?
G = C14.372+ (1+4) order 448 = 26·7
37th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.372+ (1+4), C4⋊C4⋊5D14, C4⋊D4⋊9D7, (C2×D4)⋊21D14, C22⋊C4⋊9D14, (C22×D7)⋊7D4, C28⋊7D4⋊44C2, C22.3(D4×D7), D14.39(C2×D4), (C2×D28)⋊7C22, (C22×C4)⋊15D14, C7⋊3(C23⋊3D4), C23⋊D14⋊22C2, C22⋊D28⋊11C2, D14⋊C4⋊66C22, (D4×C14)⋊29C22, (C2×C28).38C23, C4⋊Dic7⋊11C22, C14.64(C22×D4), (C23×D7)⋊9C22, Dic7⋊D4⋊12C2, D14.5D4⋊11C2, D14.D4⋊17C2, (C2×C14).149C24, Dic7⋊C4⋊15C22, (C22×C28)⋊30C22, C23.D7⋊21C22, C2.39(D4⋊6D14), C2.26(D4⋊8D14), (C22×C14).18C23, (C2×Dic7).70C23, C23.180(C22×D7), C22.170(C23×D7), (C22×Dic7)⋊18C22, (C22×D7).184C23, (C2×D4×D7)⋊9C2, C2.37(C2×D4×D7), (C7×C4⋊C4)⋊8C22, (C2×C14).5(C2×D4), (C2×C4×D7)⋊12C22, (C2×D14⋊C4)⋊25C2, (C7×C4⋊D4)⋊11C2, (C2×C7⋊D4)⋊13C22, (C7×C22⋊C4)⋊10C22, (C2×C4).175(C22×D7), SmallGroup(448,1058)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2124 in 346 conjugacy classes, 103 normal (31 characteristic)
C1, C2 [×3], C2 [×10], C4 [×8], C22, C22 [×2], C22 [×34], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×10], D4 [×20], C23, C23 [×2], C23 [×18], D7 [×6], C14 [×3], C14 [×4], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×17], C24 [×3], Dic7 [×4], C28 [×4], D14 [×4], D14 [×22], C2×C14, C2×C14 [×2], C2×C14 [×8], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4, C4⋊D4 [×3], C22.D4 [×4], C22×D4 [×2], C4×D7 [×4], D28 [×5], C2×Dic7 [×4], C2×Dic7, C7⋊D4 [×10], C2×C28 [×2], C2×C28 [×2], C2×C28, C7×D4 [×5], C22×D7 [×8], C22×D7 [×10], C22×C14, C22×C14 [×2], C23⋊3D4, Dic7⋊C4 [×2], C4⋊Dic7, D14⋊C4 [×8], C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×C4×D7 [×2], C2×D28, C2×D28 [×2], D4×D7 [×8], C22×Dic7, C2×C7⋊D4 [×6], C22×C28, D4×C14, D4×C14 [×2], C23×D7, C23×D7 [×2], C22⋊D28 [×2], D14.D4 [×2], D14.5D4 [×2], C2×D14⋊C4, C28⋊7D4, C23⋊D14 [×2], Dic7⋊D4 [×2], C7×C4⋊D4, C2×D4×D7 [×2], C14.372+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, 2+ (1+4) [×2], C22×D7 [×7], C23⋊3D4, D4×D7 [×2], C23×D7, C2×D4×D7, D4⋊6D14, D4⋊8D14, C14.372+ (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, dbd-1=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 67 15 84)(2 68 16 71)(3 69 17 72)(4 70 18 73)(5 57 19 74)(6 58 20 75)(7 59 21 76)(8 60 22 77)(9 61 23 78)(10 62 24 79)(11 63 25 80)(12 64 26 81)(13 65 27 82)(14 66 28 83)(29 92 54 100)(30 93 55 101)(31 94 56 102)(32 95 43 103)(33 96 44 104)(34 97 45 105)(35 98 46 106)(36 85 47 107)(37 86 48 108)(38 87 49 109)(39 88 50 110)(40 89 51 111)(41 90 52 112)(42 91 53 99)
(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)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 99)
(1 56 22 38)(2 55 23 37)(3 54 24 36)(4 53 25 35)(5 52 26 34)(6 51 27 33)(7 50 28 32)(8 49 15 31)(9 48 16 30)(10 47 17 29)(11 46 18 42)(12 45 19 41)(13 44 20 40)(14 43 21 39)(57 105 81 90)(58 104 82 89)(59 103 83 88)(60 102 84 87)(61 101 71 86)(62 100 72 85)(63 99 73 98)(64 112 74 97)(65 111 75 96)(66 110 76 95)(67 109 77 94)(68 108 78 93)(69 107 79 92)(70 106 80 91)
(1 31 8 38)(2 32 9 39)(3 33 10 40)(4 34 11 41)(5 35 12 42)(6 36 13 29)(7 37 14 30)(15 56 22 49)(16 43 23 50)(17 44 24 51)(18 45 25 52)(19 46 26 53)(20 47 27 54)(21 48 28 55)(57 91 64 98)(58 92 65 85)(59 93 66 86)(60 94 67 87)(61 95 68 88)(62 96 69 89)(63 97 70 90)(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), (1,67,15,84)(2,68,16,71)(3,69,17,72)(4,70,18,73)(5,57,19,74)(6,58,20,75)(7,59,21,76)(8,60,22,77)(9,61,23,78)(10,62,24,79)(11,63,25,80)(12,64,26,81)(13,65,27,82)(14,66,28,83)(29,92,54,100)(30,93,55,101)(31,94,56,102)(32,95,43,103)(33,96,44,104)(34,97,45,105)(35,98,46,106)(36,85,47,107)(37,86,48,108)(38,87,49,109)(39,88,50,110)(40,89,51,111)(41,90,52,112)(42,91,53,99), (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)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,99), (1,56,22,38)(2,55,23,37)(3,54,24,36)(4,53,25,35)(5,52,26,34)(6,51,27,33)(7,50,28,32)(8,49,15,31)(9,48,16,30)(10,47,17,29)(11,46,18,42)(12,45,19,41)(13,44,20,40)(14,43,21,39)(57,105,81,90)(58,104,82,89)(59,103,83,88)(60,102,84,87)(61,101,71,86)(62,100,72,85)(63,99,73,98)(64,112,74,97)(65,111,75,96)(66,110,76,95)(67,109,77,94)(68,108,78,93)(69,107,79,92)(70,106,80,91), (1,31,8,38)(2,32,9,39)(3,33,10,40)(4,34,11,41)(5,35,12,42)(6,36,13,29)(7,37,14,30)(15,56,22,49)(16,43,23,50)(17,44,24,51)(18,45,25,52)(19,46,26,53)(20,47,27,54)(21,48,28,55)(57,91,64,98)(58,92,65,85)(59,93,66,86)(60,94,67,87)(61,95,68,88)(62,96,69,89)(63,97,70,90)(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), (1,67,15,84)(2,68,16,71)(3,69,17,72)(4,70,18,73)(5,57,19,74)(6,58,20,75)(7,59,21,76)(8,60,22,77)(9,61,23,78)(10,62,24,79)(11,63,25,80)(12,64,26,81)(13,65,27,82)(14,66,28,83)(29,92,54,100)(30,93,55,101)(31,94,56,102)(32,95,43,103)(33,96,44,104)(34,97,45,105)(35,98,46,106)(36,85,47,107)(37,86,48,108)(38,87,49,109)(39,88,50,110)(40,89,51,111)(41,90,52,112)(42,91,53,99), (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)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,99), (1,56,22,38)(2,55,23,37)(3,54,24,36)(4,53,25,35)(5,52,26,34)(6,51,27,33)(7,50,28,32)(8,49,15,31)(9,48,16,30)(10,47,17,29)(11,46,18,42)(12,45,19,41)(13,44,20,40)(14,43,21,39)(57,105,81,90)(58,104,82,89)(59,103,83,88)(60,102,84,87)(61,101,71,86)(62,100,72,85)(63,99,73,98)(64,112,74,97)(65,111,75,96)(66,110,76,95)(67,109,77,94)(68,108,78,93)(69,107,79,92)(70,106,80,91), (1,31,8,38)(2,32,9,39)(3,33,10,40)(4,34,11,41)(5,35,12,42)(6,36,13,29)(7,37,14,30)(15,56,22,49)(16,43,23,50)(17,44,24,51)(18,45,25,52)(19,46,26,53)(20,47,27,54)(21,48,28,55)(57,91,64,98)(58,92,65,85)(59,93,66,86)(60,94,67,87)(61,95,68,88)(62,96,69,89)(63,97,70,90)(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)], [(1,67,15,84),(2,68,16,71),(3,69,17,72),(4,70,18,73),(5,57,19,74),(6,58,20,75),(7,59,21,76),(8,60,22,77),(9,61,23,78),(10,62,24,79),(11,63,25,80),(12,64,26,81),(13,65,27,82),(14,66,28,83),(29,92,54,100),(30,93,55,101),(31,94,56,102),(32,95,43,103),(33,96,44,104),(34,97,45,105),(35,98,46,106),(36,85,47,107),(37,86,48,108),