direct product, metacyclic, supersoluble, monomial
Aliases: Q16×C7⋊C3, C56.3C6, (C7×Q16)⋊C3, C7⋊3(C3×Q16), (C7×Q8).4C6, C28.19(C2×C6), C14.16(C3×D4), C8.(C2×C7⋊C3), C2.5(D4×C7⋊C3), (C8×C7⋊C3).3C2, Q8.2(C2×C7⋊C3), (Q8×C7⋊C3).2C2, (C2×C7⋊C3).16D4, C4.3(C22×C7⋊C3), (C4×C7⋊C3).19C22, SmallGroup(336,55)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×C7⋊C3
G = < a,b,c,d | a8=c7=d3=1, b2=a4, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Smallest permutation representation of Q16×C7⋊C3
►On 112 points
Generators in S112
(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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 48 13 44)(10 47 14 43)(11 46 15 42)(12 45 16 41)(17 61 21 57)(18 60 22 64)(19 59 23 63)(20 58 24 62)(33 83 37 87)(34 82 38 86)(35 81 39 85)(36 88 40 84)(49 110 53 106)(50 109 54 105)(51 108 55 112)(52 107 56 111)(65 94 69 90)(66 93 70 89)(67 92 71 96)(68 91 72 95)(73 99 77 103)(74 98 78 102)(75 97 79 101)(76 104 80 100)
(1 36 61 107 89 14 102)(2 37 62 108 90 15 103)(3 38 63 109 91 16 104)(4 39 64 110 92 9 97)(5 40 57 111 93 10 98)(6 33 58 112 94 11 99)(7 34 59 105 95 12 100)(8 35 60 106 96 13 101)(17 52 70 47 78 29 84)(18 53 71 48 79 30 85)(19 54 72 41 80 31 86)(20 55 65 42 73 32 87)(21 56 66 43 74 25 88)(22 49 67 44 75 26 81)(23 50 68 45 76 27 82)(24 51 69 46 77 28 83)
(9 110 97)(10 111 98)(11 112 99)(12 105 100)(13 106 101)(14 107 102)(15 108 103)(16 109 104)(17 70 84)(18 71 85)(19 72 86)(20 65 87)(21 66 88)(22 67 81)(23 68 82)(24 69 83)(33 58 94)(34 59 95)(35 60 96)(36 61 89)(37 62 90)(38 63 91)(39 64 92)(40 57 93)(41 54 80)(42 55 73)(43 56 74)(44 49 75)(45 50 76)(46 51 77)(47 52 78)(48 53 79)
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), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,48,13,44)(10,47,14,43)(11,46,15,42)(12,45,16,41)(17,61,21,57)(18,60,22,64)(19,59,23,63)(20,58,24,62)(33,83,37,87)(34,82,38,86)(35,81,39,85)(36,88,40,84)(49,110,53,106)(50,109,54,105)(51,108,55,112)(52,107,56,111)(65,94,69,90)(66,93,70,89)(67,92,71,96)(68,91,72,95)(73,99,77,103)(74,98,78,102)(75,97,79,101)(76,104,80,100), (1,36,61,107,89,14,102)(2,37,62,108,90,15,103)(3,38,63,109,91,16,104)(4,39,64,110,92,9,97)(5,40,57,111,93,10,98)(6,33,58,112,94,11,99)(7,34,59,105,95,12,100)(8,35,60,106,96,13,101)(17,52,70,47,78,29,84)(18,53,71,48,79,30,85)(19,54,72,41,80,31,86)(20,55,65,42,73,32,87)(21,56,66,43,74,25,88)(22,49,67,44,75,26,81)(23,50,68,45,76,27,82)(24,51,69,46,77,28,83), (9,110,97)(10,111,98)(11,112,99)(12,105,100)(13,106,101)(14,107,102)(15,108,103)(16,109,104)(17,70,84)(18,71,85)(19,72,86)(20,65,87)(21,66,88)(22,67,81)(23,68,82)(24,69,83)(33,58,94)(34,59,95)(35,60,96)(36,61,89)(37,62,90)(38,63,91)(39,64,92)(40,57,93)(41,54,80)(42,55,73)(43,56,74)(44,49,75)(45,50,76)(46,51,77)(47,52,78)(48,53,79)>;
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), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,48,13,44)(10,47,14,43)(11,46,15,42)(12,45,16,41)(17,61,21,57)(18,60,22,64)(19,59,23,63)(20,58,24,62)(33,83,37,87)(34,82,38,86)(35,81,39,85)(36,88,40,84)(49,110,53,106)(50,109,54,105)(51,108,55,112)(52,107,56,111)(65,94,69,90)(66,93,70,89)(67,92,71,96)(68,91,72,95)(73,99,77,103)(74,98,78,102)(75,97,79,101)(76,104,80,100), (1,36,61,107,89,14,102)(2,37,62,108,90,15,103)(3,38,63,109,91,16,104)(4,39,64,110,92,9,97)(5,40,57,111,93,10,98)(6,33,58,112,94,11,99)(7,34,59,105,95,12,100)(8,35,60,106,96,13,101)(17,52,70,47,78,29,84)(18,53,71,48,79,30,85)(19,54,72,41,80,31,86)(20,55,65,42,73,32,87)(21,56,66,43,74,25,88)(22,49,67,44,75,26,81)(23,50,68,45,76,27,82)(24,51,69,46,77,28,83), (9,110,97)(10,111,98)(11,112,99)(12,105,100)(13,106,101)(14,107,102)(15,108,103)(16,109,104)(17,70,84)(18,71,85)(19,72,86)(20,65,87)(21,66,88)(22,67,81)(23,68,82)(24,69,83)(33,58,94)(34,59,95)(35,60,96)(36,61,89)(37,62,90)(38,63,91)(39,64,92)(40,57,93)(41,54,80)(42,55,73)(43,56,74)(44,49,75)(45,50,76)(46,51,77)(47,52,78)(48,53,79) );
