metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.17D4, C23.7D14, (C2×D4).6D7, (C4×Dic7)⋊5C2, (D4×C14).5C2, (C2×C4).50D14, C14.48(C2×D4), C4.7(C7⋊D4), C7⋊3(C4.4D4), C23.D7⋊9C2, (C2×Dic14)⋊10C2, C14.30(C4○D4), (C2×C28).33C22, (C2×C14).51C23, C2.16(D4⋊2D7), C22.58(C22×D7), (C22×C14).19C22, (C2×Dic7).18C22, C2.12(C2×C7⋊D4), SmallGroup(224,131)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.17D4
G = < a,b,c | a28=b4=1, c2=a14, bab-1=a13, cac-1=a-1, cbc-1=a14b-1 >
Subgroups: 254 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C42, C22⋊C4, C2×D4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C4.4D4, Dic14, C2×Dic7, C2×C28, C7×D4, C22×C14, C4×Dic7, C23.D7, C2×Dic14, D4×C14, C28.17D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, C7⋊D4, C22×D7, D4⋊2D7, C2×C7⋊D4, C28.17D4
►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 83 49 112)(2 68 50 97)(3 81 51 110)(4 66 52 95)(5 79 53 108)(6 64 54 93)(7 77 55 106)(8 62 56 91)(9 75 29 104)(10 60 30 89)(11 73 31 102)(12 58 32 87)(13 71 33 100)(14 84 34 85)(15 69 35 98)(16 82 36 111)(17 67 37 96)(18 80 38 109)(19 65 39 94)(20 78 40 107)(21 63 41 92)(22 76 42 105)(23 61 43 90)(24 74 44 103)(25 59 45 88)(26 72 46 101)(27 57 47 86)(28 70 48 99)
(1 62 15 76)(2 61 16 75)(3 60 17 74)(4 59 18 73)(5 58 19 72)(6 57 20 71)(7 84 21 70)(8 83 22 69)(9 82 23 68)(10 81 24 67)(11 80 25 66)(12 79 26 65)(13 78 27 64)(14 77 28 63)(29 111 43 97)(30 110 44 96)(31 109 45 95)(32 108 46 94)(33 107 47 93)(34 106 48 92)(35 105 49 91)(36 104 50 90)(37 103 51 89)(38 102 52 88)(39 101 53 87)(40 100 54 86)(41 99 55 85)(42 98 56 112)
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), (1,83,49,112)(2,68,50,97)(3,81,51,110)(4,66,52,95)(5,79,53,108)(6,64,54,93)(7,77,55,106)(8,62,56,91)(9,75,29,104)(10,60,30,89)(11,73,31,102)(12,58,32,87)(13,71,33,100)(14,84,34,85)(15,69,35,98)(16,82,36,111)(17,67,37,96)(18,80,38,109)(19,65,39,94)(20,78,40,107)(21,63,41,92)(22,76,42,105)(23,61,43,90)(24,74,44,103)(25,59,45,88)(26,72,46,101)(27,57,47,86)(28,70,48,99), (1,62,15,76)(2,61,16,75)(3,60,17,74)(4,59,18,73)(5,58,19,72)(6,57,20,71)(7,84,21,70)(8,83,22,69)(9,82,23,68)(10,81,24,67)(11,80,25,66)(12,79,26,65)(13,78,27,64)(14,77,28,63)(29,111,43,97)(30,110,44,96)(31,109,45,95)(32,108,46,94)(33,107,47,93)(34,106,48,92)(35,105,49,91)(36,104,50,90)(37,103,51,89)(38,102,52,88)(39,101,53,87)(40,100,54,86)(41,99,55,85)(42,98,56,112)>;
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), (1,83,49,112)(2,68,50,97)(3,81,51,110)(4,66,52,95)(5,79,53,108)(6,64,54,93)(7,77,55,106)(8,62,56,91)(9,75,29,104)(10,60,30,89)(11,73,31,102)(12,58,32,87)(13,71,33,100)(14,84,34,85)(15,69,35,98)(16,82,36,111)(17,67,37,96)(18,80,38,109)(19,65,39,94)(20,78,40,107)(21,63,41,92)(22,76,42,105)(23,61,43,90)(24,74,44,103)(25,59,45,88)(26,72,46,101)(27,57,47,86)(28,70,48,99), (1,62,15,76)(2,61,16,75)(3,60,17,74)(4,59,18,73)(5,58,19,72)(6,57,20,71)(7,84,21,70)(8,83,22,69)(9,82,23,68)(10,81,24,67)(11,80,25,66)(12,79,26,65)(13,78,27,64)(14,77,28,63)(29,111,43,97)(30,110,44,96)(31,109,45,95)(32,108,46,94)(33,107,47,93)(34,106,48,92)(35,105,49,91)(36,104,50,90)(37,103,51,89)(38,102,52,88)(39,101,53,87)(40,100,54,86)(41,99,55,85)(42,98,56,112) );
