G = C14.612+ 1+4 order 448 = 26·7
61st non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.612+ 1+4, C4⋊C4⋊29D14, C28⋊2D4⋊30C2, C28⋊7D4⋊21C2, C22⋊C4⋊16D14, (C2×D4).98D14, (C22×C4)⋊24D14, C22⋊D28⋊21C2, D14⋊D4⋊32C2, C23⋊D14⋊17C2, D28⋊C4⋊33C2, D14⋊C4⋊28C22, D14⋊2Q8⋊31C2, D14.6(C4○D4), C4⋊Dic7⋊15C22, C22.D4⋊6D7, D14.5D4⋊29C2, (C2×C28).179C23, (C2×C14).201C24, Dic7⋊C4⋊54C22, (C22×C28)⋊18C22, C7⋊7(C22.32C24), (C4×Dic7)⋊33C22, (C2×D28).32C22, C2.63(D4⋊6D14), C23.D7⋊53C22, C2.42(D4⋊8D14), Dic7.D4⋊32C2, (C2×Dic14)⋊29C22, (D4×C14).139C22, C23.D14⋊30C2, (C23×D7).58C22, (C22×D7).85C23, C23.129(C22×D7), C22.222(C23×D7), (C22×C14).221C23, (C2×Dic7).244C23, (C4×C7⋊D4)⋊6C2, C2.63(D7×C4○D4), (C2×C4×D7)⋊23C22, C4⋊C4⋊D7⋊27C2, (C7×C4⋊C4)⋊27C22, (D7×C22⋊C4)⋊13C2, C14.175(C2×C4○D4), (C2×C7⋊D4)⋊20C22, (C2×C4).64(C22×D7), (C7×C22⋊C4)⋊23C22, (C7×C22.D4)⋊9C2, SmallGroup(448,1110)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.612+ 1+4
G = < a,b,c,d,e | a14=b4=c2=e2=1, d2=a7b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a7b-1, dbd-1=ebe=a7b, dcd-1=ece=a7c, ede=a7b2d >
Subgroups: 1388 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, 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, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C42⋊2C2, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22.32C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C23.D14, D7×C22⋊C4, C22⋊D28, D14⋊D4, Dic7.D4, D28⋊C4, D14.5D4, D14⋊2Q8, C4⋊C4⋊D7, C4×C7⋊D4, C28⋊7D4, C23⋊D14, C28⋊2D4, C7×C22.D4, C14.612+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.32C24, C23×D7, D4⋊6D14, D7×C4○D4, D4⋊8D14, C14.612+ 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 67 20 71)(2 68 21 72)(3 69 22 73)(4 70 23 74)(5 57 24 75)(6 58 25 76)(7 59 26 77)(8 60 27 78)(9 61 28 79)(10 62 15 80)(11 63 16 81)(12 64 17 82)(13 65 18 83)(14 66 19 84)(29 88 50 100)(30 89 51 101)(31 90 52 102)(32 91 53 103)(33 92 54 104)(34 93 55 105)(35 94 56 106)(36 95 43 107)(37 96 44 108)(38 97 45 109)(39 98 46 110)(40 85 47 111)(41 86 48 112)(42 87 49 99)
(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 82)(58 83)(59 84)(60 71)(61 72)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(85 111)(86 112)(87 99)(88 100)(89 101)(90 102)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)
(1 45 27 31)(2 44 28 30)(3 43 15 29)(4 56 16 42)(5 55 17 41)(6 54 18 40)(7 53 19 39)(8 52 20 38)(9 51 21 37)(10 50 22 36)(11 49 23 35)(12 48 24 34)(13 47 25 33)(14 46 26 32)(57 112 82 93)(58 111 83 92)(59 110 84 91)(60 109 71 90)(61 108 72 89)(62 107 73 88)(63 106 74 87)(64 105 75 86)(65 104 76 85)(66 103 77 98)(67 102 78 97)(68 101 79 96)(69 100 80 95)(70 99 81 94)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 54)(16 55)(17 56)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(57 94)(58 95)(59 96)(60 97)(61 98)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 91)(69 92)(70 93)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 112)(82 99)(83 100)(84 101)
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), (1,67,20,71)(2,68,21,72)(3,69,22,73)(4,70,23,74)(5,57,24,75)(6,58,25,76)(7,59,26,77)(8,60,27,78)(9,61,28,79)(10,62,15,80)(11,63,16,81)(12,64,17,82)(13,65,18,83)(14,66,19,84)(29,88,50,100)(30,89,51,101)(31,90,52,102)(32,91,53,103)(33,92,54,104)(34,93,55,105)(35,94,56,106)(36,95,43,107)(37,96,44,108)(38,97,45,109)(39,98,46,110)(40,85,47,111)(41,86,48,112)(42,87,49,99), (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)(57,82)(58,83)(59,84)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(85,111)(86,112)(87,99)(88,100)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110), (1,45,27,31)(2,44,28,30)(3,43,15,29)(4,56,16,42)(5,55,17,41)(6,54,18,40)(7,53,19,39)(8,52,20,38)(9,51,21,37)(10,50,22,36)(11,49,23,35)(12,48,24,34)(13,47,25,33)(14,46,26,32)(57,112,82,93)(58,111,83,92)(59,110,84,91)(60,109,71,90)(61,108,72,89)(62,107,73,88)(63,106,74,87)(64,105,75,86)(65,104,76,85)(66,103,77,98)(67,102,78,97)(68,101,79,96)(69,100,80,95)(70,99,81,94), (1,38)(2,39)(3,40)(4,41)(5,42)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,54)(16,55)(17,56)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(57,94)(58,95)(59,96)(60,97)(61,98)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,99)(83,100)(84,101)>;
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), (1,67,20,71)(2,68,21,72)(3,69,22,73)(4,70,23,74)(5,57,24,75)(6,58,25,76)(7,59,26,77)(8,60,27,78)(9,61,28,79)(10,62,15,80)(11,63,16,81)(12,64,17,82)(13,65,18,83)(14,66,19,84)(29,88,50,100)(30,89,51,101)(31,90,52,102)(32,91,53,103)(33,92,54,104)(34,93,55,105)(35,94,56,106)(36,95,43,107)(37,96,44,108)(38,97,45,109)(39,98,46,110)(40,85,47,111)(41,86,48,112)(42,87,49,99), (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)(57,82)(58,83)(59,84)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(85,111)(86,112)(87,99)(88,100)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110), (1,45,27,31)(2,44,28,30)(3,43,15,29)(4,56,16,42)(5,55,17,41)(6,54,18,40)(7,53,19,39)(8,52,20,38)(9,51,21,37)(10,50,22,36)(11,49,23,35)(12,48,24,34)(13,47,25,33)(14,46,26,32)(57,112,82,93)(58,111,83,92)(59,110,84,91)(60,109,71,90)(61,108,72,89)(62,107,73,88)(63,106,74,87)(64,105,75,86)(65,104,76,85)(66,103,77,98)(67,102,78,97)(68,101,79,96)(69,100,80,95)(70,99,81,94), (1,38)(2,39)(3,40)(4,41)(5,42)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,54)(16,55)(17,56)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(57,94)(58,95)(59,96)(60,97)(61,98)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,99)(83,100)(84,101) );
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)], [(1,67,20,71),(2,68,21,72),(3,69,22,73),(4,70,23,74),(5,57,24,75),(6,58,25,76),(7,59,26,77),(8,60,27,78),(9,61,28,79),(10,62,15,80),(11,63,16,81),(12,64,17,82),(13,65,18,83),(14,66,19,84),(29,88,50,100),(30,89,51,101),(31,90,52,102),(32,91,53,103),(33,92,54,104),(34,93,55,105),(35,94,56,106),(36,95,43,107),(37,96,44,108),(38,97,45,109),(39,98,46,110),(40,85,47,111),(41,86,48,112),(42,87,49,99)], [(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),(57,82),(58,83),(59,84),(60,71),(61,72),(62,73),(63,74),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(85,111),(86,112),(87,99),(88,100),(89,101),(90,102