(38,87,49,109),(39,88,50,110),(40,89,51,111),(41,90,52,112),(42,91,53,99)], [(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),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(97,112),(98,99)], [(1,56,22,38),(2,55,23,37),(3,54,24,36),(4,53,25,35),(5,52,26,34),(6,51,27,33),(7,50,28,32),(8,49,15,31),(9,48,16,30),(10,47,17,29),(11,46,18,42),(12,45,19,41),(13,44,20,40),(14,43,21,39),(57,105,81,90),(58,104,82,89),(59,103,83,88),(60,102,84,87),(61,101,71,86),(62,100,72,85),(63,99,73,98),(64,112,74,97),(65,111,75,96),(66,110,76,95),(67,109,77,94),(68,108,78,93),(69,107,79,92),(70,106,80,91)], [(1,31,8,38),(2,32,9,39),(3,33,10,40),(4,34,11,41),(5,35,12,42),(6,36,13,29),(7,37,14,30),(15,56,22,49),(16,43,23,50),(17,44,24,51),(18,45,25,52),(19,46,26,53),(20,47,27,54),(21,48,28,55),(57,91,64,98),(58,92,65,85),(59,93,66,86),(60,94,67,87),(61,95,68,88),(62,96,69,89),(63,97,70,90),(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)])
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 | 9 | 10 |
| 0 | 0 | 0 | 0 | 9 | 23 |
| 22 | 3 | 0 | 0 | 0 | 0 |
| 13 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 5 | 14 |
| 0 | 0 | 0 | 28 | 15 | 0 |
| 0 | 0 | 0 | 25 | 1 | 0 |
| 0 | 0 | 4 | 18 | 0 | 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 | 4 | 28 | 0 |
| 0 | 0 | 25 | 11 | 0 | 28 |
| 13 | 24 | 0 | 0 | 0 | 0 |
| 5 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 15 | 0 | 0 |
| 0 | 0 | 10 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 19 |
| 0 | 0 | 0 | 0 | 5 | 7 |
| 16 | 5 | 0 | 0 | 0 | 0 |
| 24 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 15 | 0 | 0 |
| 0 | 0 | 14 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 15 |
| 0 | 0 | 0 | 0 | 28 | 25 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,19,0,0,0,0,10,22,0,0,0,0,0,0,9,9,0,0,0,0,10,23],[22,13,0,0,0,0,3,7,0,0,0,0,0,0,28,0,0,4,0,0,0,28,25,18,0,0,5,15,1,0,0,0,14,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,25,0,0,0,1,4,11,0,0,0,0,28,0,0,0,0,0,0,28],[13,5,0,0,0,0,24,16,0,0,0,0,0,0,20,10,0,0,0,0,15,9,0,0,0,0,0,0,22,5,0,0,0,0,19,7],[16,24,0,0,0,0,5,13,0,0,0,0,0,0,20,14,0,0,0,0,15,9,0,0,0,0,0,0,4,28,0,0,0,0,15,25] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 14 | 14 | 14 | 14 | 28 | 28 | 4 | 4 | 4 | 4 | 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 | D4 | D7 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4×D7 | D4⋊6D14 | D4⋊8D14 |
| kernel | C14.372+ (1+4) | C22⋊D28 | D14.D4 | D14.5D4 | C2×D14⋊C4 | C28⋊7D4 | C23⋊D14 | Dic7⋊D4 | C7×C4⋊D4 | C2×D4×D7 | C22×D7 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 3 | 6 | 3 | 3 | 9 | 2 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{37}2_+^{(1+4)} % in TeX
G:=Group("C14.37ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1058);
// by ID
G=gap.SmallGroup(448,1058);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,675,297,136,18822]);
// 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,d*b*d^-1=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
○
≀