metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.48D4, C22⋊2Dic14, C23.20D14, (C2×C14)⋊3Q8, C4⋊Dic7⋊8C2, C7⋊4(C22⋊Q8), C14.8(C2×Q8), Dic7⋊C4⋊2C2, C14.39(C2×D4), (C2×C4).83D14, (C22×C4).5D7, (C2×Dic14)⋊6C2, C4.23(C7⋊D4), (C22×C28).6C2, C2.9(C2×Dic14), C23.D7.4C2, C2.17(C4○D28), C14.15(C4○D4), (C2×C14).42C23, (C2×C28).91C22, C22.54(C22×D7), (C22×C14).34C22, (C2×Dic7).14C22, C2.5(C2×C7⋊D4), SmallGroup(224,119)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.48D4
G = < a,b,c | a28=b4=1, c2=a14, bab-1=cac-1=a-1, cbc-1=a14b-1 >
Subgroups: 238 in 74 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊Q8, Dic14, C2×Dic7, C2×C28, C2×C28, C22×C14, Dic7⋊C4, C4⋊Dic7, C23.D7, C2×Dic14, C22×C28, C28.48D4
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, Dic14, C7⋊D4, C22×D7, C2×Dic14, C4○D28, C2×C7⋊D4, C28.48D4
►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)
(1 104 33 63)(2 103 34 62)(3 102 35 61)(4 101 36 60)(5 100 37 59)(6 99 38 58)(7 98 39 57)(8 97 40 84)(9 96 41 83)(10 95 42 82)(11 94 43 81)(12 93 44 80)(13 92 45 79)(14 91 46 78)(15 90 47 77)(16 89 48 76)(17 88 49 75)(18 87 50 74)(19 86 51 73)(20 85 52 72)(21 112 53 71)(22 111 54 70)(23 110 55 69)(24 109 56 68)(25 108 29 67)(26 107 30 66)(27 106 31 65)(28 105 32 64)
(1 77 15 63)(2 76 16 62)(3 75 17 61)(4 74 18 60)(5 73 19 59)(6 72 20 58)(7 71 21 57)(8 70 22 84)(9 69 23 83)(10 68 24 82)(11 67 25 81)(12 66 26 80)(13 65 27 79)(14 64 28 78)(29 94 43 108)(30 93 44 107)(31 92 45 106)(32 91 46 105)(33 90 47 104)(34 89 48 103)(35 88 49 102)(36 87 50 101)(37 86 51 100)(38 85 52 99)(39 112 53 98)(40 111 54 97)(41 110 55 96)(42 109 56 95)
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), (1,104,33,63)(2,103,34,62)(3,102,35,61)(4,101,36,60)(5,100,37,59)(6,99,38,58)(7,98,39,57)(8,97,40,84)(9,96,41,83)(10,95,42,82)(11,94,43,81)(12,93,44,80)(13,92,45,79)(14,91,46,78)(15,90,47,77)(16,89,48,76)(17,88,49,75)(18,87,50,74)(19,86,51,73)(20,85,52,72)(21,112,53,71)(22,111,54,70)(23,110,55,69)(24,109,56,68)(25,108,29,67)(26,107,30,66)(27,106,31,65)(28,105,32,64), (1,77,15,63)(2,76,16,62)(3,75,17,61)(4,74,18,60)(5,73,19,59)(6,72,20,58)(7,71,21,57)(8,70,22,84)(9,69,23,83)(10,68,24,82)(11,67,25,81)(12,66,26,80)(13,65,27,79)(14,64,28,78)(29,94,43,108)(30,93,44,107)(31,92,45,106)(32,91,46,105)(33,90,47,104)(34,89,48,103)(35,88,49,102)(36,87,50,101)(37,86,51,100)(38,85,52,99)(39,112,53,98)(40,111,54,97)(41,110,55,96)(42,109,56,95)>;
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), (1,104,33,63)(2,103,34,62)(3,102,35,61)(4,101,36,60)(5,100,37,59)(6,99,38,58)(7,98,39,57)(8,97,40,84)(9,96,41,83)(10,95,42,82)(11,94,43,81)(12,93,44,80)(13,92,45,79)(14,91,46,78)(15,90,47,77)(16,89,48,76)(17,88,49,75)(18,87,50,74)(19,86,51,73)(20,85,52,72)(21,112,53,71)(22,111,54,70)(23,110,55,69)(24,109,56,68)(25,108,29,67)(26,107,30,66)(27,106,31,65)(28,105,32,64), (1,77,15,63)(2,76,16,62)(3,75,17,61)(4,74,18,60)(5,73,19,59)(6,72,20,58)(7,71,21,57)(8,70,22,84)(9,69,23,83)(10,68,24,82)(11,67,25,81)(12,66,26,80)(13,65,27,79)(14,64,28,78)(29,94,43,108)(30,93,44,107)(31,92,45,106)(32,91,46,105)(33,90,47,104)(34,89,48,103)(35,88,49,102)(36,87,50,101)(37,86,51,100)(38,85,52,99)(39,112,53,98)(40,111,54,97)(41,110,55,96)(42,109,56,95) );
