G = C14.462+ 1+4 order 448 = 26·7
46th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.462+ 1+4, C4⋊C4⋊8D14, (C2×D4)⋊10D14, C4⋊D4⋊20D7, C22⋊C4⋊12D14, (D4×Dic7)⋊24C2, C23⋊D14⋊13C2, (D4×C14)⋊31C22, C4⋊Dic7⋊12C22, Dic7⋊D4⋊32C2, C28.48D4⋊44C2, (C2×C28).626C23, (C2×C14).161C24, Dic7⋊C4⋊34C22, C7⋊5(C22.32C24), (C4×Dic7)⋊25C22, (C2×Dic14)⋊8C22, (C22×C4).228D14, C2.48(D4⋊6D14), C2.29(D4⋊8D14), C23.D7⋊26C22, C23.21(C22×D7), D14⋊C4.146C22, Dic7.D4⋊22C2, C22.D28⋊11C2, C22.7(D4⋊2D7), (C22×C14).27C23, (C23×D7).50C22, (C22×D7).68C23, C22.182(C23×D7), (C22×C28).312C22, (C2×Dic7).229C23, (C22×Dic7)⋊22C22, (C2×D14⋊C4)⋊26C2, C4⋊C4⋊D7⋊14C2, (C7×C4⋊D4)⋊23C2, (C7×C4⋊C4)⋊15C22, C14.85(C2×C4○D4), C2.40(C2×D4⋊2D7), (C2×C14).23(C4○D4), (C7×C22⋊C4)⋊17C22, (C2×C4).180(C22×D7), (C2×C7⋊D4).34C22, SmallGroup(448,1070)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.462+ 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=a7b2, e2=a7, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, ebe-1=a7b, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 1292 in 250 conjugacy classes, 95 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, Dic14, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22×C14, C22.32C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C22×Dic7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C23×D7, Dic7.D4, C22.D28, C4⋊C4⋊D7, C28.48D4, C2×D14⋊C4, D4×Dic7, C23⋊D14, Dic7⋊D4, C7×C4⋊D4, C14.462+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.32C24, D4⋊2D7, C23×D7, C2×D4⋊2D7, D4⋊6D14, D4⋊8D14, C14.462+ 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 62 15 79)(2 63 16 80)(3 64 17 81)(4 65 18 82)(5 66 19 83)(6 67 20 84)(7 68 21 71)(8 69 22 72)(9 70 23 73)(10 57 24 74)(11 58 25 75)(12 59 26 76)(13 60 27 77)(14 61 28 78)(29 95 52 112)(30 96 53 99)(31 97 54 100)(32 98 55 101)(33 85 56 102)(34 86 43 103)(35 87 44 104)(36 88 45 105)(37 89 46 106)(38 90 47 107)(39 91 48 108)(40 92 49 109)(41 93 50 110)(42 94 51 111)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 81)(58 82)(59 83)(60 84)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(85 109)(86 110)(87 111)(88 112)(89 99)(90 100)(91 101)(92 102)(93 103)(94 104)(95 105)(96 106)(97 107)(98 108)
(1 51 22 35)(2 50 23 34)(3 49 24 33)(4 48 25 32)(5 47 26 31)(6 46 27 30)(7 45 28 29)(8 44 15 42)(9 43 16 41)(10 56 17 40)(11 55 18 39)(12 54 19 38)(13 53 20 37)(14 52 21 36)(57 102 81 92)(58 101 82 91)(59 100 83 90)(60 99 84 89)(61 112 71 88)(62 111 72 87)(63 110 73 86)(64 109 74 85)(65 108 75 98)(66 107 76 97)(67 106 77 96)(68 105 78 95)(69 104 79 94)(70 103 80 93)
(1 42 8 35)(2 29 9 36)(3 30 10 37)(4 31 11 38)(5 32 12 39)(6 33 13 40)(7 34 14 41)(15 51 22 44)(16 52 23 45)(17 53 24 46)(18 54 25 47)(19 55 26 48)(20 56 27 49)(21 43 28 50)(57 96 64 89)(58 97 65 90)(59 98 66 91)(60 85 67 92)(61 86 68 93)(62 87 69 94)(63 88 70 95)(71 110 78 103)(72 111 79 104)(73 112 80 105)(74 99 81 106)(75 100 82 107)(76 101 83 108)(77 102 84 109)
