?
G = C14.682+ (1+4) order 448 = 26·7
68th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.682+ (1+4), C4⋊C4⋊16D14, C28⋊D4⋊22C2, C28⋊1D4⋊32C2, C28⋊7D4⋊14C2, D14⋊C4⋊5C22, C22⋊C4⋊19D14, (C22×C4)⋊25D14, C22⋊D28⋊22C2, C23⋊D14⋊18C2, D14⋊D4⋊35C2, (C2×D4).105D14, (C2×D28)⋊27C22, C4⋊Dic7⋊16C22, D14.D4⋊38C2, (C2×C14).210C24, (C2×C28).184C23, Dic7⋊C4⋊25C22, (C22×C28)⋊13C22, (C4×Dic7)⋊34C22, C22.D4⋊15D7, C2.46(D4⋊8D14), C2.70(D4⋊6D14), C7⋊3(C22.54C24), (D4×C14).148C22, (C22×D7).91C23, (C23×D7).61C22, C22.231(C23×D7), C23.131(C22×D7), C23.D7.48C22, (C22×C14).224C23, (C2×Dic7).109C23, (C2×C4×D7)⋊24C22, C4⋊C4⋊D7⋊32C2, (C7×C4⋊C4)⋊30C22, (C2×C7⋊D4)⋊21C22, (C2×C4).71(C22×D7), (C7×C22⋊C4)⋊26C22, (C7×C22.D4)⋊18C2, SmallGroup(448,1119)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1548 in 252 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×9], C22, C22 [×22], C7, C2×C4, C2×C4 [×4], C2×C4 [×7], D4 [×12], C23 [×2], C23 [×7], D7 [×4], C14, C14 [×2], C14 [×2], C42, C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×11], C24, Dic7 [×4], C28 [×5], D14 [×16], C2×C14, C2×C14 [×6], C22≀C2 [×3], C4⋊D4 [×6], C22.D4, C22.D4 [×2], C42⋊2C2 [×2], C4⋊1D4, C4×D7 [×2], D28 [×5], C2×Dic7 [×4], C7⋊D4 [×6], C2×C28, C2×C28 [×4], C2×C28, C7×D4, C22×D7 [×4], C22×D7 [×3], C22×C14 [×2], C22.54C24, C4×Dic7, Dic7⋊C4 [×2], C4⋊Dic7 [×2], D14⋊C4 [×8], C23.D7, C7×C22⋊C4, C7×C22⋊C4 [×2], C7×C4⋊C4 [×2], C2×C4×D7 [×2], C2×D28, C2×D28 [×4], C2×C7⋊D4 [×6], C22×C28, D4×C14, C23×D7, C22⋊D28 [×2], D14.D4 [×2], D14⋊D4 [×2], C28⋊1D4 [×2], C4⋊C4⋊D7 [×2], C28⋊7D4 [×2], C23⋊D14, C28⋊D4, C7×C22.D4, C14.682+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ (1+4) [×3], C22×D7 [×7], C22.54C24, C23×D7, D4⋊6D14, D4⋊8D14 [×2], C14.682+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a7b-1, bd=db, ebe=a7b, dcd-1=ece=a7c, ede=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 58 28 82)(2 59 15 83)(3 60 16 84)(4 61 17 71)(5 62 18 72)(6 63 19 73)(7 64 20 74)(8 65 21 75)(9 66 22 76)(10 67 23 77)(11 68 24 78)(12 69 25 79)(13 70 26 80)(14 57 27 81)(29 95 52 101)(30 96 53 102)(31 97 54 103)(32 98 55 104)(33 85 56 105)(34 86 43 106)(35 87 44 107)(36 88 45 108)(37 89 46 109)(38 90 47 110)(39 91 48 111)(40 92 49 112)(41 93 50 99)(42 94 51 100)
(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 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 81)(65 82)(66 83)(67 84)(68 71)(69 72)(70 73)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 99)(94 100)(95 101)(96 102)(97 103)(98 104)
(1 30 28 53)(2 29 15 52)(3 42 16 51)(4 41 17 50)(5 40 18 49)(6 39 19 48)(7 38 20 47)(8 37 21 46)(9 36 22 45)(10 35 23 44)(11 34 24 43)(12 33 25 56)(13 32 26 55)(14 31 27 54)(57 97 81 103)(58 96 82 102)(59 95 83 101)(60 94 84 100)(61 93 71 99)(62 92 72 112)(63 91 73 111)(64 90 74 110)(65 89 75 109)(66 88 76 108)(67 87 77 107)(68 86 78 106)(69 85 79 105)(70 98 80 104)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 29)(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 88)(58 89)(59 90)(60 91)(61 92)(62 93)(63 94)(64 95)(65 96)(66 97)(67 98)(68 85)(69 86)(70 87)(71 112)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(82 109)(83 110)(84 111)
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), (1,58,28,82)(2,59,15,83)(3,60,16,84)(4,61,17,71)(5,62,18,72)(6,63,19,73)(7,64,20,74)(8,65,21,75)(9,66,22,76)(10,67,23,77)(11,68,24,78)(12,69,25,79)(13,70,26,80)(14,57,27,81)(29,95,52,101)(30,96,53,102)(31,97,54,103)(32,98,55,104)(33,85,56,105)(34,86,43,106)(35,87,44,107)(36,88,45,108)(37,89,46,109)(38,90,47,110)(39,91,48,111)(40,92,49,112)(41,93,50,99)(42,94,51,100), (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)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,81)(65,82)(66,83)(67,84)(68,71)(69,72)(70,73)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,99)(94,100)(95,101)(96,102)(97,103)(98,104), (1,30,28,53)(2,29,15,52)(3,42,16,51)(4,41,17,50)(5,40,18,49)(6,39,19,48)(7,38,20,47)(8,37,21,46)(9,36,22,45)(10,35,23,44)(11,34,24,43)(12,33,25,56)(13,32,26,55)(14,31,27,54)(57,97,81,103)(58,96,82,102)(59,95,83,101)(60,94,84,100)(61,93,71,99)(62,92,72,112)(63,91,73,111)(64,90,74,110)(65,89,75,109)(66,88,76,108)(67,87,77,107)(68,86,78,106)(69,85,79,105)(70,98,80,104), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,29)(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)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,85)(69,86)(70,87)(71,112)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108)(82,109)(83,110)(84,111)>;
