direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×C8.Q8, C16⋊1C28, C112⋊3C4, C56.10Q8, C28.44SD16, M5(2).1C14, C8.(C7×Q8), C56.86(C2×C4), C8.18(C2×C28), C28.55(C4⋊C4), C4.Q8.1C14, C4.9(C7×SD16), (C2×C28).282D4, C8.C4.2C14, (C2×C14).26SD16, C14.10(C4.Q8), (C7×M5(2)).3C2, C22.5(C7×SD16), (C2×C56).268C22, C4.6(C7×C4⋊C4), C2.3(C7×C4.Q8), (C2×C4).13(C7×D4), (C7×C4.Q8).6C2, (C2×C8).15(C2×C14), (C7×C8.C4).5C2, SmallGroup(448,169)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8.Q8
G = < a,b,c,d | a7=b8=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b3, dcd-1=b4c3 >
Smallest permutation representation of C7×C8.Q8
►On 112 points
Generators in S112
(1 39 31 58 73 87 106)(2 40 32 59 74 88 107)(3 41 17 60 75 89 108)(4 42 18 61 76 90 109)(5 43 19 62 77 91 110)(6 44 20 63 78 92 111)(7 45 21 64 79 93 112)(8 46 22 49 80 94 97)(9 47 23 50 65 95 98)(10 48 24 51 66 96 99)(11 33 25 52 67 81 100)(12 34 26 53 68 82 101)(13 35 27 54 69 83 102)(14 36 28 55 70 84 103)(15 37 29 56 71 85 104)(16 38 30 57 72 86 105)
(1 3 5 7 9 11 13 15)(2 12 6 16 10 4 14 8)(17 19 21 23 25 27 29 31)(18 28 22 32 26 20 30 24)(33 35 37 39 41 43 45 47)(34 44 38 48 42 36 46 40)(49 59 53 63 57 51 61 55)(50 52 54 56 58 60 62 64)(65 67 69 71 73 75 77 79)(66 76 70 80 74 68 78 72)(81 83 85 87 89 91 93 95)(82 92 86 96 90 84 94 88)(97 107 101 111 105 99 109 103)(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)
(2 4 10 12)(3 7)(5 13)(6 16 14 8)(11 15)(17 21)(18 24 26 32)(19 27)(20 30 28 22)(25 29)(33 37)(34 40 42 48)(35 43)(36 46 44 38)(41 45)(49 63 57 55)(51 53 59 61)(52 56)(54 62)(60 64)(66 68 74 76)(67 71)(69 77)(70 80 78 72)(75 79)(81 85)(82 88 90 96)(83 91)(84 94 92 86)(89 93)(97 111 105 103)(99 101 107 109)(100 104)(102 110)(108 112)
G:=sub<Sym(112)| (1,39,31,58,73,87,106)(2,40,32,59,74,88,107)(3,41,17,60,75,89,108)(4,42,18,61,76,90,109)(5,43,19,62,77,91,110)(6,44,20,63,78,92,111)(7,45,21,64,79,93,112)(8,46,22,49,80,94,97)(9,47,23,50,65,95,98)(10,48,24,51,66,96,99)(11,33,25,52,67,81,100)(12,34,26,53,68,82,101)(13,35,27,54,69,83,102)(14,36,28,55,70,84,103)(15,37,29,56,71,85,104)(16,38,30,57,72,86,105), (1,3,5,7,9,11,13,15)(2,12,6,16,10,4,14,8)(17,19,21,23,25,27,29,31)(18,28,22,32,26,20,30,24)(33,35,37,39,41,43,45,47)(34,44,38,48,42,36,46,40)(49,59,53,63,57,51,61,55)(50,52,54,56,58,60,62,64)(65,67,69,71,73,75,77,79)(66,76,70,80,74,68,78,72)(81,83,85,87,89,91,93,95)(82,92,86,96,90,84,94,88)(97,107,101,111,105,99,109,103)(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), (2,4,10,12)(3,7)(5,13)(6,16,14,8)(11,15)(17,21)(18,24,26,32)(19,27)(20,30,28,22)(25,29)(33,37)(34,40,42,48)(35,43)(36,46,44,38)(41,45)(49,63,57,55)(51,53,59,61)(52,56)(54,62)(60,64)(66,68,74,76)(67,71)(69,77)(70,80,78,72)(75,79)(81,85)(82,88,90,96)(83,91)(84,94,92,86)(89,93)(97,111,105,103)(99,101,107,109)(100,104)(102,110)(108,112)>;
G:=Group( (1,39,31,58,73,87,106)(2,40,32,59,74,88,107)(3,41,17,60,75,89,108)(4,42,18,61,76,90,109)(5,43,19,62,77,91,110)(6,44,20,63,78,92,111)(7,45,21,64,79,93,112)(8,46,22,49,80,94,97)(9,47,23,50,65,95,98)(10,48,24,51,66,96,99)(11,33,25,52,67,81,100)(12,34,26,53,68,82,101)(13,35,27,54,69,83,102)(14,36,28,55,70,84,103)(15,37,29,56,71,85,104)(16,38,30,57,72,86,105), (1,3,5,7,9,11,13,15)(2,12,6,16,10,4,14,8)(17,19,21,23,25,27,29,31)(18,28,22,32,26,20,30,24)(33,35,37,39,41,43,45,47)(34,44,38,48,42,36,46,40)(49,59,53,63,57,51,61,55