83,89)(78,94,84,88), (1,30,96,45)(2,25,85,40)(3,32,86,47)(4,27,87,42)(5,34,88,37)(6,29,89,44)(7,36,90,39)(8,31,91,46)(9,26,92,41)(10,33,93,48)(11,28,94,43)(12,35,95,38)(13,74,68,60)(14,81,69,55)(15,76,70,50)(16,83,71,57)(17,78,72,52)(18,73,61,59)(19,80,62,54)(20,75,63,49)(21,82,64,56)(22,77,65,51)(23,84,66,58)(24,79,67,53), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,73)(37,57)(38,58)(39,59)(40,60)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)>;
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), (1,56,7,50)(2,55,8,49)(3,54,9,60)(4,53,10,59)(5,52,11,58)(6,51,12,57)(13,32,19,26)(14,31,20,25)(15,30,21,36)(16,29,22,35)(17,28,23,34)(18,27,24,33)(37,72,43,66)(38,71,44,65)(39,70,45,64)(40,69,46,63)(41,68,47,62)(42,67,48,61)(73,87,79,93)(74,86,80,92)(75,85,81,91)(76,96,82,90)(77,95,83,89)(78,94,84,88), (1,30,96,45)(2,25,85,40)(3,32,86,47)(4,27,87,42)(5,34,88,37)(6,29,89,44)(7,36,90,39)(8,31,91,46)(9,26,92,41)(10,33,93,48)(11,28,94,43)(12,35,95,38)(13,74,68,60)(14,81,69,55)(15,76,70,50)(16,83,71,57)(17,78,72,52)(18,73,61,59)(19,80,62,54)(20,75,63,49)(21,82,64,56)(22,77,65,51)(23,84,66,58)(24,79,67,53), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,73)(37,57)(38,58)(39,59)(40,60)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56) );
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)], [(1,56,7,50),(2,55,8,49),(3,54,9,60),(4,53,10,59),(5,52,11,58),(6,51,12,57),(13,32,19,26),(14,31,20,25),(15,30,21,36),(16,29,22,35),(17,28,23,34),(18,27,24,33),(37,72,43,66),(38,71,44,65),(39,70,45,64),(40,69,46,63),(41,68,47,62),(42,67,48,61),(73,87,79,93),(74,86,80,92),(75,85,81,91),(76,96,82,90),(77,95,83,89),(78,94,84,88)], [(1,30,96,45),(2,25,85,40),(3,32,86,47),(4,27,87,42),(5,34,88,37),(6,29,89,44),(7,36,90,39),(8,31,91,46),(9,26,92,41),(10,33,93,48),(11,28,94,43),(12,35,95,38),(13,74,68,60),(14,81,69,55),(15,76,70,50),(16,83,71,57),(17,78,72,52),(18,73,61,59),(19,80,62,54),(20,75,63,49),(21,82,64,56),(22,77,65,51),(23,84,66,58),(24,79,67,53)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,81),(33,82),(34,83),(35,84),(36,73),(37,57),(38,58),(39,59),(40,60),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 10 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,5,0,0,0,0,0,5,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,10,8,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,2,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 4 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 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 | D4⋊2S3 | Q8○D12 |
| kernel | Dic6⋊19D4 | Dic3.D4 | C23.11D6 | Dic6⋊C4 | C4.D12 | C4×C3⋊D4 | D4×Dic3 | C23.12D6 | C23.14D6 | C3×C4⋊D4 | C22×Dic6 | C2×D4⋊2S3 | C4⋊D4 | Dic6 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C2×C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 1 | 3 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
Dic_6\rtimes_{19}D_4 % in TeX
G:=Group("Dic6:19D4"); // GroupNames label
G:=SmallGroup(192,1157);
// by ID
G=gap.SmallGroup(192,1157);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,477,232,570,185,6278]);
// 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=a^7,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations