direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic3⋊4D4, C24.64D6, C6⋊2(C4×D4), C23⋊5(C4×S3), C22⋊C4⋊50D6, Dic3⋊9(C2×D4), D6⋊2(C22×C4), C6.8(C23×C4), D6⋊C4⋊56C22, (C2×Dic3)⋊23D4, (C2×C6).29C24, C6.34(C22×D4), Dic3⋊1(C22×C4), (C23×Dic3)⋊3C2, (C22×C4).328D6, C22.124(S3×D4), (C2×C12).570C23, Dic3⋊C4⋊57C22, (C4×Dic3)⋊73C22, C22.18(S3×C23), (C23×C6).55C22, (S3×C23).94C22, (C22×C6).121C23, C23.329(C22×S3), C22.66(D4⋊2S3), (C22×S3).149C23, (C22×C12).351C22, (C2×Dic3).303C23, (C22×Dic3)⋊41C22, C3⋊2(C2×C4×D4), C2.2(C2×S3×D4), C22⋊3(S3×C2×C4), (C2×C3⋊D4)⋊8C4, C3⋊D4⋊5(C2×C4), (C2×D6⋊C4)⋊30C2, (S3×C2×C4)⋊64C22, (S3×C22×C4)⋊16C2, (C2×C6)⋊3(C22×C4), (C22×C6)⋊9(C2×C4), (C2×C4×Dic3)⋊30C2, C6.67(C2×C4○D4), C2.10(S3×C22×C4), (C2×C22⋊C4)⋊19S3, (C6×C22⋊C4)⋊25C2, C2.2(C2×D4⋊2S3), (C2×C6).380(C2×D4), (C2×Dic3⋊C4)⋊35C2, (C22×S3)⋊10(C2×C4), (C2×Dic3)⋊16(C2×C4), (C22×C3⋊D4).8C2, (C2×C6).167(C4○D4), (C3×C22⋊C4)⋊60C22, (C2×C4).256(C22×S3), (C2×C3⋊D4).87C22, SmallGroup(192,1044)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic3⋊4D4
G = < a,b,c,d,e | a2=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 968 in 426 conjugacy classes, 175 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×Dic3, C22×Dic3, C2×C3⋊D4, C22×C12, S3×C23, C23×C6, C2×C4×D4, Dic3⋊4D4, C2×C4×Dic3, C2×Dic3⋊C4, C2×D6⋊C4, C6×C22⋊C4, S3×C22×C4, C23×Dic3, C22×C3⋊D4, C2×Dic3⋊4D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C24, C4×S3, C22×S3, C4×D4, C23×C4, C22×D4, C2×C4○D4, S3×C2×C4, S3×D4, D4⋊2S3, S3×C23, C2×C4×D4, Dic3⋊4D4, S3×C22×C4, C2×S3×D4, C2×D4⋊2S3, C2×Dic3⋊4D4
►On 96 points
(1 65)(2 66)(3 61)(4 62)(5 63)(6 64)(7 44)(8 45)(9 46)(10 47)(11 48)(12 43)(13 60)(14 55)(15 56)(16 57)(17 58)(18 59)(19 74)(20 75)(21 76)(22 77)(23 78)(24 73)(25 72)(26 67)(27 68)(28 69)(29 70)(30 71)(31 86)(32 87)(33 88)(34 89)(35 90)(36 85)(37 84)(38 79)(39 80)(40 81)(41 82)(42 83)(49 96)(50 91)(51 92)(52 93)(53 94)(54 95)
(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 26 4 29)(2 25 5 28)(3 30 6 27)(7 84 10 81)(8 83 11 80)(9 82 12 79)(13 23 16 20)(14 22 17 19)(15 21 18 24)(31 53 34 50)(32 52 35 49)(33 51 36 54)(37 47 40 44)(38 46 41 43)(39 45 42 48)(55 77 58 74)(56 76 59 73)(57 75 60 78)(61 71 64 68)(62 70 65 67)(63 69 66 72)(85 95 88 92)(86 94 89 91)(87 93 90 96)
(1 82 17 89)(2 81 18 88)(3 80 13 87)(4 79 14 86)(5 84 15 85)(6 83 16 90)(7 24 92 25)(8 23 93 30)(9 22 94 29)(10 21 95 28)(11 20 96 27)(12 19 91 26)(31 62 38 55)(32 61 39 60)(33 66 40 59)(34 65 41 58)(35 64 42 57)(36 63 37 56)(43 74 50 67)(44 73 51 72)(45 78 52 71)(46 77 53 70)(47 76 54 69)(48 75 49 68)
(1 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 75)(8 76)(9 77)(10 78)(11 73)(12 74)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(55 79)(56 80)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)
G:=sub<Sym(96)| (1,65)(2,66)(3,61)(4,62)(5,63)(6,64)(7,44)(8,45)(9,46)(10,47)(11,48)(12,43)(13,60)(14,55)(15,56)(16,57)(17,58)(18,59)(19,74)(20,75)(21,76)(22,77)(23,78)(24,73)(25,72)(26,67)(27,68)(28,69)(29,70)(30,71)(31,86)(32,87)(33,88)(34,89)(35,90)(36,85)(37,84)(38,79)(39,80)(40,81)(41,82)(42,83)(49,96)(50,91)(51,92)(52,93)(53,94)(54,95), (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), (1,26,4,29)(2,25,5,28)(3,30,6,27)(7,84,10,81)(8,83,11,80)(9,82,12,79)(13,23,16,20)(14,22,17,19)(15,21,18,24)(31,53,34,50)(32,52,35,49)(33,51,36,54)(37,47,40,44)(38,46,41,43)(39,45,42,48)(55,77,58,74)(56,76,59,73)(57,75,60,78)(61,71,64,68)(62,70,65,67)(63,69,66,72)(85,95,88,92)(86,94,89,91)(87,93,90,96), (1,82,17,89)(2,81,18,88)(3,80,13,87)(4,79,14,86)(5,84,15,85)(6,83,16,90)(7,24,92,25)(8,23,93,30)(9,22,94,29)(10,21,95,28)(11,20,96,27)(12,19,91,26)(31,62,38,55)(32,61,39,60)(33,66,40,59)(34,65,41,58)(35,64,42,57)(36,63,37,56)(43,74,50,67)(44,73,51,72)(45,78,52,71)(46,77,53,70)(47,76,54,69)(48,75,49,68), (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,75)(8,76)(9,77)(10,78)(11,73)(12,74)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)>;