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)], [(1,83,49,112),(2,68,50,97),(3,81,51,110),(4,66,52,95),(5,79,53,108),(6,64,54,93),(7,77,55,106),(8,62,56,91),(9,75,29,104),(10,60,30,89),(11,73,31,102),(12,58,32,87),(13,71,33,100),(14,84,34,85),(15,69,35,98),(16,82,36,111),(17,67,37,96),(18,80,38,109),(19,65,39,94),(20,78,40,107),(21,63,41,92),(22,76,42,105),(23,61,43,90),(24,74,44,103),(25,59,45,88),(26,72,46,101),(27,57,47,86),(28,70,48,99)], [(1,62,15,76),(2,61,16,75),(3,60,17,74),(4,59,18,73),(5,58,19,72),(6,57,20,71),(7,84,21,70),(8,83,22,69),(9,82,23,68),(10,81,24,67),(11,80,25,66),(12,79,26,65),(13,78,27,64),(14,77,28,63),(29,111,43,97),(30,110,44,96),(31,109,45,95),(32,108,46,94),(33,107,47,93),(34,106,48,92),(35,105,49,91),(36,104,50,90),(37,103,51,89),(38,102,52,88),(39,101,53,87),(40,100,54,86),(41,99,55,85),(42,98,56,112)]])
C28.17D4 is a maximal subgroup of
C23.D28 D28.1D4 C28⋊Q8⋊C2 Dic14.D4 (C8×Dic7)⋊C2 D28.D4 (C2×D8).D7 C56⋊11D4 C56.22D4 (C7×Q8).D4 C56.31D4 C56.43D4 D28.38D4 2+ 1+4.D7 C42.106D14 C42.229D14 C42.114D14 C42.115D14 C24.32D14 C24.35D14 C24⋊4D14 Dic14⋊19D4 C4⋊C4.178D14 C14.712- 1+4 D28⋊20D4 C14.422+ 1+4 C14.452+ 1+4 C14.492+ 1+4 C14.602+ 1+4 C14.622+ 1+4 C14.842- 1+4 C42.139D14 D7×C4.4D4 C42.141D14 C42.166D14 C42⋊26D14 C42.238D14 C24.41D14 C24.42D14 C14.1052- 1+4 C14.1072- 1+4 (C2×C28)⋊17D4
C28.17D4 is a maximal quotient of
C24.4D14 C23⋊Dic14 C24.10D14 (C2×Dic7)⋊6Q8 (C4×Dic7)⋊9C4 (C2×C28).288D4 C42.62D14 C42.213D14 C28.16D8 C42.72D14 C28.Q16 C42.77D14 C24.19D14 C24.20D14
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | D4⋊2D7 |
| kernel | C28.17D4 | C4×Dic7 | C23.D7 | C2×Dic14 | D4×C14 | C28 | C2×D4 | C14 | C2×C4 | C23 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 3 | 4 | 3 | 6 | 12 | 6 |
Matrix representation of C28.17D4 ►in GL4(𝔽29) generated by
| 24 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 28 | 22 |
| 0 | 0 | 21 | 1 |
| 0 | 1 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 17 | 3 |
| 0 | 0 | 20 | 12 |
| 0 | 28 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 9 | 17 |
G:=sub<GL(4,GF(29))| [24,0,0,0,0,23,0,0,0,0,28,21,0,0,22,1],[0,28,0,0,1,0,0,0,0,0,17,20,0,0,3,12],[0,28,0,0,28,0,0,0,0,0,12,9,0,0,0,17] >;
C28.17D4 in GAP, Magma, Sage, TeX
C_{28}._{17}D_4 % in TeX
G:=Group("C28.17D4"); // GroupNames label
G:=SmallGroup(224,131);
// by ID
G=gap.SmallGroup(224,131);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,55,506,116,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=1,c^2=a^14,b*a*b^-1=a^13,c*a*c^-1=a^-1,c*b*c^-1=a^14*b^-1>;
// generators/relations