metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.55D4, C14.6M4(2), C23.2Dic7, C22:(C7:C8), (C2xC14):2C8, C7:2(C22:C8), (C2xC28).4C4, C14.10(C2xC8), (C2xC4).94D14, (C2xC4).4Dic7, (C22xC4).1D7, C4.30(C7:D4), (C22xC14).5C4, (C22xC28).11C2, C2.3(C4.Dic7), C2.1(C23.D7), C22.9(C2xDic7), C14.12(C22:C4), (C2xC28).108C22, C2.5(C2xC7:C8), (C2xC7:C8):10C2, (C2xC14).27(C2xC4), SmallGroup(224,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.55D4
G = < a,b,c | a28=1, b4=a14, c2=a21, bab-1=cac-1=a13, cbc-1=a7b3 >
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, D7, C22:C4, C2xC8, M4(2), Dic7, D14, C22:C8, C7:C8, C2xDic7, C7:D4, C2xC7:C8, C4.Dic7, C23.D7, C28.55D4
Smallest permutation representation of C28.55D4
►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 68 102 36 15 82 88 50)(2 81 103 49 16 67 89 35)(3 66 104 34 17 80 90 48)(4 79 105 47 18 65 91 33)(5 64 106 32 19 78 92 46)(6 77 107 45 20 63 93 31)(7 62 108 30 21 76 94 44)(8 75 109 43 22 61 95 29)(9 60 110 56 23 74 96 42)(10 73 111 41 24 59 97 55)(11 58 112 54 25 72 98 40)(12 71 85 39 26 57 99 53)(13 84 86 52 27 70 100 38)(14 69 87 37 28 83 101 51)
(1 43 22 36 15 29 8 50)(2 56 23 49 16 42 9 35)(3 41 24 34 17 55 10 48)(4 54 25 47 18 40 11 33)(5 39 26 32 19 53 12 46)(6 52 27 45 20 38 13 31)(7 37 28 30 21 51 14 44)(57 99 78 92 71 85 64 106)(58 112 79 105 72 98 65 91)(59 97 80 90 73 111 66 104)(60 110 81 103 74 96 67 89)(61 95 82 88 75 109 68 102)(62 108 83 101 76 94 69 87)(63 93 84 86 77 107 70 100)
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), (1,68,102,36,15,82,88,50)(2,81,103,49,16,67,89,35)(3,66,104,34,17,80,90,48)(4,79,105,47,18,65,91,33)(5,64,106,32,19,78,92,46)(6,77,107,45,20,63,93,31)(7,62,108,30,21,76,94,44)(8,75,109,43,22,61,95,29)(9,60,110,56,23,74,96,42)(10,73,111,41,24,59,97,55)(11,58,112,54,25,72,98,40)(12,71,85,39,26,57,99,53)(13,84,86,52,27,70,100,38)(14,69,87,37,28,83,101,51), (1,43,22,36,15,29,8,50)(2,56,23,49,16,42,9,35)(3,41,24,34,17,55,10,48)(4,54,25,47,18,40,11,33)(5,39,26,32,19,53,12,46)(6,52,27,45,20,38,13,31)(7,37,28,30,21,51,14,44)(57,99,78,92,71,85,64,106)(58,112,79,105,72,98,65,91)(59,97,80,90,73,111,66,104)(60,110,81,103,74,96,67,89)(61,95,82,88,75,109,68,102)(62,108,83,101,76,94,69,87)(63,93,84,86,77,107,70,100)>;
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), (1,68,102,36,15,82,88,50)(2,81,103,49,16,67,89,35)(3,66,104,34,17,80,90,48)(4,79,105,47,18,65,91,33)(5,64,106,32,19,78,92,46)(6,77,107,45,20,63,93,31)(7,62,108,30,21,76,94,44)(8,75,109,43,22,61,95,29)(9,60,110,56,23,74,96,42)(10,73,111,41,24,59,97,55)(11,58,112,54,25,72,98,40)(12,71,85,39,26,57,99,53)(13,84,86,52,27,70,100,38)(14,69,87,37,28,83,101,51), (1,43,22,36,15,29,8,50)(2,56,23,49,16,42,9,35)(3,41,24,34,17,55,10,48)(4,54,25,47,18,40,11,33)(5,39,26,32,19,53,12,46)(6,52,27,45,20,38,13,31)(7,37,28,30,21,51,14,44)(57,99,78,92,71,85,64,106)(58,112,79,105,72,98,65,91)(59,97,80,90,73,111,66,104)(60,110,81,103,74,96,67,89)(61,95,82,88,75,109,68,102)(62,108,83,101,76,94,69,87)(63,93,84,86,77,107,70,100) );