G:=Group( (1,65)(2,66)(3,61)(4,62)(5,63)(6,64)(7,44)(8,45)(9,46)(10,47)(11,48)(12,43)(13,60)(14,55)(15,56)(16,57)(17,58)(18,59)(19,74)(20,75)(21,76)(22,77)(23,78)(24,73)(25,72)(26,67)(27,68)(28,69)(29,70)(30,71)(31,86)(32,87)(33,88)(34,89)(35,90)(36,85)(37,84)(38,79)(39,80)(40,81)(41,82)(42,83)(49,96)(50,91)(51,92)(52,93)(53,94)(54,95), (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), (1,26,4,29)(2,25,5,28)(3,30,6,27)(7,84,10,81)(8,83,11,80)(9,82,12,79)(13,23,16,20)(14,22,17,19)(15,21,18,24)(31,53,34,50)(32,52,35,49)(33,51,36,54)(37,47,40,44)(38,46,41,43)(39,45,42,48)(55,77,58,74)(56,76,59,73)(57,75,60,78)(61,71,64,68)(62,70,65,67)(63,69,66,72)(85,95,88,92)(86,94,89,91)(87,93,90,96), (1,82,17,89)(2,81,18,88)(3,80,13,87)(4,79,14,86)(5,84,15,85)(6,83,16,90)(7,24,92,25)(8,23,93,30)(9,22,94,29)(10,21,95,28)(11,20,96,27)(12,19,91,26)(31,62,38,55)(32,61,39,60)(33,66,40,59)(34,65,41,58)(35,64,42,57)(36,63,37,56)(43,74,50,67)(44,73,51,72)(45,78,52,71)(46,77,53,70)(47,76,54,69)(48,75,49,68), (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,75)(8,76)(9,77)(10,78)(11,73)(12,74)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96) );
G=PermutationGroup([[(1,65),(2,66),(3,61),(4,62),(5,63),(6,64),(7,44),(8,45),(9,46),(10,47),(11,48),(12,43),(13,60),(14,55),(15,56),(16,57),(17,58),(18,59),(19,74),(20,75),(21,76),(22,77),(23,78),(24,73),(25,72),(26,67),(27,68),(28,69),(29,70),(30,71),(31,86),(32,87),(33,88),(34,89),(35,90),(36,85),(37,84),(38,79),(39,80),(40,81),(41,82),(42,83),(49,96),(50,91),(51,92),(52,93),(53,94),(54,95)], [(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)], [(1,26,4,29),(2,25,5,28),(3,30,6,27),(7,84,10,81),(8,83,11,80),(9,82,12,79),(13,23,16,20),(14,22,17,19),(15,21,18,24),(31,53,34,50),(32,52,35,49),(33,51,36,54),(37,47,40,44),(38,46,41,43),(39,45,42,48),(55,77,58,74),(56,76,59,73),(57,75,60,78),(61,71,64,68),(62,70,65,67),(63,69,66,72),(85,95,88,92),(86,94,89,91),(87,93,90,96)], [(1,82,17,89),(2,81,18,88),(3,80,13,87),(4,79,14,86),(5,84,15,85),(6,83,16,90),(7,24,92,25),(8,23,93,30),(9,22,94,29),(10,21,95,28),(11,20,96,27),(12,19,91,26),(31,62,38,55),(32,61,39,60),(33,66,40,59),(34,65,41,58),(35,64,42,57),(36,63,37,56),(43,74,50,67),(44,73,51,72),(45,78,52,71),(46,77,53,70),(47,76,54,69),(48,75,49,68)], [(1,34),(2,35),(3,36),(4,31),(5,32),(6,33),(7,75),(8,76),(9,77),(10,78),(11,73),(12,74),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(55,79),(56,80),(57,81),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4X | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C4×S3 | S3×D4 | D4⋊2S3 |
| kernel | C2×Dic3⋊4D4 | Dic3⋊4D4 | C2×C4×Dic3 | C2×Dic3⋊C4 | C2×D6⋊C4 | C6×C22⋊C4 | S3×C22×C4 | C23×Dic3 | C22×C3⋊D4 | C2×C3⋊D4 | C2×C22⋊C4 | C2×Dic3 | C22⋊C4 | C22×C4 | C24 | C2×C6 | C23 | C22 | C22 |
| # reps | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 16 | 1 | 4 | 4 | 2 | 1 | 4 | 8 | 2 | 2 |
Matrix representation of C2×Dic3⋊4D4 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 5 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 8 |
| 0 | 0 | 0 | 3 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 3 | 1 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,5,0,0,0,0,8,8,0,0,0,0,0,5,0,0,0,0,0,5],[12,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,12,3,0,0,0,8,1],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,3,0,0,0,0,1] >;
C2×Dic3⋊4D4 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_3\rtimes_4D_4 % in TeX
G:=Group("C2xDic3:4D4"); // GroupNames label
G:=SmallGroup(192,1044);
// by ID
G=gap.SmallGroup(192,1044);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=d^4=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations