G = C14.1212+ 1+4 order 448 = 26·7
30th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.1212+ 1+4, C7⋊D4⋊6D4, C4⋊C4⋊15D14, C7⋊8(D4⋊5D4), C4⋊D28⋊30C2, C22⋊C4⋊15D14, D14.22(C2×D4), (C22×C4)⋊23D14, D14⋊10(C4○D4), D14⋊D4⋊29C2, C22⋊D28⋊19C2, D28⋊C4⋊31C2, C22.12(D4×D7), D14⋊C4⋊22C22, D14⋊Q8⋊27C2, (C2×D4).163D14, (C2×D28)⋊26C22, (C2×C28).71C23, Dic7.26(C2×D4), C22.D4⋊4D7, C14.83(C22×D4), Dic7⋊D4⋊20C2, Dic7⋊4D4⋊18C2, D14.5D4⋊27C2, (C2×C14).198C24, Dic7⋊C4⋊22C22, (C22×C28)⋊17C22, (C4×Dic7)⋊31C22, C2.41(D4⋊8D14), C23.D7⋊30C22, C23.26(C22×D7), Dic7.D4⋊31C2, (C2×Dic14)⋊55C22, (D4×C14).136C22, (C22×C14).33C23, (C23×D7).57C22, C22.219(C23×D7), (C2×Dic7).102C23, (C22×Dic7)⋊24C22, (C22×D7).206C23, (C2×D4×D7)⋊15C2, C2.56(C2×D4×D7), C2.60(D7×C4○D4), (C2×C4×D7)⋊21C22, (C2×D14⋊C4)⋊23C2, (C2×C4○D28)⋊11C2, (C7×C4⋊C4)⋊25C22, (D7×C22⋊C4)⋊12C2, (C2×C14).59(C2×D4), C14.172(C2×C4○D4), (C2×C7⋊D4)⋊41C22, (C2×C4).61(C22×D7), (C7×C22⋊C4)⋊21C22, (C7×C22.D4)⋊6C2, SmallGroup(448,1107)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.1212+ 1+4
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a7b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
Subgroups: 1932 in 334 conjugacy classes, 105 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, D4⋊5D4, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, D4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, D7×C22⋊C4, Dic7⋊4D4, C22⋊D28, D14⋊D4, Dic7.D4, D28⋊C4, D14.5D4, C4⋊D28, D14⋊Q8, C2×D14⋊C4, Dic7⋊D4, C7×C22.D4, C2×C4○D28, C2×D4×D7, C14.1212+ 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2+ 1+4, C22×D7, D4⋊5D4, D4×D7, C23×D7, C2×D4×D7, D7×C4○D4, D4⋊8D14, C14.1212+ 1+4
►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)
(1 100 77 51)(2 101 78 52)(3 102 79 53)(4 103 80 54)(5 104 81 55)(6 105 82 56)(7 106 83 43)(8 107 84 44)(9 108 71 45)(10 109 72 46)(11 110 73 47)(12 111 74 48)(13 112 75 49)(14 99 76 50)(15 94 41 61)(16 95 42 62)(17 96 29 63)(18 97 30 64)(19 98 31 65)(20 85 32 66)(21 86 33 67)(22 87 34 68)(23 88 35 69)(24 89 36 70)(25 90 37 57)(26 91 38 58)(27 92 39 59)(28 93 40 60)
(1 107 8 100)(2 106 9 99)(3 105 10 112)(4 104 11 111)(5 103 12 110)(6 102 13 109)(7 101 14 108)(15 69 22 62)(16 68 23 61)(17 67 24 60)(18 66 25 59)(19 65 26 58)(20 64 27 57)(21 63 28 70)(29 86 36 93)(30 85 37 92)(31 98 38 91)(32 97 39 90)(33 96 40 89)(34 95 41 88)(35 94 42 87)(43 71 50 78)(44 84 51 77)(45 83 52 76)(46 82 53 75)(47 81 54 74)(48 80 55 73)(49 79 56 72)
(1 58 77 91)(2 57 78 90)(3 70 79 89)(4 69 80 88)(5 68 81 87)(6 67 82 86)(7 66 83 85)(8 65 84 98)(9 64 71 97)(10 63 72 96)(11 62 73 95)(12 61 74 94)(13 60 75 93)(14 59 76 92)(15 48 41 111)(16 47 42 110)(17 46 29 109)(18 45 30 108)(19 44 31 107)(20 43 32 106)(21 56 33 105)(22 55 34 104)(23 54 35 103)(24 53 36 102)(25 52 37 101)(26 51 38 100)(27 50 39 99)(28 49 40 112)
(1 98)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 91)(9 92)(10 93)(11 94)(12 95)(13 96)(14 97)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 43)(26 44)(27 45)(28 46)(29 112)(30 99)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 110)(42 111)(57 83)(58 84)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 81)(70 82)
