metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C28.1C8, C56.4C4, C7⋊2M5(2), C8.22D14, C8.2Dic7, C56.22C22, C4.(C7⋊C8), C7⋊C16⋊5C2, C22.(C7⋊C8), (C2×C8).7D7, (C2×C14).3C8, C14.9(C2×C8), (C2×C28).8C4, C28.39(C2×C4), (C2×C56).10C2, (C2×C4).5Dic7, C4.11(C2×Dic7), C2.4(C2×C7⋊C8), SmallGroup(224,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.C8
G = < a,b | a56=1, b4=a42, bab-1=a13 >
Smallest permutation representation of C28.C8
►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 85 50 106 43 71 36 92 29 57 22 78 15 99 8 64)(2 98 51 63 44 84 37 105 30 70 23 91 16 112 9 77)(3 111 52 76 45 97 38 62 31 83 24 104 17 69 10 90)(4 68 53 89 46 110 39 75 32 96 25 61 18 82 11 103)(5 81 54 102 47 67 40 88 33 109 26 74 19 95 12 60)(6 94 55 59 48 80 41 101 34 66 27 87 20 108 13 73)(7 107 56 72 49 93 42 58 35 79 28 100 21 65 14 86)
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), (1,85,50,106,43,71,36,92,29,57,22,78,15,99,8,64)(2,98,51,63,44,84,37,105,30,70,23,91,16,112,9,77)(3,111,52,76,45,97,38,62,31,83,24,104,17,69,10,90)(4,68,53,89,46,110,39,75,32,96,25,61,18,82,11,103)(5,81,54,102,47,67,40,88,33,109,26,74,19,95,12,60)(6,94,55,59,48,80,41,101,34,66,27,87,20,108,13,73)(7,107,56,72,49,93,42,58,35,79,28,100,21,65,14,86)>;
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), (1,85,50,106,43,71,36,92,29,57,22,78,15,99,8,64)(2,98,51,63,44,84,37,105,30,70,23,91,16,112,9,77)(3,111,52,76,45,97,38,62,31,83,24,104,17,69,10,90)(4,68,53,89,46,110,39,75,32,96,25,61,18,82,11,103)(5,81,54,102,47,67,40,88,33,109,26,74,19,95,12,60)(6,94,55,59,48,80,41,101,34,66,27,87,20,108,13,73)(7,107,56,72,49,93,42,58,35,79,28,100,21,65,14,86) );
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)], [(1,85,50,106,43,71,36,92,29,57,22,78,15,99,8,64),(2,98,51,63,44,84,37,105,30,70,23,91,16,112,9,77),(3,111,52,76,45,97,38,62,31,83,24,104,17,69,10,90),(4,68,53,89,46,110,39,75,32,96,25,61,18,82,11,103),(5,81,54,102,47,67,40,88,33,109,26,74,19,95,12,60),(6,94,55,59,48,80,41,101,34,66,27,87,20,108,13,73),(7,107,56,72,49,93,42,58,35,79,28,100,21,65,14,86)]])
C28.C8 is a maximal subgroup of
C56.16Q8 C28.15C42 C56.Q8 D56⋊8C4 C8.Dic14 D56.C4 C56.8D4 D28.C8 C56.9Q8 C112⋊C4 M5(2)⋊D7 C56.D4 C56.92D4 D8.Dic7 Q16.Dic7 D8⋊2Dic7 D28.4C8 D7×M5(2) C56.70C23 D8.D14 Q16.D14 Q16⋊D14 C56.31C23
C28.C8 is a maximal quotient of
C56.C8 C28⋊C16 C56.91D4
68 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 14A | ··· | 14I | 16A | ··· | 16H | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | D7 | Dic7 | D14 | Dic7 | M5(2) | C7⋊C8 | C7⋊C8 | C28.C8 |
| kernel | C28.C8 | C7⋊C16 | C2×C56 | C56 | C2×C28 | C28 | C2×C14 | C2×C8 | C8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 24 |
Matrix representation of C28.C8 ►in GL2(𝔽113) generated by
| 52 | 0 |
| 51 | 72 |
| 74 | 111 |
| 17 | 39 |
G:=sub<GL(2,GF(113))| [52,51,0,72],[74,17,111,39] >;
C28.C8 in GAP, Magma, Sage, TeX
C_{28}.C_8 % in TeX
G:=Group("C28.C8"); // GroupNames label
G:=SmallGroup(224,18);
// by ID
G=gap.SmallGroup(224,18);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,217,50,69,6917]);
// Polycyclic
G:=Group<a,b|a^56=1,b^4=a^42,b*a*b^-1=a^13>;
// generators/relations
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