direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8.C4, C8.1C28, C56.5C4, C28.68D4, M4(2).2C14, (C2×C8).5C14, C4.8(C2×C28), C22.(C7×Q8), C4.19(C7×D4), (C2×C14).2Q8, (C2×C56).15C2, C28.45(C2×C4), C14.14(C4⋊C4), (C7×M4(2)).4C2, (C2×C28).119C22, C2.5(C7×C4⋊C4), (C2×C4).22(C2×C14), SmallGroup(224,57)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8.C4
G = < a,b,c | a7=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×C8.C4
►On 112 points
Generators in S112
(1 57 38 53 16 45 23)(2 58 39 54 9 46 24)(3 59 40 55 10 47 17)(4 60 33 56 11 48 18)(5 61 34 49 12 41 19)(6 62 35 50 13 42 20)(7 63 36 51 14 43 21)(8 64 37 52 15 44 22)(25 79 108 91 88 71 100)(26 80 109 92 81 72 101)(27 73 110 93 82 65 102)(28 74 111 94 83 66 103)(29 75 112 95 84 67 104)(30 76 105 96 85 68 97)(31 77 106 89 86 69 98)(32 78 107 90 87 70 99)
(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 97 3 103 5 101 7 99)(2 104 4 102 6 100 8 98)(9 95 11 93 13 91 15 89)(10 94 12 92 14 90 16 96)(17 66 19 72 21 70 23 68)(18 65 20 71 22 69 24 67)(25 64 31 58 29 60 27 62)(26 63 32 57 30 59 28 61)(33 73 35 79 37 77 39 75)(34 80 36 78 38 76 40 74)(41 81 43 87 45 85 47 83)(42 88 44 86 46 84 48 82)(49 109 51 107 53 105 55 111)(50 108 52 106 54 112 56 110)
G:=sub<Sym(112)| (1,57,38,53,16,45,23)(2,58,39,54,9,46,24)(3,59,40,55,10,47,17)(4,60,33,56,11,48,18)(5,61,34,49,12,41,19)(6,62,35,50,13,42,20)(7,63,36,51,14,43,21)(8,64,37,52,15,44,22)(25,79,108,91,88,71,100)(26,80,109,92,81,72,101)(27,73,110,93,82,65,102)(28,74,111,94,83,66,103)(29,75,112,95,84,67,104)(30,76,105,96,85,68,97)(31,77,106,89,86,69,98)(32,78,107,90,87,70,99), (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), (1,97,3,103,5,101,7,99)(2,104,4,102,6,100,8,98)(9,95,11,93,13,91,15,89)(10,94,12,92,14,90,16,96)(17,66,19,72,21,70,23,68)(18,65,20,71,22,69,24,67)(25,64,31,58,29,60,27,62)(26,63,32,57,30,59,28,61)(33,73,35,79,37,77,39,75)(34,80,36,78,38,76,40,74)(41,81,43,87,45,85,47,83)(42,88,44,86,46,84,48,82)(49,109,51,107,53,105,55,111)(50,108,52,106,54,112,56,110)>;
G:=Group( (1,57,38,53,16,45,23)(2,58,39,54,9,46,24)(3,59,40,55,10,47,17)(4,60,33,56,11,48,18)(5,61,34,49,12,41,19)(6,62,35,50,13,42,20)(7,63,36,51,14,43,21)(8,64,37,52,15,44,22)(25,79,108,91,88,71,100)(26,80,109,92,81,72,101)(27,73,110,93,82,65,102)(28,74,111,94,83,66,103)(29,75,112,95,84,67,104)(30,76,105,96,85,68,97)(31,77,106,89,86,69,98)(32,78,107,90,87,70,99), (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), (1,97,3,103,5,101,7,99)(2,104,4,102,6,100,8,98)(9,95,11,93,13,91,15,89)(10,94,12,92,14,90,16,96)(17,66,19,72,21,70,23,68)(18,65,20,71,22,69,24,67)(25,64,31,58,29,60,27,62)(26,63,32,57,30,59,28,61)(33,73,35,79,37,77,39,75)(34,80,36,78,38,76,40,74)(41,81,43,87,45,85,47,83)(42,88,44,86,46,84,48,82)(49,109,51,107,53,105,55,111)(50,108,52,106,54,112,56,110) );
G=PermutationGroup([[(1,57,38,53,16,45,23),(2,58,39,54,9,46,24),(3,59,40,55,10,47,17),(4,60,33,56,11,48,18),(5,61,34,49,12,41,19),(6,62,35,50,13,42,20),(7,63,36,51,14,43,21),(8,64,37,52,15,44,22),(25,79,108,91,88,71,100),(26,80,109,92,81,72,101),(27,73,110,93,82,65,102),(28,74,111,94,83,66,103),(29,75,112,95,84,67,104),(30,76,105,96,85,68,97),(31,77,106,89,86,69,98),(32,78,107,90,87,70,99)], [(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)], [(1,97,3,103,5,101,7,99),(2,104,4,102,6,100,8,98),(9,95,11,93,13,91,15,89),(10,94,12,92,14,90,16,96),(17,66,19,72,21,70,23,68),(18,65,20,71,22,69,24,67),(25,64,31,58,29,60,27,62),(26,63,32,57,30,59,28,61),(33,73,35,79,37,77,39,75),(34,80,36,78,38,76,40,74),(41,81,43,87,45,85,47,83),(42,88,44,86,46,84,48,82),(49,109,51,107,53,105,55,111),(50,108,52,106,54,112,56,110)]])
C7×C8.C4 is a maximal subgroup of
C8.7Dic14 C8.Dic14 D56.C4 C56.8D4 Dic28.C4 M4(2).25D14 D56⋊10C4 D56⋊7C4 C8.20D28 C8.21D28 C8.24D28
98 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14F | 14G | ··· | 14L | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X | 56Y | ··· | 56AV |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | D4 | Q8 | C8.C4 | C7×D4 | C7×Q8 | C7×C8.C4 |
| kernel | C7×C8.C4 | C2×C56 | C7×M4(2) | C56 | C8.C4 | C2×C8 | M4(2) | C8 | C28 | C2×C14 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 4 | 6 | 6 | 12 | 24 | 1 | 1 | 4 | 6 | 6 | 24 |
Matrix representation of C7×C8.C4 ►in GL2(𝔽113) generated by
| 28 | 0 |
| 0 | 28 |
| 44 | 0 |
| 0 | 18 |
| 0 | 1 |
| 15 | 0 |
G:=sub<GL(2,GF(113))| [28,0,0,28],[44,0,0,18],[0,15,1,0] >;
C7×C8.C4 in GAP, Magma, Sage, TeX
C_7\times C_8.C_4
% in TeX
G:=Group("C7xC8.C4"); // GroupNames label
G:=SmallGroup(224,57);
// by ID
G=gap.SmallGroup(224,57);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,175,3363,117,88]);
// Polycyclic
G:=Group<a,b,c|a^7=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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