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)], [(1,104,33,63),(2,103,34,62),(3,102,35,61),(4,101,36,60),(5,100,37,59),(6,99,38,58),(7,98,39,57),(8,97,40,84),(9,96,41,83),(10,95,42,82),(11,94,43,81),(12,93,44,80),(13,92,45,79),(14,91,46,78),(15,90,47,77),(16,89,48,76),(17,88,49,75),(18,87,50,74),(19,86,51,73),(20,85,52,72),(21,112,53,71),(22,111,54,70),(23,110,55,69),(24,109,56,68),(25,108,29,67),(26,107,30,66),(27,106,31,65),(28,105,32,64)], [(1,77,15,63),(2,76,16,62),(3,75,17,61),(4,74,18,60),(5,73,19,59),(6,72,20,58),(7,71,21,57),(8,70,22,84),(9,69,23,83),(10,68,24,82),(11,67,25,81),(12,66,26,80),(13,65,27,79),(14,64,28,78),(29,94,43,108),(30,93,44,107),(31,92,45,106),(32,91,46,105),(33,90,47,104),(34,89,48,103),(35,88,49,102),(36,87,50,101),(37,86,51,100),(38,85,52,99),(39,112,53,98),(40,111,54,97),(41,110,55,96),(42,109,56,95)]])
C28.48D4 is a maximal subgroup of
C4⋊Dic7⋊C4 C23.30D28 C23.34D28 C23.35D28 C23.10D28 D28.31D4 D28.32D4 C22⋊Dic28 C4⋊C4.230D14 C4⋊C4.231D14 C4⋊C4.233D14 C7⋊C8⋊23D4 C7⋊C8⋊5D4 C7⋊C8.29D4 C7⋊C8.6D4 C56⋊30D4 C56.82D4 C56⋊2D4 C56.4D4 (C7×D4).31D4 (C2×C14)⋊8Q16 (C7×D4).32D4 C42.274D14 C42.277D14 C23⋊2Dic14 C24.30D14 C24.31D14 C14.72+ 1+4 C14.102+ 1+4 C14.52- 1+4 C14.62- 1+4 C42.89D14 C42.94D14 C42.98D14 C42.99D14 D4×Dic14 D4⋊5Dic14 C42.105D14 C42.106D14 D4⋊6Dic14 D28⋊23D4 D28⋊24D4 Dic14⋊23D4 C42⋊16D14 C42.115D14 C42.118D14 C4⋊C4.178D14 C14.352+ 1+4 C14.712- 1+4 C4⋊C4⋊21D14 C14.722- 1+4 C14.462+ 1+4 C14.492+ 1+4 (Q8×Dic7)⋊C2 C14.752- 1+4 C14.152- 1+4 D7×C22⋊Q8 C14.162- 1+4 C14.512+ 1+4 C14.582+ 1+4 C14.602+ 1+4 C14.832- 1+4 C14.842- 1+4 C14.862- 1+4 C24.72D14 C24.42D14 Q8×C7⋊D4 C14.1042- 1+4 C14.1052- 1+4 C14.1072- 1+4 C14.1082- 1+4
C28.48D4 is a maximal quotient of
C28⋊4(C4⋊C4) (C2×C28)⋊10Q8 (C2×C42).D7 C23⋊Dic14 C24.6D14 C24.7D14 (C2×C4)⋊Dic14 (C2×C28).54D4 (C2×C28).55D4 C28.50D8 C28.38SD16 D4.3Dic14 C28.48SD16 C28.23Q16 Q8.3Dic14 C24.62D14 C23.27D28
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | Dic14 | C4○D28 |
| kernel | C28.48D4 | Dic7⋊C4 | C4⋊Dic7 | C23.D7 | C2×Dic14 | C22×C28 | C28 | C2×C14 | C22×C4 | C14 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 3 | 2 | 6 | 3 | 12 | 12 | 12 |
Matrix representation of C28.48D4 ►in GL4(𝔽29) generated by
| 21 | 0 | 0 | 0 |
| 23 | 18 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 20 |
| 22 | 11 | 0 | 0 |
| 22 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 |
| 22 | 11 | 0 | 0 |
| 6 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(29))| [21,23,0,0,0,18,0,0,0,0,16,0,0,0,0,20],[22,22,0,0,11,7,0,0,0,0,0,28,0,0,1,0],[22,6,0,0,11,7,0,0,0,0,0,1,0,0,1,0] >;
C28.48D4 in GAP, Magma, Sage, TeX
C_{28}._{48}D_4 % in TeX
G:=Group("C28.48D4"); // GroupNames label
G:=SmallGroup(224,119);
// by ID
G=gap.SmallGroup(224,119);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,217,103,218,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=1,c^2=a^14,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^14*b^-1>;
// generators/relations