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), (1,62,15,79)(2,63,16,80)(3,64,17,81)(4,65,18,82)(5,66,19,83)(6,67,20,84)(7,68,21,71)(8,69,22,72)(9,70,23,73)(10,57,24,74)(11,58,25,75)(12,59,26,76)(13,60,27,77)(14,61,28,78)(29,95,52,112)(30,96,53,99)(31,97,54,100)(32,98,55,101)(33,85,56,102)(34,86,43,103)(35,87,44,104)(36,88,45,105)(37,89,46,106)(38,90,47,107)(39,91,48,108)(40,92,49,109)(41,93,50,110)(42,94,51,111), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,81)(58,82)(59,83)(60,84)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(85,109)(86,110)(87,111)(88,112)(89,99)(90,100)(91,101)(92,102)(93,103)(94,104)(95,105)(96,106)(97,107)(98,108), (1,51,22,35)(2,50,23,34)(3,49,24,33)(4,48,25,32)(5,47,26,31)(6,46,27,30)(7,45,28,29)(8,44,15,42)(9,43,16,41)(10,56,17,40)(11,55,18,39)(12,54,19,38)(13,53,20,37)(14,52,21,36)(57,102,81,92)(58,101,82,91)(59,100,83,90)(60,99,84,89)(61,112,71,88)(62,111,72,87)(63,110,73,86)(64,109,74,85)(65,108,75,98)(66,107,76,97)(67,106,77,96)(68,105,78,95)(69,104,79,94)(70,103,80,93), (1,42,8,35)(2,29,9,36)(3,30,10,37)(4,31,11,38)(5,32,12,39)(6,33,13,40)(7,34,14,41)(15,51,22,44)(16,52,23,45)(17,53,24,46)(18,54,25,47)(19,55,26,48)(20,56,27,49)(21,43,28,50)(57,96,64,89)(58,97,65,90)(59,98,66,91)(60,85,67,92)(61,86,68,93)(62,87,69,94)(63,88,70,95)(71,110,78,103)(72,111,79,104)(73,112,80,105)(74,99,81,106)(75,100,82,107)(76,101,83,108)(77,102,84,109)>;
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), (1,62,15,79)(2,63,16,80)(3,64,17,81)(4,65,18,82)(5,66,19,83)(6,67,20,84)(7,68,21,71)(8,69,22,72)(9,70,23,73)(10,57,24,74)(11,58,25,75)(12,59,26,76)(13,60,27,77)(14,61,28,78)(29,95,52,112)(30,96,53,99)(31,97,54,100)(32,98,55,101)(33,85,56,102)(34,86,43,103)(35,87,44,104)(36,88,45,105)(37,89,46,106)(38,90,47,107)(39,91,48,108)(40,92,49,109)(41,93,50,110)(42,94,51,111), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,81)(58,82)(59,83)(60,84)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(85,109)(86,110)(87,111)(88,112)(89,99)(90,100)(91,101)(92,102)(93,103)(94,104)(95,105)(96,106)(97,107)(98,108), (1,51,22,35)(2,50,23,34)(3,49,24,33)(4,48,25,32)(5,47,26,31)(6,46,27,30)(7,45,28,29)(8,44,15,42)(9,43,16,41)(10,56,17,40)(11,55,18,39)(12,54,19,38)(13,53,20,37)(14,52,21,36)(57,102,81,92)(58,101,82,91)(59,100,83,90)(60,99,84,89)(61,112,71,88)(62,111,72,87)(63,110,73,86)(64,109,74,85)(65,108,75,98)(66,107,76,97)(67,106,77,96)(68,105,78,95)(69,104,79,94)(70,103,80,93), (1,42,8,35)(2,29,9,36)(3,30,10,37)(4,31,11,38)(5,32,12,39)(6,33,13,40)(7,34,14,41)(15,51,22,44)(16,52,23,45)(17,53,24,46)(18,54,25,47)(19,55,26,48)(20,56,27,49)(21,43,28,50)(57,96,64,89)(58,97,65,90)(59,98,66,91)(60,85,67,92)(61,86,68,93)(62,87,69,94)(63,88,70,95)(71,110,78,103)(72,111,79,104)(73,112,80,105)(74,99,81,106)(75,100,82,107)(76,101,83,108)(77,102,84,109) );
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)], [(1,62,15,79),(2,63,16,80),(3,64,17,81),(4,65,18,82),(5,66,19,83),(6,67,20,84),(7,68,21,71),(8,69,22,72),(9,70,23,73),(10,57,24,74),(11,58,25,75),(12,59,26,76),(13,60,27,77),(14,61,28,78),(29,95,52,112),(30,96,53,99),(31,97,54,100),(32,98,55,101),(33,85,56,102),(34,86,43,103),(35,87,44,104),(36,88,45,105),(37,89,46,106),(38,90,47,107),(39,91,48,108),(40,92,49,109),(41,93,50,110),(42,94,51,111)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,81),(58,82),(59,83),(60,84),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(85,109),(86,110),(87,111),(88,112),(89,99),(90,100),(91,101),(92,102),(93,103),(94,104),(95,105),(96,106),(97,107),(98,108)], [(1,51,22,35),(2,50,23,34),(3,49,24,33),(4,48,25,32),(5,47,26,31),(6,46,27,30),(7,45,28,29),(8,44,15,42),(9,43,16,41),(10,56,17,40),(11,55,18,39),(12,54,19,38),(13,53,20,37),(14,52,21,36),(57,102,81,92),(58,101,82,91),(59,100,83,90),(60,99,84,89),(61,112,71,88),(62,111,72,87),(63,110,73,86),(64,109,74,85),(65,108,75,98),(66,107,76,97),(67,106,77,96),(68,105,78,95),(69,104,79,94),(70,103,80,93)], [(1,42,8,35),(2,29,9,36),(3,30,10,37),(4,31,11,38),(5,32,12,39),(6,33,13,40),(7,34,14,41),(15,51,22,44),(16,52,23,45),(17,53,24,46),(18,54,25,47),(19,55,26,48),(20,56,27,49),(21,43,28,50),(57,96,64,89),(58,97,65,90),(59,98,66,91),(60,85,67,92),(61,86,68,93),(62,87,69,94),(63,88,70,95),(71,110,78,103),(72,111,79,104),(73,112,80,105),(74,99,81,106),(75,100,82,107),(76,101,83,108),(77,102,84,109)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 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 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 28 | 28 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | D4⋊2D7 | D4⋊6D14 | D4⋊8D14 |
| kernel | C14.462+ 1+4 | Dic7.D4 | C22.D28 | C4⋊C4⋊D7 | C28.48D4 | C2×D14⋊C4 | D4×Dic7 | C23⋊D14 | Dic7⋊D4 | C7×C4⋊D4 | C4⋊D4 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 3 | 4 | 6 | 3 | 3 | 9 | 2 | 6 | 6 | 6 |
Matrix representation of C14.462+ 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 19 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 19 | 22 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 14 | 0 | 0 |
| 0 | 0 | 19 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 14 |
| 0 | 0 | 0 | 0 | 19 | 20 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 14 | 0 | 0 |
| 0 | 0 | 15 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 14 |
| 0 | 0 | 0 | 0 | 15 | 20 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,19,0,0,0,0,10,22,0,0,0,0,0,0,10,19,0,0,0,0,10,22],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,0,28,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,19,0,0,0,0,14,20,0,0,0,0,0,0,9,19,0,0,0,0,14,20],[17,0,0,0,0,0,0,12,0,0,0,0,0,0,9,15,0,0,0,0,14,20,0,0,0,0,0,0,9,15,0,0,0,0,14,20] >;
C14.462+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._{46}2_+^{1+4} % in TeX
G:=Group("C14.46ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1070);
// by ID
G=gap.SmallGroup(448,1070);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,675,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=a^7*b^2,e^2=a^7,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations