direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C7⋊C24, C14⋊C24, C28.3C12, C7⋊C8⋊7C6, C7⋊2(C2×C24), (C2×C4).5F7, C4.3(C7⋊C12), (C2×C28).4C6, C4.14(C2×F7), C14.6(C2×C12), (C2×C14).1C12, C28.15(C2×C6), C22.2(C7⋊C12), (C2×C7⋊C8)⋊C3, (C2×C7⋊C3)⋊C8, C7⋊C3⋊2(C2×C8), (C4×C7⋊C3).3C4, C2.1(C2×C7⋊C12), (C22×C7⋊C3).1C4, (C4×C7⋊C3).15C22, (C2×C4×C7⋊C3).4C2, (C2×C7⋊C3).5(C2×C4), SmallGroup(336,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C2×C7⋊C24 |
Generators and relations for C2×C7⋊C24
G = < a,b,c | a2=b7=c24=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C2×C7⋊C24
►On 112 points
Generators in S112
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(65 112)(66 89)(67 90)(68 91)(69 92)(70 93)(71 94)(72 95)(73 96)(74 97)(75 98)(76 99)(77 100)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)
(1 96 112 29 104 37 21)(2 30 22 89 38 97 105)(3 90 106 23 98 31 39)(4 24 40 107 32 91 99)(5 108 100 17 92 25 33)(6 18 34 101 26 109 93)(7 102 94 35 110 19 27)(8 36 28 95 20 103 111)(9 79 71 45 87 53 61)(10 46 62 72 54 80 88)(11 73 65 63 81 47 55)(12 64 56 66 48 74 82)(13 67 83 57 75 41 49)(14 58 50 84 42 68 76)(15 85 77 51 69 59 43)(16 52 44 78 60 86 70)
(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)
G:=sub<Sym(112)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50)(65,112)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95)(73,96)(74,97)(75,98)(76,99)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111), (1,96,112,29,104,37,21)(2,30,22,89,38,97,105)(3,90,106,23,98,31,39)(4,24,40,107,32,91,99)(5,108,100,17,92,25,33)(6,18,34,101,26,109,93)(7,102,94,35,110,19,27)(8,36,28,95,20,103,111)(9,79,71,45,87,53,61)(10,46,62,72,54,80,88)(11,73,65,63,81,47,55)(12,64,56,66,48,74,82)(13,67,83,57,75,41,49)(14,58,50,84,42,68,76)(15,85,77,51,69,59,43)(16,52,44,78,60,86,70), (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)>;
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50)(65,112)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95)(73,96)(74,97)(75,98)(76,99)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111), (1,96,112,29,104,37,21)(2,30,22,89,38,97,105)(3,90,106,23,98,31,39)(4,24,40,107,32,91,99)(5,108,100,17,92,25,33)(6,18,34,101,26,109,93)(7,102,94,35,110,19,27)(8,36,28,95,20,103,111)(9,79,71,45,87,53,61)(10,46,62,72,54,80,88)(11,73,65,63,81,47,55)(12,64,56,66,48,74,82)(13,67,83,57,75,41,49)(14,58,50,84,42,68,76)(15,85,77,51,69,59,43)(16,52,44,78,60,86,70), (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) );
G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,57),(24,58),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,49),(40,50),(65,112),(66,89),(67,90),(68,91),(69,92),(70,93),(71,94),(72,95),(73,96),(74,97),(75,98),(76,99),(77,100),(78,101),(79,102),(80,103),(81,104),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111)], [(1,96,112,29,104,37,21),(2,30,22,89,38,97,105),(3,90,106,23,98,31,39),(4,24,40,107,32,91,99),(5,108,100,17,92,25,33),(6,18,34,101,26,109,93),(7,102,94,35,110,19,27),(8,36,28,95,20,103,111),(9,79,71,45,87,53,61),(10,46,62,72,54,80,88),(11,73,65,63,81,47,55),(12,64,56,66,48,74,82),(13,67,83,57,75,41,49),(14,58,50,84,42,68,76),(15,85,77,51,69,59,43),(16,52,44,78,60,86,70)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 7 | 8A | ··· | 8H | 12A | ··· | 12H | 14A | 14B | 14C | 24A | ··· | 24P | 28A | 28B | 28C | 28D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 7 | 8 | ··· | 8 | 12 | ··· | 12 | 14 | 14 | 14 | 24 | ··· | 24 | 28 | 28 | 28 | 28 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 6 | 7 | ··· | 7 | 7 | ··· | 7 | 6 | 6 | 6 | 7 | ··· | 7 | 6 | 6 | 6 | 6 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | - | ||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | F7 | C7⋊C12 | C2×F7 | C7⋊C12 | C7⋊C24 |
| kernel | C2×C7⋊C24 | C7⋊C24 | C2×C4×C7⋊C3 | C2×C7⋊C8 | C4×C7⋊C3 | C22×C7⋊C3 | C7⋊C8 | C2×C28 | C2×C7⋊C3 | C28 | C2×C14 | C14 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 4 |
Matrix representation of C2×C7⋊C24 ►in GL7(𝔽337)
| 336 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 336 | 336 | 336 | 336 | 336 | 336 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 129 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 152 | 152 | 229 | 0 | 0 | 152 |
| 0 | 0 | 152 | 0 | 152 | 152 | 229 |
| 0 | 0 | 77 | 185 | 185 | 0 | 185 |
| 0 | 152 | 0 | 152 | 152 | 229 | 0 |
| 0 | 77 | 185 | 185 | 0 | 185 | 0 |
| 0 | 185 | 0 | 0 | 77 | 185 | 185 |
G:=sub<GL(7,GF(337))| [336,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,336,1,0,0,0,0,0,336,0,1,0,0,0,0,336,0,0,1,0,0,0,336,0,0,0,1,0,0,336,0,0,0,0,1,0,336,0,0,0,0,0],[129,0,0,0,0,0,0,0,152,0,0,152,77,185,0,152,152,77,0,185,0,0,229,0,185,152,185,0,0,0,152,185,152,0,77,0,0,152,0,229,185,185,0,152,229,185,0,0,185] >;
C2×C7⋊C24 in GAP, Magma, Sage, TeX
C_2\times C_7\rtimes C_{24} % in TeX
G:=Group("C2xC7:C24"); // GroupNames label
G:=SmallGroup(336,12);
// by ID
G=gap.SmallGroup(336,12);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,72,69,10373,1745]);
// Polycyclic
G:=Group<a,b,c|a^2=b^7=c^24=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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