?
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×C14).161C24, (C2×C28).626C23, Dic7⋊C4⋊34C22, C7⋊5(C22.32C24), (C2×Dic14)⋊8C22, (C4×Dic7)⋊25C22, (C22×C4).228D14, C2.48(D4⋊6D14), C2.29(D4⋊8D14), C23.D7⋊26C22, C23.21(C22×D7), D14⋊C4.146C22, Dic7.D4⋊22C2, C22.7(D4⋊2D7), C22.D28⋊11C2, (C22×C14).27C23, (C22×D7).68C23, (C23×D7).50C22, 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
Subgroups: 1292 in 250 conjugacy classes, 95 normal (31 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×2], C22 [×18], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×10], D4 [×9], Q8, C23, C23 [×2], C23 [×6], D7 [×2], C14 [×3], C14 [×4], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×5], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C24, Dic7 [×6], C28 [×4], D14 [×10], C2×C14, C2×C14 [×2], C2×C14 [×8], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C42⋊2C2 [×2], Dic14, C2×Dic7 [×6], C2×Dic7 [×3], C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×2], C2×C28, C7×D4 [×5], C22×D7 [×2], C22×D7 [×4], C22×C14, C22×C14 [×2], C22.32C24, C4×Dic7 [×2], Dic7⋊C4 [×2], C4⋊Dic7, C4⋊Dic7 [×2], D14⋊C4 [×8], C23.D7 [×4], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14, C22×Dic7, C22×Dic7 [×2], C2×C7⋊D4 [×4], C22×C28, D4×C14, D4×C14 [×2], C23×D7, Dic7.D4 [×2], C22.D28 [×2], C4⋊C4⋊D7 [×2], C28.48D4, C2×D14⋊C4, D4×Dic7 [×2], C23⋊D14 [×2], Dic7⋊D4 [×2], C7×C4⋊D4, C14.462+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D7 [×7], C22.32C24, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, D4⋊6D14, D4⋊8D14, C14.462+ (1+4)
Generators and relations
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 >
►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 68 27 81)(2 69 28 82)(3 70 15 83)(4 57 16 84)(5 58 17 71)(6 59 18 72)(7 60 19 73)(8 61 20 74)(9 62 21 75)(10 63 22 76)(11 64 23 77)(12 65 24 78)(13 66 25 79)(14 67 26 80)(29 87 55 104)(30 88 56 105)(31 89 43 106)(32 90 44 107)(33 91 45 108)(34 92 46 109)(35 93 47 110)(36 94 48 111)(37 95 49 112)(38 96 50 99)(39 97 51 100)(40 98 52 101)(41 85 53 102)(42 86 54 103)
(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 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 71)(66 72)(67 73)(68 74)(69 75)(70 76)(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 20 32)(2 50 21 31)(3 49 22 30)(4 48 23 29)(5 47 24 42)(6 46 25 41)(7 45 26 40)(8 44 27 39)(9 43 28 38)(10 56 15 37)(11 55 16 36)(12 54 17 35)(13 53 18 34)(14 52 19 33)(57 111 77 87)(58 110 78 86)(59 109 79 85)(60 108 80 98)(61 107 81 97)(62 106 82 96)(63 105 83 95)(64 104 84 94)(65 103 71 93)(66 102 72 92)(67 101 73 91)(68 100 74 90)(69 99 75 89)(70 112 76 88)
(1 39 8 32)(2 40 9 33)(3 41 10 34)(4 42 11 35)(5 29 12 36)(6 30 13 37)(7 31 14 38)(15 53 22 46)(16 54 23 47)(17 55 24 48)(18 56 25 49)(19 43 26 50)(20 44 27 51)(21 45 28 52)(57 93 64 86)(58 94 65 87)(59 95 66 88)(60 96 67 89)(61 97 68 90)(62 98 69 91)(63 85 70 92)(71 111 78 104)(72 112 79 105)(73 99 80 106)(74 100 81 107)(75 101 82 108)(76 102 83 109)(77 103 84 110)
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), (1,68,27,81)(2,69,28,82)(3,70,15,83)(4,57,16,84)(5,58,17,71)(6,59,18,72)(7,60,19,73)(8,61,20,74)(9,62,21,75)(10,63,22,76)(11,64,23,77)(12,65,24,78)(13,66,25,79)(14,67,26,80)(29,87,55,104)(30,88,56,105)(31,89,43,106)(32,90,44,107)(33,91,45,108)(34,92,46,109)(35,93,47,110)(36,94,48,111)(37,95,49,112)(38,96,50,99)(39,97,51,100)(40,98,52,101)(41,85,53,102)(42,86,54,103), (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)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76)(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), (1,