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)], [(1,25,5,29),(2,32,6,28),(3,31,7,27),(4,30,8,26),(9,48,13,44),(10,47,14,43),(11,46,15,42),(12,45,16,41),(17,61,21,57),(18,60,22,64),(19,59,23,63),(20,58,24,62),(33,83,37,87),(34,82,38,86),(35,81,39,85),(36,88,40,84),(49,110,53,106),(50,109,54,105),(51,108,55,112),(52,107,56,111),(65,94,69,90),(66,93,70,89),(67,92,71,96),(68,91,72,95),(73,99,77,103),(74,98,78,102),(75,97,79,101),(76,104,80,100)], [(1,36,61,107,89,14,102),(2,37,62,108,90,15,103),(3,38,63,109,91,16,104),(4,39,64,110,92,9,97),(5,40,57,111,93,10,98),(6,33,58,112,94,11,99),(7,34,59,105,95,12,100),(8,35,60,106,96,13,101),(17,52,70,47,78,29,84),(18,53,71,48,79,30,85),(19,54,72,41,80,31,86),(20,55,65,42,73,32,87),(21,56,66,43,74,25,88),(22,49,67,44,75,26,81),(23,50,68,45,76,27,82),(24,51,69,46,77,28,83)], [(9,110,97),(10,111,98),(11,112,99),(12,105,100),(13,106,101),(14,107,102),(15,108,103),(16,109,104),(17,70,84),(18,71,85),(19,72,86),(20,65,87),(21,66,88),(22,67,81),(23,68,82),(24,69,83),(33,58,94),(34,59,95),(35,60,96),(36,61,89),(37,62,90),(38,63,91),(39,64,92),(40,57,93),(41,54,80),(42,55,73),(43,56,74),(44,49,75),(45,50,76),(46,51,77),(47,52,78),(48,53,79)]])
35 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 7A | 7B | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 14A | 14B | 24A | 24B | 24C | 24D | 28A | 28B | 28C | 28D | 28E | 28F | 56A | 56B | 56C | 56D |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 7 | 7 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 14 | 14 | 24 | 24 | 24 | 24 | 28 | 28 | 28 | 28 | 28 | 28 | 56 | 56 | 56 | 56 |
| size | 1 | 1 | 7 | 7 | 2 | 4 | 4 | 7 | 7 | 3 | 3 | 2 | 2 | 14 | 14 | 28 | 28 | 28 | 28 | 3 | 3 | 14 | 14 | 14 | 14 | 6 | 6 | 12 | 12 | 12 | 12 | 6 | 6 | 6 | 6 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | Q16 | C3×D4 | C3×Q16 | C7⋊C3 | C2×C7⋊C3 | C2×C7⋊C3 | D4×C7⋊C3 | Q16×C7⋊C3 |
| kernel | Q16×C7⋊C3 | C8×C7⋊C3 | Q8×C7⋊C3 | C7×Q16 | C56 | C7×Q8 | C2×C7⋊C3 | C7⋊C3 | C14 | C7 | Q16 | C8 | Q8 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 2 | 4 |
Matrix representation of Q16×C7⋊C3 ►in GL5(𝔽337)
| 0 | 311 | 0 | 0 | 0 |
| 13 | 311 | 0 | 0 | 0 |
| 0 | 0 | 336 | 0 | 0 |
| 0 | 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 0 | 336 |
| 333 | 255 | 0 | 0 | 0 |
| 292 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 336 | 212 | 1 |
| 0 | 0 | 0 | 212 | 1 |
| 0 | 0 | 336 | 213 | 1 |
| 208 | 0 | 0 | 0 | 0 |
| 0 | 208 | 0 | 0 | 0 |
| 0 | 0 | 213 | 1 | 125 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 124 |
G:=sub<GL(5,GF(337))| [0,13,0,0,0,311,311,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[333,292,0,0,0,255,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,336,0,336,0,0,212,212,213,0,0,1,1,1],[208,0,0,0,0,0,208,0,0,0,0,0,213,1,1,0,0,1,0,1,0,0,125,0,124] >;
Q16×C7⋊C3 in GAP, Magma, Sage, TeX
Q_{16}\times C_7\rtimes C_3 % in TeX
G:=Group("Q16xC7:C3"); // GroupNames label
G:=SmallGroup(336,55);
// by ID
G=gap.SmallGroup(336,55);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,169,151,867,441,69,881]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^7=d^3=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations
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