metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.2C42, C4⋊C4⋊4Dic7, C14.13C4≀C2, C28.8(C4⋊C4), (C2×C28).7Q8, (C4×Dic7)⋊5C4, C4.Dic7⋊3C4, C4.2(C4×Dic7), C7⋊2(C42⋊6C4), (C2×C4).126D28, (C2×C28).488D4, C4.2(C4⋊Dic7), C4.46(D14⋊C4), (C2×C4).24Dic14, C42⋊C2.2D7, (C22×C14).43D4, C28.61(C22⋊C4), C4.30(Dic7⋊C4), (C22×C4).324D14, C23.47(C7⋊D4), C22.18(D14⋊C4), C2.1(D4⋊2Dic7), C22.4(Dic7⋊C4), C14.8(C2.C42), C2.9(C14.C42), (C22×C28).121C22, C22.28(C23.D7), (C7×C4⋊C4)⋊6C4, (C2×C4).68(C4×D7), (C2×C4×Dic7).1C2, (C2×C14).4(C4⋊C4), (C2×C28).57(C2×C4), (C2×C4).36(C2×Dic7), (C2×C4.Dic7).8C2, (C2×C4).268(C7⋊D4), (C7×C42⋊C2).2C2, (C2×C14).90(C22⋊C4), SmallGroup(448,89)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.2C42
G = < a,b,c | a28=c4=1, b4=a14, bab-1=a-1, cac-1=a15, cbc-1=a21b >
Subgroups: 404 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2×C42, C42⋊C2, C2×M4(2), C7⋊C8, C2×Dic7, C2×C28, C2×C28, C22×C14, C42⋊6C4, C2×C7⋊C8, C4.Dic7, C4.Dic7, C4×Dic7, C4×Dic7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×Dic7, C22×C28, C2×C4.Dic7, C2×C4×Dic7, C7×C42⋊C2, C28.2C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, C4≀C2, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C42⋊6C4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C14.C42, D4⋊2Dic7, C28.2C42
►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 30 86 15 69 44 100)(2 82 31 85 16 68 45 99)(3 81 32 112 17 67 46 98)(4 80 33 111 18 66 47 97)(5 79 34 110 19 65 48 96)(6 78 35 109 20 64 49 95)(7 77 36 108 21 63 50 94)(8 76 37 107 22 62 51 93)(9 75 38 106 23 61 52 92)(10 74 39 105 24 60 53 91)(11 73 40 104 25 59 54 90)(12 72 41 103 26 58 55 89)(13 71 42 102 27 57 56 88)(14 70 43 101 28 84 29 87)
(1 100 51 62)(2 87 52 77)(3 102 53 64)(4 89 54 79)(5 104 55 66)(6 91 56 81)(7 106 29 68)(8 93 30 83)(9 108 31 70)(10 95 32 57)(11 110 33 72)(12 97 34 59)(13 112 35 74)(14 99 36 61)(15 86 37 76)(16 101 38 63)(17 88 39 78)(18 103 40 65)(19 90 41 80)(20 105 42 67)(21 92 43 82)(22 107 44 69)(23 94 45 84)(24 109 46 71)(25 96 47 58)(26 111 48 73)(27 98 49 60)(28 85 50 75)
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), (1,83,30,86,15,69,44,100)(2,82,31,85,16,68,45,99)(3,81,32,112,17,67,46,98)(4,80,33,111,18,66,47,97)(5,79,34,110,19,65,48,96)(6,78,35,109,20,64,49,95)(7,77,36,108,21,63,50,94)(8,76,37,107,22,62,51,93)(9,75,38,106,23,61,52,92)(10,74,39,105,24,60,53,91)(11,73,40,104,25,59,54,90)(12,72,41,103,26,58,55,89)(13,71,42,102,27,57,56,88)(14,70,43,101,28,84,29,87), (1,100,51,62)(2,87,52,77)(3,102,53,64)(4,89,54,79)(5,104,55,66)(6,91,56,81)(7,106,29,68)(8,93,30,83)(9,108,31,70)(10,95,32,57)(11,110,33,72)(12,97,34,59)(13,112,35,74)(14,99,36,61)(15,86,37,76)(16,101,38,63)(17,88,39,78)(18,103,40,65)(19,90,41,80)(20,105,42,67)(21,92,43,82)(22,107,44,69)(23,94,45,84)(24,109,46,71)(25,96,47,58)(26,111,48,73)(27,98,49,60)(28,85,50,75)>;
