direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C4×C7⋊C12, C28⋊2C12, Dic7⋊2C12, C7⋊(C4×C12), C7⋊C3⋊C42, (C4×Dic7)⋊C3, (C2×C4).6F7, C2.2(C4×F7), (C2×C28).5C6, C14.3(C2×C12), C22.3(C2×F7), (C2×Dic7).4C6, (C4×C7⋊C3)⋊2C4, C2.2(C2×C7⋊C12), (C2×C7⋊C12).4C2, (C2×C14).2(C2×C6), (C22×C7⋊C3).2C22, (C2×C4×C7⋊C3).5C2, (C2×C7⋊C3).7(C2×C4), SmallGroup(336,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C4×C7⋊C12 |
Generators and relations for C4×C7⋊C12
G = < a,b,c | a4=b7=c12=1, ab=ba, ac=ca, cbc-1=b5 >
Subgroups: 192 in 60 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C7, C2×C4, C2×C4, C12, C2×C6, C14, C14, C42, C7⋊C3, C2×C12, Dic7, C28, C2×C14, C2×C7⋊C3, C2×C7⋊C3, C4×C12, C2×Dic7, C2×C28, C7⋊C12, C4×C7⋊C3, C22×C7⋊C3, C4×Dic7, C2×C7⋊C12, C2×C4×C7⋊C3, C4×C7⋊C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, C42, C2×C12, F7, C4×C12, C7⋊C12, C2×F7, C4×F7, C2×C7⋊C12, C4×C7⋊C12
►On 112 points
Generators in S112
(1 12 6 14)(2 9 7 15)(3 10 8 16)(4 11 5 13)(17 49 85 105)(18 50 86 106)(19 51 87 107)(20 52 88 108)(21 41 77 109)(22 42 78 110)(23 43 79 111)(24 44 80 112)(25 45 81 101)(26 46 82 102)(27 47 83 103)(28 48 84 104)(29 75 57 91)(30 76 58 92)(31 65 59 93)(32 66 60 94)(33 67 61 95)(34 68 62 96)(35 69 63 97)(36 70 64 98)(37 71 53 99)(38 72 54 100)(39 73 55 89)(40 74 56 90)
(1 98 94 26 90 18 22)(2 19 27 99 23 91 95)(3 92 100 20 96 24 28)(4 25 21 93 17 97 89)(5 81 77 65 85 69 73)(6 70 66 82 74 86 78)(7 87 83 71 79 75 67)(8 76 72 88 68 80 84)(9 51 47 37 43 29 33)(10 30 38 52 34 44 48)(11 45 41 31 49 35 39)(12 36 32 46 40 50 42)(13 101 109 59 105 63 55)(14 64 60 102 56 106 110)(15 107 103 53 111 57 61)(16 58 54 108 62 112 104)
(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)| (1,12,6,14)(2,9,7,15)(3,10,8,16)(4,11,5,13)(17,49,85,105)(18,50,86,106)(19,51,87,107)(20,52,88,108)(21,41,77,109)(22,42,78,110)(23,43,79,111)(24,44,80,112)(25,45,81,101)(26,46,82,102)(27,47,83,103)(28,48,84,104)(29,75,57,91)(30,76,58,92)(31,65,59,93)(32,66,60,94)(33,67,61,95)(34,68,62,96)(35,69,63,97)(36,70,64,98)(37,71,53,99)(38,72,54,100)(39,73,55,89)(40,74,56,90), (1,98,94,26,90,18,22)(2,19,27,99,23,91,95)(3,92,100,20,96,24,28)(4,25,21,93,17,97,89)(5,81,77,65,85,69,73)(6,70,66,82,74,86,78)(7,87,83,71,79,75,67)(8,76,72,88,68,80,84)(9,51,47,37,43,29,33)(10,30,38,52,34,44,48)(11,45,41,31,49,35,39)(12,36,32,46,40,50,42)(13,101,109,59,105,63,55)(14,64,60,102,56,106,110)(15,107,103,53,111,57,61)(16,58,54,108,62,112,104), (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( (1,12,6,14)(2,9,7,15)(3,10,8,16)(4,11,5,13)(17,49,85,105)(18,50,86,106)(19,51,87,107)(20,52,88,108)(21,41,77,109)(22,42,78,110)(23,43,79,111)(24,44,80,112)(25,45,81,101)(26,46,82,102)(27,47,83,103)(28,48,84,104)(29,75,57,91)(30,76,58,92)(31,65,59,93)(32,66,60,94)(33,67,61,95)(34,68,62,96)(35,69,63,97)(36,70,64,98)(37,71,53,99)(38,72,54,100)(39,73,55,89)(40,74,56,90), (1,98,94,26,90,18,22)(2,19,27,99,23,91,95)(3,92,100,20,96,24,28)(4,25,21,93,17,97,89)(5,81,77,65,85,69,73)(6,70,66,82,74