),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108),(97,109),(98,110)], [(1,45,27,31),(2,44,28,30),(3,43,15,29),(4,56,16,42),(5,55,17,41),(6,54,18,40),(7,53,19,39),(8,52,20,38),(9,51,21,37),(10,50,22,36),(11,49,23,35),(12,48,24,34),(13,47,25,33),(14,46,26,32),(57,112,82,93),(58,111,83,92),(59,110,84,91),(60,109,71,90),(61,108,72,89),(62,107,73,88),(63,106,74,87),(64,105,75,86),(65,104,76,85),(66,103,77,98),(67,102,78,97),(68,101,79,96),(69,100,80,95),(70,99,81,94)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,37),(15,54),(16,55),(17,56),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(57,94),(58,95),(59,96),(60,97),(61,98),(62,85),(63,86),(64,87),(65,88),(66,89),(67,90),(68,91),(69,92),(70,93),(71,102),(72,103),(73,104),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111),(81,112),(82,99),(83,100),(84,101)]])
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 | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| 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 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | D4⋊6D14 | D7×C4○D4 | D4⋊8D14 |
| kernel | C14.612+ 1+4 | C23.D14 | D7×C22⋊C4 | C22⋊D28 | D14⋊D4 | Dic7.D4 | D28⋊C4 | D14.5D4 | D14⋊2Q8 | C4⋊C4⋊D7 | C4×C7⋊D4 | C28⋊7D4 | C23⋊D14 | C28⋊2D4 | C7×C22.D4 | C22.D4 | D14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 9 | 6 | 3 | 3 | 2 | 6 | 6 | 6 |
Matrix representation of C14.612+ 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 19 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 19 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 23 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 6 | 13 | 17 |
| 0 | 0 | 23 | 21 | 12 | 16 |
| 0 | 0 | 8 | 6 | 21 | 23 |
| 0 | 0 | 23 | 21 | 6 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 15 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 28 | 0 |
| 0 | 0 | 0 | 1 | 0 | 28 |
| 23 | 24 | 0 | 0 | 0 | 0 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 2 | 0 |
| 0 | 0 | 22 | 1 | 14 | 27 |
| 0 | 0 | 28 | 0 | 1 | 0 |
| 0 | 0 | 22 | 1 | 7 | 28 |
| 23 | 24 | 0 | 0 | 0 | 0 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 27 | 0 |
| 0 | 0 | 0 | 1 | 0 | 27 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,19,10,0,0,0,0,19,7,0,0,0,0,0,0,19,10,0,0,0,0,19,7],[17,23,0,0,0,0,0,12,0,0,0,0,0,0,8,23,8,23,0,0,6,21,6,21,0,0,13,12,21,6,0,0,17,16,23,8],[1,15,0,0,0,0,0,28,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,0,0,28,0,0,0,0,0,0,28],[23,7,0,0,0,0,24,6,0,0,0,0,0,0,28,22,28,22,0,0,0,1,0,1,0,0,2,14,1,7,0,0,0,27,0,28],[23,7,0,0,0,0,24,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,27,0,28,0,0,0,0,27,0,28] >;
C14.612+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._{61}2_+^{1+4} % in TeX
G:=Group("C14.61ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1110);
// by ID
G=gap.SmallGroup(448,1110);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,675,570,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=e^2=1,d^2=a^7*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=a^7*b^-1,d*b*d^-1=e*b*e=a^7*b,d*c*d^-1=e*c*e=a^7*c,e*d*e=a^7*b^2*d>;
// generators/relations