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), (1,83,30,86,15,69,44,100)(2,82,31,85,16,68,45,99)(3,81,32,112,17,67,46,98)(4,80,33,111,18,66,47,97)(5,79,34,110,19,65,48,96)(6,78,35,109,20,64,49,95)(7,77,36,108,21,63,50,94)(8,76,37,107,22,62,51,93)(9,75,38,106,23,61,52,92)(10,74,39,105,24,60,53,91)(11,73,40,104,25,59,54,90)(12,72,41,103,26,58,55,89)(13,71,42,102,27,57,56,88)(14,70,43,101,28,84,29,87), (1,100,51,62)(2,87,52,77)(3,102,53,64)(4,89,54,79)(5,104,55,66)(6,91,56,81)(7,106,29,68)(8,93,30,83)(9,108,31,70)(10,95,32,57)(11,110,33,72)(12,97,34,59)(13,112,35,74)(14,99,36,61)(15,86,37,76)(16,101,38,63)(17,88,39,78)(18,103,40,65)(19,90,41,80)(20,105,42,67)(21,92,43,82)(22,107,44,69)(23,94,45,84)(24,109,46,71)(25,96,47,58)(26,111,48,73)(27,98,49,60)(28,85,50,75) );
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)], [(1,83,30,86,15,69,44,100),(2,82,31,85,16,68,45,99),(3,81,32,112,17,67,46,98),(4,80,33,111,18,66,47,97),(5,79,34,110,19,65,48,96),(6,78,35,109,20,64,49,95),(7,77,36,108,21,63,50,94),(8,76,37,107,22,62,51,93),(9,75,38,106,23,61,52,92),(10,74,39,105,24,60,53,91),(11,73,40,104,25,59,54,90),(12,72,41,103,26,58,55,89),(13,71,42,102,27,57,56,88),(14,70,43,101,28,84,29,87)], [(1,100,51,62),(2,87,52,77),(3,102,53,64),(4,89,54,79),(5,104,55,66),(6,91,56,81),(7,106,29,68),(8,93,30,83),(9,108,31,70),(10,95,32,57),(11,110,33,72),(12,97,34,59),(13,112,35,74),(14,99,36,61),(15,86,37,76),(16,101,38,63),(17,88,39,78),(18,103,40,65),(19,90,41,80),(20,105,42,67),(21,92,43,82),(22,107,44,69),(23,94,45,84),(24,109,46,71),(25,96,47,58),(26,111,48,73),(27,98,49,60),(28,85,50,75)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4R | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | D7 | Dic7 | D14 | C4≀C2 | Dic14 | C4×D7 | D28 | C7⋊D4 | C7⋊D4 | D4⋊2Dic7 |
| kernel | C28.2C42 | C2×C4.Dic7 | C2×C4×Dic7 | C7×C42⋊C2 | C4.Dic7 | C4×Dic7 | C7×C4⋊C4 | C2×C28 | C2×C28 | C22×C14 | C42⋊C2 | C4⋊C4 | C22×C4 | C14 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 3 | 6 | 3 | 8 | 6 | 12 | 6 | 6 | 6 | 12 |
Matrix representation of C28.2C42 ►in GL4(𝔽113) generated by
| 15 | 0 | 0 | 0 |
| 98 | 98 | 0 | 0 |
| 0 | 0 | 0 | 24 |
| 0 | 0 | 80 | 9 |
| 1 | 2 | 0 | 0 |
| 7 | 112 | 0 | 0 |
| 0 | 0 | 98 | 76 |
| 0 | 0 | 0 | 15 |
| 112 | 111 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
G:=sub<GL(4,GF(113))| [15,98,0,0,0,98,0,0,0,0,0,80,0,0,24,9],[1,7,0,0,2,112,0,0,0,0,98,0,0,0,76,15],[112,1,0,0,111,1,0,0,0,0,15,0,0,0,0,15] >;
C28.2C42 in GAP, Magma, Sage, TeX
C_{28}._2C_4^2 % in TeX
G:=Group("C28.2C4^2"); // GroupNames label
G:=SmallGroup(448,89);
// by ID
G=gap.SmallGroup(448,89);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,184,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=c^4=1,b^4=a^14,b*a*b^-1=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^21*b>;
// generators/relations