51,20,32)(2,50,21,31)(3,49,22,30)(4,48,23,29)(5,47,24,42)(6,46,25,41)(7,45,26,40)(8,44,27,39)(9,43,28,38)(10,56,15,37)(11,55,16,36)(12,54,17,35)(13,53,18,34)(14,52,19,33)(57,111,77,87)(58,110,78,86)(59,109,79,85)(60,108,80,98)(61,107,81,97)(62,106,82,96)(63,105,83,95)(64,104,84,94)(65,103,71,93)(66,102,72,92)(67,101,73,91)(68,100,74,90)(69,99,75,89)(70,112,76,88), (1,39,8,32)(2,40,9,33)(3,41,10,34)(4,42,11,35)(5,29,12,36)(6,30,13,37)(7,31,14,38)(15,53,22,46)(16,54,23,47)(17,55,24,48)(18,56,25,49)(19,43,26,50)(20,44,27,51)(21,45,28,52)(57,93,64,86)(58,94,65,87)(59,95,66,88)(60,96,67,89)(61,97,68,90)(62,98,69,91)(63,85,70,92)(71,111,78,104)(72,112,79,105)(73,99,80,106)(74,100,81,107)(75,101,82,108)(76,102,83,109)(77,103,84,110)>;
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), (1,68,27,81)(2,69,28,82)(3,70,15,83)(4,57,16,84)(5,58,17,71)(6,59,18,72)(7,60,19,73)(8,61,20,74)(9,62,21,75)(10,63,22,76)(11,64,23,77)(12,65,24,78)(13,66,25,79)(14,67,26,80)(29,87,55,104)(30,88,56,105)(31,89,43,106)(32,90,44,107)(33,91,45,108)(34,92,46,109)(35,93,47,110)(36,94,48,111)(37,95,49,112)(38,96,50,99)(39,97,51,100)(40,98,52,101)(41,85,53,102)(42,86,54,103), (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)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76)(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), (1,51,20,32)(2,50,21,31)(3,49,22,30)(4,48,23,29)(5,47,24,42)(6,46,25,41)(7,45,26,40)(8,44,27,39)(9,43,28,38)(10,56,15,37)(11,55,16,36)(12,54,17,35)(13,53,18,34)(14,52,19,33)(57,111,77,87)(58,110,78,86)(59,109,79,85)(60,108,80,98)(61,107,81,97)(62,106,82,96)(63,105,83,95)(64,104,84,94)(65,103,71,93)(66,102,72,92)(67,101,73,91)(68,100,74,90)(69,99,75,89)(70,112,76,88), (1,39,8,32)(2,40,9,33)(3,41,10,34)(4,42,11,35)(5,29,12,36)(6,30,13,37)(7,31,14,38)(15,53,22,46)(16,54,23,47)(17,55,24,48)(18,56,25,49)(19,43,26,50)(20,44,27,51)(21,45,28,52)(57,93,64,86)(58,94,65,87)(59,95,66,88)(60,96,67,89)(61,97,68,90)(62,98,69,91)(63,85,70,92)(71,111,78,104)(72,112,79,105)(73,99,80,106)(74,100,81,107)(75,101,82,108)(76,102,83,109)(77,103,84,110) );
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)], [(1,68,27,81),(2,69,28,82),(3,70,15,83),(4,57,16,84),(5,58,17,71),(6,59,18,72),(7,60,19,73),(8,61,20,74),(9,62,21,75),(10,63,22,76),(11,64,23,77),(12,65,24,78),(13,66,25,79),(14,67,26,80),(29,87,55,104),(30,88,56,105),(31,89,43,106),(32,90,44,107),(33,91,45,108),(34,92,46,109),(35,93,47,110),(36,94,48,111),(37,95,49,112),(38,96,50,99),(39,97,51,100),(40,98,52,101),(41,85,53,102),(42,86,54,103)], [(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),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,71),(66,72),(67,73),(68,74),(69,75),(70,76),(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)], [(1,51,20,32),(2,50,21,31),(3,49,22,30),(4,48,23,29),(5,47,24,42),(6,46,25,41),(7,45,26,40),(8,44,27,39),(9,43,28,38),(10,56,15,37),(11,55,16,36),(12,54,17,35),(13,53,18,34),(14,52,19,33),(57,111,77,87),(58,110,78,86),(59,109,79,85),(60,108,80,98),(61,107,81,97),(62,106,82,96),(63,105,83,95),(64,104,84,94),(65,103,71,93),(66,102,72,92),(67,101,73,91),(68,100,74,90),(69,99,75,89),(70,112,76,88)], [(1,39,8,32),(2,40,9,33),(3,41,10,34),(4,42,11,35),(5,29,12,36),(6,30,13,37),(7,31,14,38),(15,53,22,46),(16,54,23,47),(17,55,24,48),(18,56,25,49),(19,43,26,50),(20,44,27,51),(21,45,28,52),(57,93,64,86),(58,94,65,87),(59,95,66,88),(60,96,67,89),(61,97,68,90),(62,98,69,91),(63,85,70,92),(71,111,78,104),(72,112,79,105),(73,99,80,106),(74,100,81,107),(75,101,82,108),(76,102,83,109),(77,103,84,110)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