)(50,52,54,56,58,60,62,64)(65,67,69,71,73,75,77,79)(66,76,70,80,74,68,78,72)(81,83,85,87,89,91,93,95)(82,92,86,96,90,84,94,88)(97,107,101,111,105,99,109,103)(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), (2,4,10,12)(3,7)(5,13)(6,16,14,8)(11,15)(17,21)(18,24,26,32)(19,27)(20,30,28,22)(25,29)(33,37)(34,40,42,48)(35,43)(36,46,44,38)(41,45)(49,63,57,55)(51,53,59,61)(52,56)(54,62)(60,64)(66,68,74,76)(67,71)(69,77)(70,80,78,72)(75,79)(81,85)(82,88,90,96)(83,91)(84,94,92,86)(89,93)(97,111,105,103)(99,101,107,109)(100,104)(102,110)(108,112) );
G=PermutationGroup([[(1,39,31,58,73,87,106),(2,40,32,59,74,88,107),(3,41,17,60,75,89,108),(4,42,18,61,76,90,109),(5,43,19,62,77,91,110),(6,44,20,63,78,92,111),(7,45,21,64,79,93,112),(8,46,22,49,80,94,97),(9,47,23,50,65,95,98),(10,48,24,51,66,96,99),(11,33,25,52,67,81,100),(12,34,26,53,68,82,101),(13,35,27,54,69,83,102),(14,36,28,55,70,84,103),(15,37,29,56,71,85,104),(16,38,30,57,72,86,105)], [(1,3,5,7,9,11,13,15),(2,12,6,16,10,4,14,8),(17,19,21,23,25,27,29,31),(18,28,22,32,26,20,30,24),(33,35,37,39,41,43,45,47),(34,44,38,48,42,36,46,40),(49,59,53,63,57,51,61,55),(50,52,54,56,58,60,62,64),(65,67,69,71,73,75,77,79),(66,76,70,80,74,68,78,72),(81,83,85,87,89,91,93,95),(82,92,86,96,90,84,94,88),(97,107,101,111,105,99,109,103),(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)], [(2,4,10,12),(3,7),(5,13),(6,16,14,8),(11,15),(17,21),(18,24,26,32),(19,27),(20,30,28,22),(25,29),(33,37),(34,40,42,48),(35,43),(36,46,44,38),(41,45),(49,63,57,55),(51,53,59,61),(52,56),(54,62),(60,64),(66,68,74,76),(67,71),(69,77),(70,80,78,72),(75,79),(81,85),(82,88,90,96),(83,91),(84,94,92,86),(89,93),(97,111,105,103),(99,101,107,109),(100,104),(102,110),(108,112)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 14A | ··· | 14F | 14G | ··· | 14L | 16A | 16B | 16C | 16D | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56L | 56M | ··· | 56R | 56S | ··· | 56AD | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 4 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C28 | Q8 | D4 | SD16 | SD16 | C7×Q8 | C7×D4 | C7×SD16 | C7×SD16 | C8.Q8 | C7×C8.Q8 |
| kernel | C7×C8.Q8 | C7×C4.Q8 | C7×C8.C4 | C7×M5(2) | C112 | C8.Q8 | C4.Q8 | C8.C4 | M5(2) | C16 | C56 | C2×C28 | C28 | C2×C14 | C8 | C2×C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 6 | 6 | 24 | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 2 | 12 |
Matrix representation of C7×C8.Q8 ►in GL4(𝔽113) generated by
| 109 | 0 | 0 | 0 |
| 0 | 109 | 0 | 0 |
| 0 | 0 | 109 | 0 |
| 0 | 0 | 0 | 109 |
| 13 | 100 | 0 | 0 |
| 13 | 13 | 0 | 0 |
| 0 | 0 | 100 | 13 |
| 0 | 0 | 100 | 100 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 100 | 0 | 0 |
| 13 | 13 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 100 | 13 |
| 0 | 0 | 13 | 13 |
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,109,0,0,0,0,109],[13,13,0,0,100,13,0,0,0,0,100,100,0,0,13,100],[0,0,13,13,0,0,100,13,1,0,0,0,0,1,0,0],[1,0,0,0,0,112,0,0,0,0,100,13,0,0,13,13] >;
C7×C8.Q8 in GAP, Magma, Sage, TeX
C_7\times C_8.Q_8
% in TeX
G:=Group("C7xC8.Q8"); // GroupNames label
G:=SmallGroup(448,169);
// by ID
G=gap.SmallGroup(448,169);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,204,3923,136,4911,14117,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^3,d*c*d^-1=b^4*c^3>;
// generators/relations
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