metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊2D4, D14⋊3D4, C23.8D14, (C2×D4)⋊4D7, (D4×C14)⋊3C2, C7⋊4(C4⋊D4), C4⋊2(C7⋊D4), C2.26(D4×D7), C4⋊Dic7⋊14C2, (C2×C4).51D14, C14.50(C2×D4), C23.D7⋊11C2, C14.31(C4○D4), (C2×C14).53C23, (C2×C28).34C22, C2.17(D4⋊2D7), C22.60(C22×D7), (C22×C14).20C22, (C2×Dic7).19C22, (C22×D7).26C22, (C2×C4×D7)⋊2C2, (C2×C7⋊D4)⋊5C2, C2.14(C2×C7⋊D4), SmallGroup(224,133)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊2D4
G = < a,b,c | a28=b4=c2=1, bab-1=a-1, cac=a13, cbc=b-1 >
Subgroups: 382 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, Dic7, C28, D14, D14, C2×C14, C2×C14, C4⋊D4, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, C4⋊Dic7, C23.D7, C2×C4×D7, C2×C7⋊D4, D4×C14, C28⋊2D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C7⋊D4, C22×D7, D4×D7, D4⋊2D7, C2×C7⋊D4, C28⋊2D4
►On 112 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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 53 99 65)(2 52 100 64)(3 51 101 63)(4 50 102 62)(5 49 103 61)(6 48 104 60)(7 47 105 59)(8 46 106 58)(9 45 107 57)(10 44 108 84)(11 43 109 83)(12 42 110 82)(13 41 111 81)(14 40 112 80)(15 39 85 79)(16 38 86 78)(17 37 87 77)(18 36 88 76)(19 35 89 75)(20 34 90 74)(21 33 91 73)(22 32 92 72)(23 31 93 71)(24 30 94 70)(25 29 95 69)(26 56 96 68)(27 55 97 67)(28 54 98 66)
(1 15)(2 28)(3 13)(4 26)(5 11)(6 24)(7 9)(8 22)(10 20)(12 18)(14 16)(17 27)(19 25)(21 23)(29 75)(30 60)(31 73)(32 58)(33 71)(34 84)(35 69)(36 82)(37 67)(38 80)(39 65)(40 78)(41 63)(42 76)(43 61)(44 74)(45 59)(46 72)(47 57)(48 70)(49 83)(50 68)(51 81)(52 66)(53 79)(54 64)(55 77)(56 62)(85 99)(86 112)(87 97)(88 110)(89 95)(90 108)(91 93)(92 106)(94 104)(96 102)(98 100)(101 111)(103 109)(105 107)
G:=sub<Sym(112)| (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,53,99,65)(2,52,100,64)(3,51,101,63)(4,50,102,62)(5,49,103,61)(6,48,104,60)(7,47,105,59)(8,46,106,58)(9,45,107,57)(10,44,108,84)(11,43,109,83)(12,42,110,82)(13,41,111,81)(14,40,112,80)(15,39,85,79)(16,38,86,78)(17,37,87,77)(18,36,88,76)(19,35,89,75)(20,34,90,74)(21,33,91,73)(22,32,92,72)(23,31,93,71)(24,30,94,70)(25,29,95,69)(26,56,96,68)(27,55,97,67)(28,54,98,66), (1,15)(2,28)(3,13)(4,26)(5,11)(6,24)(7,9)(8,22)(10,20)(12,18)(14,16)(17,27)(19,25)(21,23)(29,75)(30,60)(31,73)(32,58)(33,71)(34,84)(35,69)(36,82)(37,67)(38,80)(39,65)(40,78)(41,63)(42,76)(43,61)(44,74)(45,59)(46,72)(47,57)(48,70)(49,83)(50,68)(51,81)(52,66)(53,79)(54,64)(55,77)(56,62)(85,99)(86,112)(87,97)(88,110)(89,95)(90,108)(91,93)(92,106)(94,104)(96,102)(98,100)(101,111)(103,109)(105,107)>;
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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,53,99,65)(2,52,100,64)(3,51,101,63)(4,50,102,62)(5,49,103,61)(6,48,104,60)(7,47,105,59)(8,46,106,58)(9,45,107,57)(10,44,108,84)(11,43,109,83)(12,42,110,82)(13,41,111,81)(14,40,112,80)(15,39,85,79)(16,38,86,78)(17,37,87,77)(18,36,88,76)(19,35,89,75)(20,34,90,74)(21,33,91,73)(22,32,92,72)(23,31,93,71)(24,30,94,70)(25,29,95,69)(26,56,96,68)(27,55,97,67)(28,54,98,66), (1,15)(2,28)(3,13)(4,26)(5,11)(6,24)(7,9)(8,22)(10,20)(12,18)(14,16)(17,27)(19,25)(21,23)(29,75)(30,60)(31,73)(32,58)(33,71)(34,84)(35,69)(36,82)(37,67)(38,80)(39,65)(40,78)(41,63)(42,76)(43,61)(44,74)(45,59)(46,72)(47,57)(48,70)(49,83)(50,68)(51,81)(52,66)(53,79)(54,64)(55,77)(56,62)(85,99)(86,112)(87,97)(88,110)(89,95)(90,108)(91,93)(92,106)(94,104)(96,102)(98,100)(101,111)(103,109)(105,107) );
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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,53,99,65),(2,52,100,64),(3,51,101,63),(4,50,102,62),(5,49,103,61),(6,48,104,60),(7,47,105,59),(8,46,106,58),(9,45,107,57),(10,44,108,84),(11,43,109,83),(12,42,110,82),(13,41,111,81),(14,40,112,80),(15,39,85,79),(16,38,86,78),(17,37,87,77),(18,36,88,76),(19,35,89,75),(20,34,90,74),(21,33,91,73),(22,32,92,72),(23,31,93,71),(24,30,94,70),(25,29,95,69),(26,56,96,68),(27,55,97,67),(28,54,98,66)], [(1,15),(2,28),(3,13),(4,26),(5,11),(6,24),(7,9),(8,22),(10,20),(12,18),(14,16),(17,27),(19,25),(21,23),(29,75),(30,60),(31,73),(32,58),(33,71),(34,84),(35,69),(36,82),(37,67),(38,80),(39,65),(40,78),(41,63),(42,76),(43,61),(44,74),(45,59),(46,72),(47,57),(48,70),(49,83),(50,68),(51,81),(52,66),(53,79),(54,64),(55,77),(56,62),(85,99),(86,112),(87,97),(88,110),(89,95),(90,108),(91,93),(92,106),(94,104),(96,102),(98,100),(101,111),(103,109),(105,107)]])
C28⋊2D4 is a maximal subgroup of
D14.D8 D14⋊D8 D14.SD16 D14⋊SD16 C8⋊Dic7⋊C2 C7⋊C8⋊1D4 C7⋊C8⋊D4 C56⋊1C4⋊C2 D28⋊D4 C56⋊6D4 Dic14⋊D4 C56⋊12D4 Dic14⋊7D4 C56⋊14D4 D28⋊7D4 C56⋊8D4 C42.228D14 D28⋊24D4 C42.229D14 C42.113D14 C42.115D14 C42.116D14 C42.117D14 C24⋊2D14 C24.33D14 C24.35D14 C24.36D14 D7×C4⋊D4 C4⋊C4⋊21D14 C14.382+ 1+4 C14.722- 1+4 C14.732- 1+4 D28⋊20D4 C14.422+ 1+4 C14.432+ 1+4 C14.442+ 1+4 C14.452+ 1+4 C14.1152+ 1+4 C14.472+ 1+4 C14.482+ 1+4 C14.492+ 1+4 C14.612+ 1+4 C14.832- 1+4 C14.642+ 1+4 C14.862- 1+4 D28⋊10D4 Dic14⋊10D4 C42.234D14 C42.144D14 C42.238D14 D28⋊11D4 Dic14⋊11D4 C42.168D14 D4×C7⋊D4 C24.41D14 C24.42D14 (C2×C28)⋊15D4 C14.1072- 1+4 C14.1082- 1+4 C14.1482+ 1+4
C28⋊2D4 is a maximal quotient of
C24.3D14 C24.6D14 C24.12D14 C23.16D28 C28⋊(C4⋊C4) (C2×C28).287D4 C4⋊(D14⋊C4) (C2×C28).290D4 C42.61D14 D28.23D4 C28⋊2D8 Dic14⋊9D4 C28⋊5SD16 C28⋊Q16 C56⋊6D4 C56⋊12D4 C56.23D4 C56⋊14D4 C56⋊8D4 C56.44D4 D14⋊3Q16 C56.36D4 C56.29D4 C24.19D14 C24.20D14 C24.21D14
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 2 | 2 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | D4×D7 | D4⋊2D7 |
| kernel | C28⋊2D4 | C4⋊Dic7 | C23.D7 | C2×C4×D7 | C2×C7⋊D4 | D4×C14 | C28 | D14 | C2×D4 | C14 | C2×C4 | C23 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 3 | 2 | 3 | 6 | 12 | 3 | 3 |
Matrix representation of C28⋊2D4 ►in GL4(𝔽29) generated by
| 1 | 22 | 0 | 0 |
| 7 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 17 | 17 |
| 20 | 15 | 0 | 0 |
| 10 | 9 | 0 | 0 |
| 0 | 0 | 28 | 27 |
| 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 7 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(29))| [1,7,0,0,22,10,0,0,0,0,12,17,0,0,0,17],[20,10,0,0,15,9,0,0,0,0,28,1,0,0,27,1],[1,7,0,0,0,28,0,0,0,0,28,1,0,0,0,1] >;
C28⋊2D4 in GAP, Magma, Sage, TeX
C_{28}\rtimes_2D_4 % in TeX
G:=Group("C28:2D4"); // GroupNames label
G:=SmallGroup(224,133);
// by ID
G=gap.SmallGroup(224,133);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^13,c*b*c=b^-1>;
// generators/relations