,86,78)(7,87,83,71,79,75,67)(8,76,72,88,68,80,84)(9,51,47,37,43,29,33)(10,30,38,52,34,44,48)(11,45,41,31,49,35,39)(12,36,32,46,40,50,42)(13,101,109,59,105,63,55)(14,64,60,102,56,106,110)(15,107,103,53,111,57,61)(16,58,54,108,62,112,104), (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([[(1,12,6,14),(2,9,7,15),(3,10,8,16),(4,11,5,13),(17,49,85,105),(18,50,86,106),(19,51,87,107),(20,52,88,108),(21,41,77,109),(22,42,78,110),(23,43,79,111),(24,44,80,112),(25,45,81,101),(26,46,82,102),(27,47,83,103),(28,48,84,104),(29,75,57,91),(30,76,58,92),(31,65,59,93),(32,66,60,94),(33,67,61,95),(34,68,62,96),(35,69,63,97),(36,70,64,98),(37,71,53,99),(38,72,54,100),(39,73,55,89),(40,74,56,90)], [(1,98,94,26,90,18,22),(2,19,27,99,23,91,95),(3,92,100,20,96,24,28),(4,25,21,93,17,97,89),(5,81,77,65,85,69,73),(6,70,66,82,74,86,78),(7,87,83,71,79,75,67),(8,76,72,88,68,80,84),(9,51,47,37,43,29,33),(10,30,38,52,34,44,48),(11,45,41,31,49,35,39),(12,36,32,46,40,50,42),(13,101,109,59,105,63,55),(14,64,60,102,56,106,110),(15,107,103,53,111,57,61),(16,58,54,108,62,112,104)], [(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 | 4E | ··· | 4L | 6A | ··· | 6F | 7 | 12A | ··· | 12X | 14A | 14B | 14C | 28A | 28B | 28C | 28D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 7 | 12 | ··· | 12 | 14 | 14 | 14 | 28 | 28 | 28 | 28 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 7 | ··· | 7 | 6 | 7 | ··· | 7 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | F7 | C7⋊C12 | C2×F7 | C4×F7 |
| kernel | C4×C7⋊C12 | C2×C7⋊C12 | C2×C4×C7⋊C3 | C4×Dic7 | C7⋊C12 | C4×C7⋊C3 | C2×Dic7 | C2×C28 | Dic7 | C28 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 8 | 4 | 4 | 2 | 16 | 8 | 1 | 2 | 1 | 4 |
Matrix representation of C4×C7⋊C12 ►in GL7(𝔽337)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 189 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 189 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 189 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 189 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 189 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 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 | 1 | 336 |
| 265 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 8 | 240 | 329 |
| 0 | 8 | 8 | 240 | 0 | 329 | 0 |
| 0 | 248 | 0 | 329 | 8 | 329 | 0 |
| 0 | 0 | 8 | 329 | 8 | 0 | 240 |
| 0 | 0 | 8 | 0 | 248 | 329 | 329 |
| 0 | 8 | 248 | 329 | 0 | 0 | 329 |
G:=sub<GL(7,GF(337))| [1,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,336,336,336,336,336,336],[265,0,0,0,0,0,0,0,8,8,248,0,0,8,0,0,8,0,8,8,248,0,0,240,329,329,0,329,0,8,0,8,8,248,0,0,240,329,329,0,329,0,0,329,0,0,240,329,329] >;
C4×C7⋊C12 in GAP, Magma, Sage, TeX
C_4\times C_7\rtimes C_{12} % in TeX
G:=Group("C4xC7:C12"); // GroupNames label
G:=SmallGroup(336,14);
// by ID
G=gap.SmallGroup(336,14);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,72,151,10373,1745]);
// Polycyclic
G:=Group<a,b,c|a^4=b^7=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations