direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C16×C7⋊C3, C112⋊C3, C7⋊2C48, C56.4C6, C28.5C12, C14.2C24, C2.(C8×C7⋊C3), C4.2(C4×C7⋊C3), C8.2(C2×C7⋊C3), (C2×C7⋊C3).2C8, (C4×C7⋊C3).5C4, (C8×C7⋊C3).4C2, SmallGroup(336,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C16×C7⋊C3 |
Generators and relations for C16×C7⋊C3
G = < a,b,c | a16=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C16×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 74 54 84 22 42 107)(2 75 55 85 23 43 108)(3 76 56 86 24 44 109)(4 77 57 87 25 45 110)(5 78 58 88 26 46 111)(6 79 59 89 27 47 112)(7 80 60 90 28 48 97)(8 65 61 91 29 33 98)(9 66 62 92 30 34 99)(10 67 63 93 31 35 100)(11 68 64 94 32 36 101)(12 69 49 95 17 37 102)(13 70 50 96 18 38 103)(14 71 51 81 19 39 104)(15 72 52 82 20 40 105)(16 73 53 83 21 41 106)
(17 69 49)(18 70 50)(19 71 51)(20 72 52)(21 73 53)(22 74 54)(23 75 55)(24 76 56)(25 77 57)(26 78 58)(27 79 59)(28 80 60)(29 65 61)(30 66 62)(31 67 63)(32 68 64)(33 91 98)(34 92 99)(35 93 100)(36 94 101)(37 95 102)(38 96 103)(39 81 104)(40 82 105)(41 83 106)(42 84 107)(43 85 108)(44 86 109)(45 87 110)(46 88 111)(47 89 112)(48 90 97)
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), (1,74,54,84,22,42,107)(2,75,55,85,23,43,108)(3,76,56,86,24,44,109)(4,77,57,87,25,45,110)(5,78,58,88,26,46,111)(6,79,59,89,27,47,112)(7,80,60,90,28,48,97)(8,65,61,91,29,33,98)(9,66,62,92,30,34,99)(10,67,63,93,31,35,100)(11,68,64,94,32,36,101)(12,69,49,95,17,37,102)(13,70,50,96,18,38,103)(14,71,51,81,19,39,104)(15,72,52,82,20,40,105)(16,73,53,83,21,41,106), (17,69,49)(18,70,50)(19,71,51)(20,72,52)(21,73,53)(22,74,54)(23,75,55)(24,76,56)(25,77,57)(26,78,58)(27,79,59)(28,80,60)(29,65,61)(30,66,62)(31,67,63)(32,68,64)(33,91,98)(34,92,99)(35,93,100)(36,94,101)(37,95,102)(38,96,103)(39,81,104)(40,82,105)(41,83,106)(42,84,107)(43,85,108)(44,86,109)(45,87,110)(46,88,111)(47,89,112)(48,90,97)>;
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), (1,74,54,84,22,42,107)(2,75,55,85,23,43,108)(3,76,56,86,24,44,109)(4,77,57,87,25,45,110)(5,78,58,88,26,46,111)(6,79,59,89,27,47,112)(7,80,60,90,28,48,97)(8,65,61,91,29,33,98)(9,66,62,92,30,34,99)(10,67,63,93,31,35,100)(11,68,64,94,32,36,101)(12,69,49,95,17,37,102)(13,70,50,96,18,38,103)(14,71,51,81,19,39,104)(15,72,52,82,20,40,105)(16,73,53,83,21,41,106), (17,69,49)(18,70,50)(19,71,51)(20,72,52)(21,73,53)(22,74,54)(23,75,55)(24,76,56)(25,77,57)(26,78,58)(27,79,59)(28,80,60)(29,65,61)(30,66,62)(31,67,63)(32,68,64)(33,91,98)(34,92,99)(35,93,100)(36,94,101)(37,95,102)(38,96,103)(39,81,104)(40,82,105)(41,83,106)(42,84,107)(43,85,108)(44,86,109)(45,87,110)(46,88,111)(47,89,112)(48,90,97) );
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)], [(1,74,54,84,22,42,107),(2,75,55,85,23,43,108),(3,76,56,86,24,44,109),(4,77,57,87,25,45,110),(5,78,58,88,26,46,111),(6,79,59,89,27,47,112),(7,80,60,90,28,48,97),(8,65,61,91,29,33,98),(9,66,62,92,30,34,99),(10,67,63,93,31,35,100),(11,68,64,94,32,36,101),(12,69,49,95,17,37,102),(13,70,50,96,18,38,103),(14,71,51,81,19,39,104),(15,72,52,82,20,40,105),(16,73,53,83,21,41,106)], [(17,69,49),(18,70,50),(19,71,51),(20,72,52),(21,73,53),(22,74,54),(23,75,55),(24,76,56),(25,77,57),(26,78,58),(27,79,59),(28,80,60),(29,65,61),(30,66,62),(31,67,63),(32,68,64),(33,91,98),(34,92,99),(35,93,100),(36,94,101),(37,95,102),(38,96,103),(39,81,104),(40,82,105),(41,83,106),(42,84,107),(43,85,108),(44,86,109),(45,87,110),(46,88,111),(47,89,112),(48,90,97)]])
80 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 7A | 7B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 14A | 14B | 16A | ··· | 16H | 24A | ··· | 24H | 28A | 28B | 28C | 28D | 48A | ··· | 48P | 56A | ··· | 56H | 112A | ··· | 112P |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 7 | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 14 | 14 | 16 | ··· | 16 | 24 | ··· | 24 | 28 | 28 | 28 | 28 | 48 | ··· | 48 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 7 | 7 | 3 | 3 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 3 | 3 | 1 | ··· | 1 | 7 | ··· | 7 | 3 | 3 | 3 | 3 | 7 | ··· | 7 | 3 | ··· | 3 | 3 | ··· | 3 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | |||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C48 | C7⋊C3 | C2×C7⋊C3 | C4×C7⋊C3 | C8×C7⋊C3 | C16×C7⋊C3 |
| kernel | C16×C7⋊C3 | C8×C7⋊C3 | C112 | C4×C7⋊C3 | C56 | C2×C7⋊C3 | C28 | C7⋊C3 | C14 | C7 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 2 | 2 | 4 | 8 | 16 |
Matrix representation of C16×C7⋊C3 ►in GL4(𝔽337) generated by
| 40 | 0 | 0 | 0 |
| 0 | 252 | 0 | 0 |
| 0 | 0 | 252 | 0 |
| 0 | 0 | 0 | 252 |
| 1 | 0 | 0 | 0 |
| 0 | 124 | 125 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 128 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 212 | 336 | 336 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(337))| [40,0,0,0,0,252,0,0,0,0,252,0,0,0,0,252],[1,0,0,0,0,124,1,0,0,125,0,1,0,1,0,0],[128,0,0,0,0,1,212,0,0,0,336,1,0,0,336,0] >;
C16×C7⋊C3 in GAP, Magma, Sage, TeX
C_{16}\times C_7\rtimes C_3 % in TeX
G:=Group("C16xC7:C3"); // GroupNames label
G:=SmallGroup(336,2);
// by ID
G=gap.SmallGroup(336,2);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,-2,-7,36,50,69,1739]);
// Polycyclic
G:=Group<a,b,c|a^16=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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