metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.21C42, C23.3Dic14, (C2×C56).3C4, (C2×C4).15D28, (C2×C8).3Dic7, C7⋊(C4.10C42), (C2×C28).113D4, C4.26(C4×Dic7), C4.36(D14⋊C4), (C22×C14).9Q8, (C22×C4).65D14, C28.12(C22⋊C4), (C2×M4(2)).11D7, C4.20(C23.D7), (C14×M4(2)).15C2, C22.7(Dic7⋊C4), C22.11(C4⋊Dic7), (C22×C28).129C22, C2.19(C14.C42), C14.19(C2.C42), (C2×C7⋊C8).3C4, (C2×C28).66(C2×C4), (C2×C4).143(C4×D7), (C2×C14).10(C4⋊C4), (C2×C4).23(C7⋊D4), (C2×C4).78(C2×Dic7), (C2×C4.Dic7).14C2, SmallGroup(448,117)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.21C42
G = < a,b,c | a28=1, b4=c4=a14, bab-1=a13, ac=ca, cbc-1=a21b >
Subgroups: 260 in 86 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C2×C8, C2×C8, M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C2×M4(2), C2×M4(2), C7⋊C8, C56, C2×C28, C2×C28, C22×C14, C4.10C42, C2×C7⋊C8, C4.Dic7, C2×C56, C7×M4(2), C22×C28, C2×C4.Dic7, C14×M4(2), C28.21C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C4.10C42, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C14.C42, C28.21C42
►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 61 8 68 15 75 22 82)(2 74 9 81 16 60 23 67)(3 59 10 66 17 73 24 80)(4 72 11 79 18 58 25 65)(5 57 12 64 19 71 26 78)(6 70 13 77 20 84 27 63)(7 83 14 62 21 69 28 76)(29 102 50 95 43 88 36 109)(30 87 51 108 44 101 37 94)(31 100 52 93 45 86 38 107)(32 85 53 106 46 99 39 92)(33 98 54 91 47 112 40 105)(34 111 55 104 48 97 41 90)(35 96 56 89 49 110 42 103)
(1 38 8 45 15 52 22 31)(2 39 9 46 16 53 23 32)(3 40 10 47 17 54 24 33)(4 41 11 48 18 55 25 34)(5 42 12 49 19 56 26 35)(6 43 13 50 20 29 27 36)(7 44 14 51 21 30 28 37)(57 110 78 103 71 96 64 89)(58 111 79 104 72 97 65 90)(59 112 80 105 73 98 66 91)(60 85 81 106 74 99 67 92)(61 86 82 107 75 100 68 93)(62 87 83 108 76 101 69 94)(63 88 84 109 77 102 70 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), (1,61,8,68,15,75,22,82)(2,74,9,81,16,60,23,67)(3,59,10,66,17,73,24,80)(4,72,11,79,18,58,25,65)(5,57,12,64,19,71,26,78)(6,70,13,77,20,84,27,63)(7,83,14,62,21,69,28,76)(29,102,50,95,43,88,36,109)(30,87,51,108,44,101,37,94)(31,100,52,93,45,86,38,107)(32,85,53,106,46,99,39,92)(33,98,54,91,47,112,40,105)(34,111,55,104,48,97,41,90)(35,96,56,89,49,110,42,103), (1,38,8,45,15,52,22,31)(2,39,9,46,16,53,23,32)(3,40,10,47,17,54,24,33)(4,41,11,48,18,55,25,34)(5,42,12,49,19,56,26,35)(6,43,13,50,20,29,27,36)(7,44,14,51,21,30,28,37)(57,110,78,103,71,96,64,89)(58,111,79,104,72,97,65,90)(59,112,80,105,73,98,66,91)(60,85,81,106,74,99,67,92)(61,86,82,107,75,100,68,93)(62,87,83,108,76,101,69,94)(63,88,84,109,77,102,70,95)>;