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), (1,100,77,51)(2,101,78,52)(3,102,79,53)(4,103,80,54)(5,104,81,55)(6,105,82,56)(7,106,83,43)(8,107,84,44)(9,108,71,45)(10,109,72,46)(11,110,73,47)(12,111,74,48)(13,112,75,49)(14,99,76,50)(15,94,41,61)(16,95,42,62)(17,96,29,63)(18,97,30,64)(19,98,31,65)(20,85,32,66)(21,86,33,67)(22,87,34,68)(23,88,35,69)(24,89,36,70)(25,90,37,57)(26,91,38,58)(27,92,39,59)(28,93,40,60), (1,107,8,100)(2,106,9,99)(3,105,10,112)(4,104,11,111)(5,103,12,110)(6,102,13,109)(7,101,14,108)(15,69,22,62)(16,68,23,61)(17,67,24,60)(18,66,25,59)(19,65,26,58)(20,64,27,57)(21,63,28,70)(29,86,36,93)(30,85,37,92)(31,98,38,91)(32,97,39,90)(33,96,40,89)(34,95,41,88)(35,94,42,87)(43,71,50,78)(44,84,51,77)(45,83,52,76)(46,82,53,75)(47,81,54,74)(48,80,55,73)(49,79,56,72), (1,58,77,91)(2,57,78,90)(3,70,79,89)(4,69,80,88)(5,68,81,87)(6,67,82,86)(7,66,83,85)(8,65,84,98)(9,64,71,97)(10,63,72,96)(11,62,73,95)(12,61,74,94)(13,60,75,93)(14,59,76,92)(15,48,41,111)(16,47,42,110)(17,46,29,109)(18,45,30,108)(19,44,31,107)(20,43,32,106)(21,56,33,105)(22,55,34,104)(23,54,35,103)(24,53,36,102)(25,52,37,101)(26,51,38,100)(27,50,39,99)(28,49,40,112), (1,98)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,97)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,43)(26,44)(27,45)(28,46)(29,112)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(57,83)(58,84)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,81)(70,82)>;
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), (1,100,77,51)(2,101,78,52)(3,102,79,53)(4,103,80,54)(5,104,81,55)(6,105,82,56)(7,106,83,43)(8,107,84,44)(9,108,71,45)(10,109,72,46)(11,110,73,47)(12,111,74,48)(13,112,75,49)(14,99,76,50)(15,94,41,61)(16,95,42,62)(17,96,29,63)(18,97,30,64)(19,98,31,65)(20,85,32,66)(21,86,33,67)(22,87,34,68)(23,88,35,69)(24,89,36,70)(25,90,37,57)(26,91,38,58)(27,92,39,59)(28,93,40,60), (1,107,8,100)(2,106,9,99)(3,105,10,112)(4,104,11,111)(5,103,12,110)(6,102,13,109)(7,101,14,108)(15,69,22,62)(16,68,23,61)(17,67,24,60)(18,66,25,59)(19,65,26,58)(20,64,27,57)(21,63,28,70)(29,86,36,93)(30,85,37,92)(31,98,38,91)(32,97,39,90)(33,96,40,89)(34,95,41,88)(35,94,42,87)(43,71,50,78)(44,84,51,77)(45,83,52,76)(46,82,53,75)(47,81,54,74)(48,80,55,73)(49,79,56,72), (1,58,77,91)(2,57,78,90)(3,70,79,89)(4,69,80,88)(5,68,81,87)(6,67,82,86)(7,66,83,85)(8,65,84,98)(9,64,71,97)(10,63,72,96)(11,62,73,95)(12,61,74,94)(13,60,75,93)(14,59,76,92)(15,48,41,111)(16,47,42,110)(17,46,29,109)(18,45,30,108)(19,44,31,107)(20,43,32,106)(21,56,33,105)(22,55,34,104)(23,54,35,103)(24,53,36,102)(25,52,37,101)(26,51,38,100)(27,50,39,99)(28,49,40,112), (1,98)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,97)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,43)(26,44)(27,45)(28,46)(29,112)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(57,83)(58,84)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,81)(70,82) );
