direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4⋊Dic3, (C6×D4)⋊5C4, C6.71(C2×D8), (C2×C6).47D8, D4⋊3(C2×Dic3), (C2×D4)⋊3Dic3, C6⋊3(D4⋊C4), (C2×D4).195D6, (C2×C12).188D4, C12.202(C2×D4), C6.63(C2×SD16), (C2×C6).34SD16, (C22×D4).2S3, C12.79(C22×C4), C4⋊Dic3⋊67C22, (C22×C6).194D4, (C22×C4).368D6, C4.9(C22×Dic3), C12.30(C22⋊C4), (C2×C12).469C23, C22.25(D4⋊S3), C4.7(C6.D4), (C6×D4).237C22, C23.107(C3⋊D4), C22.12(D4.S3), (C22×C12).194C22, C22.33(C6.D4), (D4×C2×C6).1C2, C3⋊4(C2×D4⋊C4), (C22×C3⋊C8)⋊7C2, C2.4(C2×D4⋊S3), (C3×D4)⋊16(C2×C4), (C2×C3⋊C8)⋊32C22, C4.88(C2×C3⋊D4), C2.4(C2×D4.S3), (C2×C4⋊Dic3)⋊34C2, (C2×C6).551(C2×D4), C6.71(C2×C22⋊C4), (C2×C12).115(C2×C4), (C2×C4).49(C2×Dic3), C2.7(C2×C6.D4), C22.89(C2×C3⋊D4), (C2×C4).147(C3⋊D4), (C2×C4).556(C22×S3), (C2×C6).110(C22⋊C4), SmallGroup(192,773)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4⋊Dic3
G = < a,b,c,d,e | a2=b4=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >
Subgroups: 488 in 202 conjugacy classes, 87 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C22×C6, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C2×C3⋊C8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C22×Dic3, C22×C12, C6×D4, C6×D4, C23×C6, C2×D4⋊C4, D4⋊Dic3, C22×C3⋊C8, C2×C4⋊Dic3, D4×C2×C6, C2×D4⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, D8, SD16, C22×C4, C2×D4, C2×Dic3, C3⋊D4, C22×S3, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, D4⋊S3, D4.S3, C6.D4, C22×Dic3, C2×C3⋊D4, C2×D4⋊C4, D4⋊Dic3, C2×D4⋊S3, C2×D4.S3, C2×C6.D4, C2×D4⋊Dic3
►On 96 points
(1 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 39)(8 40)(9 41)(10 42)(11 37)(12 38)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)(25 53)(26 54)(27 49)(28 50)(29 51)(30 52)(55 82)(56 83)(57 84)(58 79)(59 80)(60 81)(61 88)(62 89)(63 90)(64 85)(65 86)(66 87)(67 94)(68 95)(69 96)(70 91)(71 92)(72 93)
(1 29 19 7)(2 30 20 8)(3 25 21 9)(4 26 22 10)(5 27 23 11)(6 28 24 12)(13 80 90 95)(14 81 85 96)(15 82 86 91)(16 83 87 92)(17 84 88 93)(18 79 89 94)(31 54 43 42)(32 49 44 37)(33 50 45 38)(34 51 46 39)(35 52 47 40)(36 53 48 41)(55 65 70 75)(56 66 71 76)(57 61 72 77)(58 62 67 78)(59 63 68 73)(60 64 69 74)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(19 29)(20 30)(21 25)(22 26)(23 27)(24 28)(31 42)(32 37)(33 38)(34 39)(35 40)(36 41)(43 54)(44 49)(45 50)(46 51)(47 52)(48 53)(55 70)(56 71)(57 72)(58 67)(59 68)(60 69)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)
(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 83 4 80)(2 82 5 79)(3 81 6 84)(7 87 10 90)(8 86 11 89)(9 85 12 88)(13 29 16 26)(14 28 17 25)(15 27 18 30)(19 92 22 95)(20 91 23 94)(21 96 24 93)(31 59 34 56)(32 58 35 55)(33 57 36 60)(37 62 40 65)(38 61 41 64)(39 66 42 63)(43 68 46 71)(44 67 47 70)(45 72 48 69)(49 78 52 75)(50 77 53 74)(51 76 54 73)
G:=sub<Sym(96)| (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,39)(8,40)(9,41)(10,42)(11,37)(12,38)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45)(25,53)(26,54)(27,49)(28,50)(29,51)(30,52)(55,82)(56,83)(57,84)(58,79)(59,80)(60,81)(61,88)(62,89)(63,90)(64,85)(65,86)(66,87)(67,94)(68,95)(69,96)(70,91)(71,92)(72,93), (1,29,19,7)(2,30,20,8)(3,25,21,9)(4,26,22,10)(5,27,23,11)(6,28,24,12)(13,80,90,95)(14,81,85,96)(15,82,86,91)(16,83,87,92)(17,84,88,93)(18,79,89,94)(31,54,43,42)(32,49,44,37)(33,50,45,38)(34,51,46,39)(35,52,47,40)(36,53,48,41)(55,65,70,75)(56,66,71,76)(57,61,72,77)(58,62,67,78)(59,63,68,73)(60,64,69,74), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(19,29)(20,30)(21,25)(22,26)(23,27)(24,28)(31,42)(32,37)(33,38)(34,39)(35,40)(36,41)(43,54)(44,49)(45,50)(46,51)(47,52)(48,53)(55,70)(56,71)(57,72)(58,67)(59,68)(60,69)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93), (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), (1,83,4,80)(2,82,5,79)(3,81,6,84)(7,87,10,90)(8,86,11,89)(9,85,12,88)(13,29,16,26)(14,28,17,25)(15,27,18,30)(19,92,22,95)(20,91,23,94)(21,96,24,93)(31,59,34,56)(32,58,35,55)(33,57,36,60)(37,62,40,65)(38,61,41,64)(39,66,42,63)(43,68,46,71)(44,67,47,70)(45,72,48,69)(49,78,52,75)(50,77,53,74)(51,76,54,73)>;
G:=Group( (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,39)(8,40)(9,41)(10,42)(11,37)(12,38)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45)(25,53)(26,54)(27,49)(28,50)(29,51)(30,52)(55,82)(56,83)(57,84)(58,79)(59,80)(60,81)(61,88)(62,89)(63,90)(64,85)(65,86)(66,87)(67,94)(68,95)(69,96)(70,91)(71,92)(72,93), (1,29,19,7)(2,30,20,8)(3,25,21,9)(4,26,22,10)(5,27,23,11)(6,28,24,12)(13,80,90,95)(14,81,85,96)(15,82,86,91)(16,83,87,92)(17,84,88,93)(18,79,89,94)(31,54,43,42)(32,49,44,37)(33,50,45,38)(34,51,46,39)(35,52,47,40)(36,53,48,41)(55,65,70,75)(56,66,71,76)(57,61,72,77)(58,62,67,78)(59,63,68,73)(60,64,69,74), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(19,29)(20,30)(21,25)(22,26)(23,27)(24,28)(31,42)(32,37)(33,38)(34,39)(35,40)(36,41)(43,54)(44,49)(45,50)(46,51)(47,52)(48,53)(55,70)(56,71)(57,72)(58,67)(59,68)(60,69)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93), (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), (1,83,4,80)(2,82,5,79)(3,81,6,84)(7,87,10,90)(8,86,11,89)(9,85,12,88)(13,29,16,26)(14,28,17,25)(15,27,18,30)(19,92,22,95)(20,91,23,94)(21,96,24,93)(31,59,34,56)(32,58,35,55)(33,57,36,60)(37,62,40,65)(38,61,41,64)(39,66,42,63)(43,68,46,71)(44,67,47,70)(45,72,48,69)(49,78,52,75)(50,77,53,74)(51,76,54,73) );
G=PermutationGroup([[(1,34),(2,35),(3,36),(4,31),(5,32),(6,33),(7,39),(8,40),(9,41),(10,42),(11,37),(12,38),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,46),(20,47),(21,48),(22,43),(23,44),(24,45),(25,53),(26,54),(27,49),(28,50),(29,51),(30,52),(55,82),(56,83),(57,84),(58,79),(59,80),(60,81),(61,88),(62,89),(63,90),(64,85),(65,86),(66,87),(67,94),(68,95),(69,96),(70,91),(71,92),(72,93)], [(1,29,19,7),(2,30,20,8),(3,25,21,9),(4,26,22,10),(5,27,23,11),(6,28,24,12),(13,80,90,95),(14,81,85,96),(15,82,86,91),(16,83,87,92),(17,84,88,93),(18,79,89,94),(31,54,43,42),(32,49,44,37),(33,50,45,38),(34,51,46,39),(35,52,47,40),(36,53,48,41),(55,65,70,75),(56,66,71,76),(57,61,72,77),(58,62,67,78),(59,63,68,73),(60,64,69,74)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(19,29),(20,30),(21,25),(22,26),(23,27),(24,28),(31,42),(32,37),(33,38),(34,39),(35,40),(36,41),(43,54),(44,49),(45,50),(46,51),(47,52),(48,53),(55,70),(56,71),(57,72),(58,67),(59,68),(60,69),(79,94),(80,95),(81,96),(82,91),(83,92),(84,93)], [(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)], [(1,83,4,80),(2,82,5,79),(3,81,6,84),(7,87,10,90),(8,86,11,89),(9,85,12,88),(13,29,16,26),(14,28,17,25),(15,27,18,30),(19,92,22,95),(20,91,23,94),(21,96,24,93),(31,59,34,56),(32,58,35,55),(33,57,36,60),(37,62,40,65),(38,61,41,64),(39,66,42,63),(43,68,46,71),(44,67,47,70),(45,72,48,69),(49,78,52,75),(50,77,53,74),(51,76,54,73)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6G | 6H | ··· | 6O | 8A | ··· | 8H | 12A | 12B | 12C | 12D |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | Dic3 | D6 | D8 | SD16 | C3⋊D4 | C3⋊D4 | D4⋊S3 | D4.S3 |
| kernel | C2×D4⋊Dic3 | D4⋊Dic3 | C22×C3⋊C8 | C2×C4⋊Dic3 | D4×C2×C6 | C6×D4 | C22×D4 | C2×C12 | C22×C6 | C22×C4 | C2×D4 | C2×D4 | C2×C6 | C2×C6 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 3 | 1 | 1 | 4 | 2 | 4 | 4 | 6 | 2 | 2 | 2 |
Matrix representation of C2×D4⋊Dic3 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 67 |
| 0 | 0 | 0 | 0 | 67 | 67 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,27,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,6,67,0,0,0,0,67,67] >;
C2×D4⋊Dic3 in GAP, Magma, Sage, TeX
C_2\times D_4\rtimes {\rm Dic}_3 % in TeX
G:=Group("C2xD4:Dic3"); // GroupNames label
G:=SmallGroup(192,773);
// by ID
G=gap.SmallGroup(192,773);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations