direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×M5(2), C4.C56, C112⋊7C2, C16⋊3C14, C56.6C4, C8.2C28, C28.4C8, C22.C56, C56.29C22, (C2×C4).5C28, C8.8(C2×C14), (C2×C14).1C8, (C2×C8).8C14, C2.3(C2×C56), (C2×C56).18C2, (C2×C28).14C4, C4.12(C2×C28), C14.13(C2×C8), C28.49(C2×C4), SmallGroup(224,59)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×M5(2)
G = < a,b,c | a7=b16=c2=1, ab=ba, ac=ca, cbc=b9 >
Smallest permutation representation of C7×M5(2)
►On 112 points
Generators in S112
(1 29 86 74 50 98 35)(2 30 87 75 51 99 36)(3 31 88 76 52 100 37)(4 32 89 77 53 101 38)(5 17 90 78 54 102 39)(6 18 91 79 55 103 40)(7 19 92 80 56 104 41)(8 20 93 65 57 105 42)(9 21 94 66 58 106 43)(10 22 95 67 59 107 44)(11 23 96 68 60 108 45)(12 24 81 69 61 109 46)(13 25 82 70 62 110 47)(14 26 83 71 63 111 48)(15 27 84 72 64 112 33)(16 28 85 73 49 97 34)
(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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)(81 89)(83 91)(85 93)(87 95)(97 105)(99 107)(101 109)(103 111)
G:=sub<Sym(112)| (1,29,86,74,50,98,35)(2,30,87,75,51,99,36)(3,31,88,76,52,100,37)(4,32,89,77,53,101,38)(5,17,90,78,54,102,39)(6,18,91,79,55,103,40)(7,19,92,80,56,104,41)(8,20,93,65,57,105,42)(9,21,94,66,58,106,43)(10,22,95,67,59,107,44)(11,23,96,68,60,108,45)(12,24,81,69,61,109,46)(13,25,82,70,62,110,47)(14,26,83,71,63,111,48)(15,27,84,72,64,112,33)(16,28,85,73,49,97,34), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63)(65,73)(67,75)(69,77)(71,79)(81,89)(83,91)(85,93)(87,95)(97,105)(99,107)(101,109)(103,111)>;
G:=Group( (1,29,86,74,50,98,35)(2,30,87,75,51,99,36)(3,31,88,76,52,100,37)(4,32,89,77,53,101,38)(5,17,90,78,54,102,39)(6,18,91,79,55,103,40)(7,19,92,80,56,104,41)(8,20,93,65,57,105,42)(9,21,94,66,58,106,43)(10,22,95,67,59,107,44)(11,23,96,68,60,108,45)(12,24,81,69,61,109,46)(13,25,82,70,62,110,47)(14,26,83,71,63,111,48)(15,27,84,72,64,112,33)(16,28,85,73,49,97,34), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63)(65,73)(67,75)(69,77)(71,79)(81,89)(83,91)(85,93)(87,95)(97,105)(99,107)(101,109)(103,111) );
G=PermutationGroup([[(1,29,86,74,50,98,35),(2,30,87,75,51,99,36),(3,31,88,76,52,100,37),(4,32,89,77,53,101,38),(5,17,90,78,54,102,39),(6,18,91,79,55,103,40),(7,19,92,80,56,104,41),(8,20,93,65,57,105,42),(9,21,94,66,58,106,43),(10,22,95,67,59,107,44),(11,23,96,68,60,108,45),(12,24,81,69,61,109,46),(13,25,82,70,62,110,47),(14,26,83,71,63,111,48),(15,27,84,72,64,112,33),(16,28,85,73,49,97,34)], [(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)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(34,42),(36,44),(38,46),(40,48),(49,57),(51,59),(53,61),(55,63),(65,73),(67,75),(69,77),(71,79),(81,89),(83,91),(85,93),(87,95),(97,105),(99,107),(101,109),(103,111)]])
C7×M5(2) is a maximal subgroup of
C56.9Q8 C112⋊C4 C16⋊Dic7 M5(2)⋊D7 Dic14.C8 C28.3D8 C28.4D8 D56⋊2C4 C16.12D14 C16⋊D14 C16.D14
140 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 8F | 14A | ··· | 14F | 14G | ··· | 14L | 16A | ··· | 16H | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X | 56Y | ··· | 56AJ | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C8 | C8 | C14 | C14 | C28 | C28 | C56 | C56 | M5(2) | C7×M5(2) |
| kernel | C7×M5(2) | C112 | C2×C56 | C56 | C2×C28 | M5(2) | C28 | C2×C14 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 6 | 4 | 4 | 12 | 6 | 12 | 12 | 24 | 24 | 4 | 24 |
Matrix representation of C7×M5(2) ►in GL3(𝔽113) generated by
| 28 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 112 | 0 | 0 |
| 0 | 3 | 111 |
| 0 | 52 | 110 |
| 112 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 3 | 112 |
G:=sub<GL(3,GF(113))| [28,0,0,0,1,0,0,0,1],[112,0,0,0,3,52,0,111,110],[112,0,0,0,1,3,0,0,112] >;
C7×M5(2) in GAP, Magma, Sage, TeX
C_7\times M_5(2)
% in TeX
G:=Group("C7xM5(2)"); // GroupNames label
G:=SmallGroup(224,59);
// by ID
G=gap.SmallGroup(224,59);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-2,168,1369,69,88]);
// Polycyclic
G:=Group<a,b,c|a^7=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations
Export