metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.15C42, C42.1Dic7, C28.24M4(2), C7⋊C16⋊3C4, (C4×C28).1C4, (C2×C56).1C4, C8.32(C4×D7), C7⋊1(C16⋊C4), C56.36(C2×C4), C8⋊C4.4D7, (C2×C8).1Dic7, (C2×C8).150D14, C4.22(C4×Dic7), C14.2(C8⋊C4), C28.C8.6C2, C4.6(C4.Dic7), (C2×C56).217C22, (C2×C14).20M4(2), C2.3(C42.D7), C22.3(C4.Dic7), (C7×C8⋊C4).3C2, (C2×C28).302(C2×C4), (C2×C4).70(C2×Dic7), SmallGroup(448,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.15C42
G = < a,b,c | a28=c4=1, b4=a21, bab-1=a13, ac=ca, cbc-1=a21b >
Smallest permutation representation of C28.15C42
►On 112 points
Generators in S112
(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 94 42 75 22 87 35 68 15 108 56 61 8 101 49 82)(2 107 43 60 23 100 36 81 16 93 29 74 9 86 50 67)(3 92 44 73 24 85 37 66 17 106 30 59 10 99 51 80)(4 105 45 58 25 98 38 79 18 91 31 72 11 112 52 65)(5 90 46 71 26 111 39 64 19 104 32 57 12 97 53 78)(6 103 47 84 27 96 40 77 20 89 33 70 13 110 54 63)(7 88 48 69 28 109 41 62 21 102 34 83 14 95 55 76)
(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 78 71 64)(58 79 72 65)(59 80 73 66)(60 81 74 67)(61 82 75 68)(62 83 76 69)(63 84 77 70)(85 92 99 106)(86 93 100 107)(87 94 101 108)(88 95 102 109)(89 96 103 110)(90 97 104 111)(91 98 105 112)
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), (1,94,42,75,22,87,35,68,15,108,56,61,8,101,49,82)(2,107,43,60,23,100,36,81,16,93,29,74,9,86,50,67)(3,92,44,73,24,85,37,66,17,106,30,59,10,99,51,80)(4,105,45,58,25,98,38,79,18,91,31,72,11,112,52,65)(5,90,46,71,26,111,39,64,19,104,32,57,12,97,53,78)(6,103,47,84,27,96,40,77,20,89,33,70,13,110,54,63)(7,88,48,69,28,109,41,62,21,102,34,83,14,95,55,76), (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,78,71,64)(58,79,72,65)(59,80,73,66)(60,81,74,67)(61,82,75,68)(62,83,76,69)(63,84,77,70)(85,92,99,106)(86,93,100,107)(87,94,101,108)(88,95,102,109)(89,96,103,110)(90,97,104,111)(91,98,105,112)>;
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), (1,94,42,75,22,87,35,68,15,108,56,61,8,101,49,82)(2,107,43,60,23,100,36,81,16,93,29,74,9,86,50,67)(3,92,44,73,24,85,37,66,17,106,30,59,10,99,51,80)(4,105,45,58,25,98,38,79,18,91,31,72,11,112,52,65)(5,90,46,71,26,111,39,64,19,104,32,57,12,97,53,78)(6,103,47,84,27,96,40,77,20,89,33,70,13,110,54,63)(7,88,48,69,28,109,41,62,21,102,34,83,14,95,55,76), (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,78,71,64)(58,79,72,65)(59,80,73,66)(60,81,74,67)(61,82,75,68)(62,83,76,69)(63,84,77,70)(85,92,99,106)(86,93,100,107)(87,94,101,108)(88,95,102,109)(89,96,103,110)(90,97,104,111)(91,98,105,112) );
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)], [(1,94,42,75,22,87,35,68,15,108,56,61,8,101,49,82),(2,107,43,60,23,100,36,81,16,93,29,74,9,86,50,67),(3,92,44,73,24,85,37,66,17,106,30,59,10,99,51,80),(4,105,45,58,25,98,38,79,18,91,31,72,11,112,52,65),(5,90,46,71,26,111,39,64,19,104,32,57,12,97,53,78),(6,103,47,84,27,96,40,77,20,89,33,70,13,110,54,63),(7,88,48,69,28,109,41,62,21,102,34,83,14,95,55,76)], [(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,78,71,64),(58,79,72,65),(59,80,73,66),(60,81,74,67),(61,82,75,68),(62,83,76,69),(63,84,77,70),(85,92,99,106),(86,93,100,107),(87,94,101,108),(88,95,102,109),(89,96,103,110),(90,97,104,111),(91,98,105,112)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 14A | ··· | 14I | 16A | ··· | 16H | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 28 | ··· | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | D7 | M4(2) | M4(2) | Dic7 | Dic7 | D14 | C4×D7 | C4.Dic7 | C4.Dic7 | C16⋊C4 | C28.15C42 |
| kernel | C28.15C42 | C28.C8 | C7×C8⋊C4 | C7⋊C16 | C4×C28 | C2×C56 | C8⋊C4 | C28 | C2×C14 | C42 | C2×C8 | C2×C8 | C8 | C4 | C22 | C7 | C1 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 3 | 2 | 2 | 3 | 3 | 3 | 12 | 12 | 12 | 2 | 12 |
Matrix representation of C28.15C42 ►in GL4(𝔽113) generated by
| 81 | 0 | 0 | 0 |
| 0 | 81 | 0 | 0 |
| 0 | 0 | 53 | 0 |
| 0 | 0 | 0 | 53 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 98 |
G:=sub<GL(4,GF(113))| [81,0,0,0,0,81,0,0,0,0,53,0,0,0,0,53],[0,0,0,15,0,0,1,0,1,0,0,0,0,1,0,0],[1,0,0,0,0,112,0,0,0,0,15,0,0,0,0,98] >;
C28.15C42 in GAP, Magma, Sage, TeX
C_{28}._{15}C_4^2 % in TeX
G:=Group("C28.15C4^2"); // GroupNames label
G:=SmallGroup(448,23);
// by ID
G=gap.SmallGroup(448,23);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,387,100,1123,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=c^4=1,b^4=a^21,b*a*b^-1=a^13,a*c=c*a,c*b*c^-1=a^21*b>;
// generators/relations
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