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), (1,58,28,82)(2,59,15,83)(3,60,16,84)(4,61,17,71)(5,62,18,72)(6,63,19,73)(7,64,20,74)(8,65,21,75)(9,66,22,76)(10,67,23,77)(11,68,24,78)(12,69,25,79)(13,70,26,80)(14,57,27,81)(29,95,52,101)(30,96,53,102)(31,97,54,103)(32,98,55,104)(33,85,56,105)(34,86,43,106)(35,87,44,107)(36,88,45,108)(37,89,46,109)(38,90,47,110)(39,91,48,111)(40,92,49,112)(41,93,50,99)(42,94,51,100), (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)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,81)(65,82)(66,83)(67,84)(68,71)(69,72)(70,73)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,99)(94,100)(95,101)(96,102)(97,103)(98,104), (1,30,28,53)(2,29,15,52)(3,42,16,51)(4,41,17,50)(5,40,18,49)(6,39,19,48)(7,38,20,47)(8,37,21,46)(9,36,22,45)(10,35,23,44)(11,34,24,43)(12,33,25,56)(13,32,26,55)(14,31,27,54)(57,97,81,103)(58,96,82,102)(59,95,83,101)(60,94,84,100)(61,93,71,99)(62,92,72,112)(63,91,73,111)(64,90,74,110)(65,89,75,109)(66,88,76,108)(67,87,77,107)(68,86,78,106)(69,85,79,105)(70,98,80,104), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,29)(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)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,85)(69,86)(70,87)(71,112)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108)(82,109)(83,110)(84,111) );
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)], [(1,58,28,82),(2,59,15,83),(3,60,16,84),(4,61,17,71),(5,62,18,72),(6,63,19,73),(7,64,20,74),(8,65,21,75),(9,66,22,76),(10,67,23,77),(11,68,24,78),(12,69,25,79),(13,70,26,80),(14,57,27,81),(29,95,52,101),(30,96,53,102),(31,97,54,103),(32,98,55,104),(33,85,56,105),(34,86,43,106),(35,87,44,107),(36,88,45,108),(37,89,46,109),(38,90,47,110),(39,91,48,111),(40,92,49,112),(41,93,50,99),(42,94,51,100)], [(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),(57,74),(58,75),(59,76),(60,77),(61,78),(62,79),(63,80),(64,81),(65,82),(66,83),(67,84),(68,71),(69,72),(70,73),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110),(91,111),(92,112),(93,99),(94,100),(95,101),(96,102),(97,103),(98,104)], [(1,30,28,53),(2,29,15,52),(3,42,16,51),(4,41,17,50),(5,40,18,49),(6,39,19,48),(7,38,20,47),(8,37,21,46),(9,36,22,45),(10,35,23,44),(11,34,24,43),(12,33,25,56),(13,32,26,55),(14,31,27,54),(57,97,81,103),(58,96,82,102),(59,95,83,101),(60,94,84,100),(61,93,71,99),(62,92,72,112),(63,91,73,111),(64,90,74,110),(65,89,75,109),(66,88,76,108),(67,87,77,107),(68,86,78,106),(69,85,79,105),(70,98,80,104)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,29),(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),(57,88),(58,89),(59,90),(60,91),(61,92),(62,93),(63,94),(64,95),(65,96),(66,97),(67,98),(68,85),(69,86),(70,87),(71,112),(72,99),(73,100),(74,101),(75,102),(76,103),(77,104),(78,105),(79,106),(80,107),(81,108),(82,109),(83,110),(84,111)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 22 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 21 |
| 0 | 0 | 0 | 0 | 21 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 8 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 19 |
| 0 | 0 | 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 19 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(29))| [28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,4,22,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,4,22],[0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,21,13,0,0,0,0,0,0,24,8,0,0,0,0,8,16,0,0,0,0,0,0,5,21,0,0],[1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,10,17,0,0,0,0,0,0,1,19,0,0,0,0,10,17,0,0,0,0,0,0,1,19,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4E | 4F | 4G | 4H | 4I | 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 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 28 | 28 | 28 | 28 | 4 | ··· | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊6D14 | D4⋊8D14 |
| kernel | C14.682+ (1+4) | C22⋊D28 | D14.D4 | D14⋊D4 | C28⋊1D4 | C4⋊C4⋊D7 | C28⋊7D4 | C23⋊D14 | C28⋊D4 | C7×C22.D4 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 3 | 9 | 6 | 3 | 3 | 3 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_{14}._{68}2_+^{(1+4)} % in TeX
G:=Group("C14.68ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1119);
// by ID
G=gap.SmallGroup(448,1119);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,184,1571,570,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=e^2=1,d^2=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,b*d=d*b,e*b*e=a^7*b,d*c*d^-1=e*c*e=a^7*c,e*d*e=b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