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)], [(1,68,102,36,15,82,88,50),(2,81,103,49,16,67,89,35),(3,66,104,34,17,80,90,48),(4,79,105,47,18,65,91,33),(5,64,106,32,19,78,92,46),(6,77,107,45,20,63,93,31),(7,62,108,30,21,76,94,44),(8,75,109,43,22,61,95,29),(9,60,110,56,23,74,96,42),(10,73,111,41,24,59,97,55),(11,58,112,54,25,72,98,40),(12,71,85,39,26,57,99,53),(13,84,86,52,27,70,100,38),(14,69,87,37,28,83,101,51)], [(1,43,22,36,15,29,8,50),(2,56,23,49,16,42,9,35),(3,41,24,34,17,55,10,48),(4,54,25,47,18,40,11,33),(5,39,26,32,19,53,12,46),(6,52,27,45,20,38,13,31),(7,37,28,30,21,51,14,44),(57,99,78,92,71,85,64,106),(58,112,79,105,72,98,65,91),(59,97,80,90,73,111,66,104),(60,110,81,103,74,96,67,89),(61,95,82,88,75,109,68,102),(62,108,83,101,76,94,69,87),(63,93,84,86,77,107,70,100)]])
C28.55D4 is a maximal subgroup of
C14.C4wrC2 C4:Dic7:C4 (C22xD7):C8 (C2xDic7):C8 C24.Dic7 (C2xC28):C8 (D4xC14):C4 C4:C4:Dic7 Dic7.5M4(2) Dic7.M4(2) C56:C4:C2 D7xC22:C8 D14:M4(2) D14:C8:C2 C42.6Dic7 C42.7Dic7 (C2xC14).40D8 C4:C4.228D14 C4:C4.230D14 C4:C4.231D14 C4:C4.233D14 C42.187D14 C4:C4.236D14 D4xC7:C8 C42.47D14 C28:3M4(2) (C2xC14).D8 C4:D4.D7 (C2xD4).D14 D28:16D4 D28:17D4 Dic14:17D4 C22:Q8.D7 (C2xC14).Q16 C14.(C4oD8) D28.36D4 D28.37D4 Dic14.37D4 C8xC7:D4 C56:32D4 C56:D4 C56:18D4 C24.4Dic7 (C2xC14):8D8 (C7xD4).31D4 (C7xQ8):13D4 (C2xC14):8Q16 (D4xC14).11C4 (C7xD4):14D4 (C7xD4).32D4
C28.55D4 is a maximal quotient of
(C2xC28):3C8 C24.Dic7 (C2xC28):C8 C28.57D8 C28.26Q16 C56.91D4 C56.D4 C56.92D4
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | D7 | M4(2) | Dic7 | D14 | Dic7 | C7:D4 | C7:C8 | C4.Dic7 |
| kernel | C28.55D4 | C2xC7:C8 | C22xC28 | C2xC28 | C22xC14 | C2xC14 | C28 | C22xC4 | C14 | C2xC4 | C2xC4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 3 | 2 | 3 | 3 | 3 | 12 | 12 | 12 |
Matrix representation of C28.55D4 ►in GL4(F113) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 32 | 83 |
| 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 2 | 103 |
| 0 | 0 | 81 | 111 |
| 0 | 1 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 2 | 103 |
| 0 | 0 | 81 | 111 |
G:=sub<GL(4,GF(113))| [15,0,0,0,0,15,0,0,0,0,32,0,0,0,83,60],[0,98,0,0,1,0,0,0,0,0,2,81,0,0,103,111],[0,15,0,0,1,0,0,0,0,0,2,81,0,0,103,111] >;
C28.55D4 in GAP, Magma, Sage, TeX
C_{28}._{55}D_4 % in TeX
G:=Group("C28.55D4"); // GroupNames label
G:=SmallGroup(224,36);
// by ID
G=gap.SmallGroup(224,36);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,86,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=1,b^4=a^14,c^2=a^21,b*a*b^-1=c*a*c^-1=a^13,c*b*c^-1=a^7*b^3>;
// generators/relations
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