direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8.C8, C8.1C56, C56.7C8, C56.22Q8, C56.109D4, C42.8C28, M5(2).4C14, C8.6(C7×Q8), C4.8(C2×C56), (C2×C8).9C28, C8.29(C7×D4), (C4×C8).10C14, (C4×C56).28C2, (C4×C28).23C4, C28.48(C2×C8), (C2×C56).28C4, C28.68(C4⋊C4), C14.15(C4⋊C8), (C7×M5(2)).8C2, (C2×C56).443C22, (C2×C14).17M4(2), C22.5(C7×M4(2)), C2.5(C7×C4⋊C8), C4.19(C7×C4⋊C4), (C2×C8).97(C2×C14), (C2×C4).68(C2×C28), (C2×C28).329(C2×C4), SmallGroup(448,168)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8.C8
G = < a,b,c | a7=b8=1, c8=b4, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C7×C8.C8
►On 112 points
Generators in S112
(1 55 100 79 24 37 91)(2 56 101 80 25 38 92)(3 57 102 65 26 39 93)(4 58 103 66 27 40 94)(5 59 104 67 28 41 95)(6 60 105 68 29 42 96)(7 61 106 69 30 43 81)(8 62 107 70 31 44 82)(9 63 108 71 32 45 83)(10 64 109 72 17 46 84)(11 49 110 73 18 47 85)(12 50 111 74 19 48 86)(13 51 112 75 20 33 87)(14 52 97 76 21 34 88)(15 53 98 77 22 35 89)(16 54 99 78 23 36 90)
(1 3 5 7 9 11 13 15)(2 8 14 4 10 16 6 12)(17 23 29 19 25 31 21 27)(18 20 22 24 26 28 30 32)(33 35 37 39 41 43 45 47)(34 40 46 36 42 48 38 44)(49 51 53 55 57 59 61 63)(50 56 62 52 58 64 54 60)(65 67 69 71 73 75 77 79)(66 72 78 68 74 80 70 76)(81 83 85 87 89 91 93 95)(82 88 94 84 90 96 86 92)(97 103 109 99 105 111 101 107)(98 100 102 104 106 108 110 112)
(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)
G:=sub<Sym(112)| (1,55,100,79,24,37,91)(2,56,101,80,25,38,92)(3,57,102,65,26,39,93)(4,58,103,66,27,40,94)(5,59,104,67,28,41,95)(6,60,105,68,29,42,96)(7,61,106,69,30,43,81)(8,62,107,70,31,44,82)(9,63,108,71,32,45,83)(10,64,109,72,17,46,84)(11,49,110,73,18,47,85)(12,50,111,74,19,48,86)(13,51,112,75,20,33,87)(14,52,97,76,21,34,88)(15,53,98,77,22,35,89)(16,54,99,78,23,36,90), (1,3,5,7,9,11,13,15)(2,8,14,4,10,16,6,12)(17,23,29,19,25,31,21,27)(18,20,22,24,26,28,30,32)(33,35,37,39,41,43,45,47)(34,40,46,36,42,48,38,44)(49,51,53,55,57,59,61,63)(50,56,62,52,58,64,54,60)(65,67,69,71,73,75,77,79)(66,72,78,68,74,80,70,76)(81,83,85,87,89,91,93,95)(82,88,94,84,90,96,86,92)(97,103,109,99,105,111,101,107)(98,100,102,104,106,108,110,112), (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)>;
G:=Group( (1,55,100,79,24,37,91)(2,56,101,80,25,38,92)(3,57,102,65,26,39,93)(4,58,103,66,27,40,94)(5,59,104,67,28,41,95)(6,60,105,68,29,42,96)(7,61,106,69,30,43,81)(8,62,107,70,31,44,82)(9,63,108,71,32,45,83)(10,64,109,72,17,46,84)(11,49,110,73,18,47,85)(12,50,111,74,19,48,86)(13,51,112,75,20,33,87)(14,52,97,76,21,34,88)(15,53,98,77,22,35,89)(16,54,99,78,23,36,90), (1,3,5,7,9,11,13,15)(2,8,14,4,10,16,6,12)(17,23,29,19,25,31,21,27)(18,20,22,24,26,28,30,32)(33,35,37,39,41,43,45,47)(34,40,46,36,42,48,38,44)(49,51,53,55,57,59,61,63)(50,56,62,52,58,64,54,60)(65,67,69,71,73,75,77,79)(66,72,78,68,74,80,70,76)(81,83,85,87,89,91,93,95)(82,88,94,84,90,96,86,92)(97,103,109,99,105,111,101,107)(98,100,102,104,106,108,110,112), (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) );
G=PermutationGroup([[(1,55,100,79,24,37,91),(2,56,101,80,25,38,92),(3,57,102,65,26,39,93),(4,58,103,66,27,40,94),(5,59,104,67,28,41,95),(6,60,105,68,29,42,96),(7,61,106,69,30,43,81),(8,62,107,70,31,44,82),(9,63,108,71,32,45,83),(10,64,109,72,17,46,84),(11,49,110,73,18,47,85),(12,50,111,74,19,48,86),(13,51,112,75,20,33,87),(14,52,97,76,21,34,88),(15,53,98,77,22,35,89),(16,54,99,78,23,36,90)], [(1,3,5,7,9,11,13,15),(2,8,14,4,10,16,6,12),(17,23,29,19,25,31,21,27),(18,20,22,24,26,28,30,32),(33,35,37,39,41,43,45,47),(34,40,46,36,42,48,38,44),(49,51,53,55,57,59,61,63),(50,56,62,52,58,64,54,60),(65,67,69,71,73,75,77,79),(66,72,78,68,74,80,70,76),(81,83,85,87,89,91,93,95),(82,88,94,84,90,96,86,92),(97,103,109,99,105,111,101,107),(98,100,102,104,106,108,110,112)], [(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)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | ··· | 4G | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 14A | ··· | 14F | 14G | ··· | 14L | 16A | ··· | 16H | 28A | ··· | 28L | 28M | ··· | 28AP | 56A | ··· | 56X | 56Y | ··· | 56BH | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 4 | 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 | ··· | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C8 | C14 | C14 | C28 | C28 | C56 | D4 | Q8 | M4(2) | C7×D4 | C7×Q8 | C8.C8 | C7×M4(2) | C7×C8.C8 |
| kernel | C7×C8.C8 | C4×C56 | C7×M5(2) | C4×C28 | C2×C56 | C8.C8 | C56 | C4×C8 | M5(2) | C42 | C2×C8 | C8 | C56 | C56 | C2×C14 | C8 | C8 | C7 | C22 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 8 | 6 | 12 | 12 | 12 | 48 | 1 | 1 | 2 | 6 | 6 | 8 | 12 | 48 |
Matrix representation of C7×C8.C8 ►in GL2(𝔽113) generated by
| 28 | 0 |
| 0 | 28 |
| 95 | 0 |
| 0 | 44 |
| 0 | 1 |
| 95 | 0 |
G:=sub<GL(2,GF(113))| [28,0,0,28],[95,0,0,44],[0,95,1,0] >;
C7×C8.C8 in GAP, Magma, Sage, TeX
C_7\times C_8.C_8
% in TeX
G:=Group("C7xC8.C8"); // GroupNames label
G:=SmallGroup(448,168);
// by ID
G=gap.SmallGroup(448,168);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,204,7059,248,102,124]);
// Polycyclic
G:=Group<a,b,c|a^7=b^8=1,c^8=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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