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), (1,61,8,68,15,75,22,82)(2,74,9,81,16,60,23,67)(3,59,10,66,17,73,24,80)(4,72,11,79,18,58,25,65)(5,57,12,64,19,71,26,78)(6,70,13,77,20,84,27,63)(7,83,14,62,21,69,28,76)(29,102,50,95,43,88,36,109)(30,87,51,108,44,101,37,94)(31,100,52,93,45,86,38,107)(32,85,53,106,46,99,39,92)(33,98,54,91,47,112,40,105)(34,111,55,104,48,97,41,90)(35,96,56,89,49,110,42,103), (1,38,8,45,15,52,22,31)(2,39,9,46,16,53,23,32)(3,40,10,47,17,54,24,33)(4,41,11,48,18,55,25,34)(5,42,12,49,19,56,26,35)(6,43,13,50,20,29,27,36)(7,44,14,51,21,30,28,37)(57,110,78,103,71,96,64,89)(58,111,79,104,72,97,65,90)(59,112,80,105,73,98,66,91)(60,85,81,106,74,99,67,92)(61,86,82,107,75,100,68,93)(62,87,83,108,76,101,69,94)(63,88,84,109,77,102,70,95) );
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)], [(1,61,8,68,15,75,22,82),(2,74,9,81,16,60,23,67),(3,59,10,66,17,73,24,80),(4,72,11,79,18,58,25,65),(5,57,12,64,19,71,26,78),(6,70,13,77,20,84,27,63),(7,83,14,62,21,69,28,76),(29,102,50,95,43,88,36,109),(30,87,51,108,44,101,37,94),(31,100,52,93,45,86,38,107),(32,85,53,106,46,99,39,92),(33,98,54,91,47,112,40,105),(34,111,55,104,48,97,41,90),(35,96,56,89,49,110,42,103)], [(1,38,8,45,15,52,22,31),(2,39,9,46,16,53,23,32),(3,40,10,47,17,54,24,33),(4,41,11,48,18,55,25,34),(5,42,12,49,19,56,26,35),(6,43,13,50,20,29,27,36),(7,44,14,51,21,30,28,37),(57,110,78,103,71,96,64,89),(58,111,79,104,72,97,65,90),(59,112,80,105,73,98,66,91),(60,85,81,106,74,99,67,92),(61,86,82,107,75,100,68,93),(62,87,83,108,76,101,69,94),(63,88,84,109,77,102,70,95)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D7 | Dic7 | D14 | C4×D7 | D28 | C7⋊D4 | Dic14 | C4.10C42 | C28.21C42 |
| kernel | C28.21C42 | C2×C4.Dic7 | C14×M4(2) | C2×C7⋊C8 | C2×C56 | C2×C28 | C22×C14 | C2×M4(2) | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 2 | 1 | 8 | 4 | 3 | 1 | 3 | 6 | 3 | 12 | 6 | 12 | 6 | 2 | 12 |
Matrix representation of C28.21C42 ►in GL4(𝔽113) generated by
| 81 | 0 | 0 | 0 |
| 0 | 81 | 0 | 0 |
| 25 | 0 | 53 | 0 |
| 25 | 0 | 0 | 53 |
| 45 | 0 | 40 | 0 |
| 0 | 0 | 112 | 1 |
| 62 | 0 | 68 | 0 |
| 91 | 15 | 68 | 0 |
| 1 | 40 | 0 | 0 |
| 90 | 112 | 0 | 0 |
| 44 | 68 | 0 | 15 |
| 0 | 68 | 1 | 0 |
G:=sub<GL(4,GF(113))| [81,0,25,25,0,81,0,0,0,0,53,0,0,0,0,53],[45,0,62,91,0,0,0,15,40,112,68,68,0,1,0,0],[1,90,44,0,40,112,68,68,0,0,0,1,0,0,15,0] >;
C28.21C42 in GAP, Magma, Sage, TeX
C_{28}._{21}C_4^2 % in TeX
G:=Group("C28.21C4^2"); // GroupNames label
G:=SmallGroup(448,117);
// by ID
G=gap.SmallGroup(448,117);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,387,184,1123,136,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=1,b^4=c^4=a^14,b*a*b^-1=a^13,a*c=c*a,c*b*c^-1=a^21*b>;
// generators/relations