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)], [(1,100,77,51),(2,101,78,52),(3,102,79,53),(4,103,80,54),(5,104,81,55),(6,105,82,56),(7,106,83,43),(8,107,84,44),(9,108,71,45),(10,109,72,46),(11,110,73,47),(12,111,74,48),(13,112,75,49),(14,99,76,50),(15,94,41,61),(16,95,42,62),(17,96,29,63),(18,97,30,64),(19,98,31,65),(20,85,32,66),(21,86,33,67),(22,87,34,68),(23,88,35,69),(24,89,36,70),(25,90,37,57),(26,91,38,58),(27,92,39,59),(28,93,40,60)], [(1,107,8,100),(2,106,9,99),(3,105,10,112),(4,104,11,111),(5,103,12,110),(6,102,13,109),(7,101,14,108),(15,69,22,62),(16,68,23,61),(17,67,24,60),(18,66,25,59),(19,65,26,58),(20,64,27,57),(21,63,28,70),(29,86,36,93),(30,85,37,92),(31,98,38,91),(32,97,39,90),(33,96,40,89),(34,95,41,88),(35,94,42,87),(43,71,50,78),(44,84,51,77),(45,83,52,76),(46,82,53,75),(47,81,54,74),(48,80,55,73),(49,79,56,72)], [(1,58,77,91),(2,57,78,90),(3,70,79,89),(4,69,80,88),(5,68,81,87),(6,67,82,86),(7,66,83,85),(8,65,84,98),(9,64,71,97),(10,63,72,96),(11,62,73,95),(12,61,74,94),(13,60,75,93),(14,59,76,92),(15,48,41,111),(16,47,42,110),(17,46,29,109),(18,45,30,108),(19,44,31,107),(20,43,32,106),(21,56,33,105),(22,55,34,104),(23,54,35,103),(24,53,36,102),(25,52,37,101),(26,51,38,100),(27,50,39,99),(28,49,40,112)], [(1,98),(2,85),(3,86),(4,87),(5,88),(6,89),(7,90),(8,91),(9,92),(10,93),(11,94),(12,95),(13,96),(14,97),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,43),(26,44),(27,45),(28,46),(29,112),(30,99),(31,100),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(41,110),(42,111),(57,83),(58,84),(59,71),(60,72),(61,73),(62,74),(63,75),(64,76),(65,77),(66,78),(67,79),(68,80),(69,81),(70,82)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | D4×D7 | D7×C4○D4 | D4⋊8D14 |
| kernel | C14.1212+ 1+4 | D7×C22⋊C4 | Dic7⋊4D4 | C22⋊D28 | D14⋊D4 | Dic7.D4 | D28⋊C4 | D14.5D4 | C4⋊D28 | D14⋊Q8 | C2×D14⋊C4 | Dic7⋊D4 | C7×C22.D4 | C2×C4○D28 | C2×D4×D7 | C7⋊D4 | C22.D4 | D14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 9 | 6 | 3 | 3 | 1 | 6 | 6 | 6 |
Matrix representation of C14.1212+ 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 10 | 0 | 0 |
| 0 | 0 | 9 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 5 | 7 | 0 | 0 | 0 | 0 |
| 9 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 5 | 7 | 0 | 0 | 0 | 0 |
| 17 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 22 | 0 | 0 |
| 0 | 0 | 13 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 22 | 0 | 0 |
| 0 | 0 | 13 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 2 |
| 0 | 0 | 0 | 0 | 28 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 27 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,9,9,0,0,0,0,10,23,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[5,9,0,0,0,0,7,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,17,0,0,0,0,7,24,0,0,0,0,0,0,11,13,0,0,0,0,22,18,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,11,13,0,0,0,0,22,18,0,0,0,0,0,0,28,28,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,27,28] >;
C14.1212+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._{121}2_+^{1+4} % in TeX
G:=Group("C14.121ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1107);
// by ID
G=gap.SmallGroup(448,1107);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,100,346,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^7*b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations