direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C14×C22≀C2, C25⋊3C14, C23⋊6(C7×D4), C24⋊7(C2×C14), (C24×C14)⋊1C2, C22⋊3(D4×C14), (C2×C28)⋊10C23, (C22×D4)⋊3C14, (C22×C14)⋊17D4, (D4×C14)⋊60C22, C23⋊2(C22×C14), (C22×C14)⋊3C23, (C23×C14)⋊2C22, (C2×C14).341C24, (C22×C28)⋊45C22, C14.180(C22×D4), C22.15(C23×C14), C2.4(D4×C2×C14), (D4×C2×C14)⋊18C2, (C2×D4)⋊8(C2×C14), (C2×C14)⋊15(C2×D4), (C2×C22⋊C4)⋊8C14, (C2×C4)⋊1(C22×C14), (C22×C4)⋊5(C2×C14), C22⋊C4⋊10(C2×C14), (C14×C22⋊C4)⋊28C2, (C7×C22⋊C4)⋊64C22, SmallGroup(448,1304)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1138 in 662 conjugacy classes, 210 normal (12 characteristic)
C1, C2 [×7], C2 [×14], C4 [×6], C22, C22 [×18], C22 [×78], C7, C2×C4 [×6], C2×C4 [×6], D4 [×24], C23, C23 [×20], C23 [×74], C14 [×7], C14 [×14], C22⋊C4 [×12], C22×C4 [×3], C2×D4 [×12], C2×D4 [×12], C24, C24 [×7], C24 [×12], C28 [×6], C2×C14, C2×C14 [×18], C2×C14 [×78], C2×C22⋊C4 [×3], C22≀C2 [×8], C22×D4 [×3], C25, C2×C28 [×6], C2×C28 [×6], C7×D4 [×24], C22×C14, C22×C14 [×20], C22×C14 [×74], C2×C22≀C2, C7×C22⋊C4 [×12], C22×C28 [×3], D4×C14 [×12], D4×C14 [×12], C23×C14, C23×C14 [×7], C23×C14 [×12], C14×C22⋊C4 [×3], C7×C22≀C2 [×8], D4×C2×C14 [×3], C24×C14, C14×C22≀C2
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×12], C23 [×15], C14 [×15], C2×D4 [×18], C24, C2×C14 [×35], C22≀C2 [×4], C22×D4 [×3], C7×D4 [×12], C22×C14 [×15], C2×C22≀C2, D4×C14 [×18], C23×C14, C7×C22≀C2 [×4], D4×C2×C14 [×3], C14×C22≀C2
Generators and relations
G = < a,b,c,d,e,f | a14=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >
►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)
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 39)(16 40)(17 41)(18 42)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(51 94)(52 95)(53 96)(54 97)(55 98)(56 85)(57 83)(58 84)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 81)(70 82)
(1 82)(2 83)(3 84)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 97)(16 98)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 43)(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)(63 112)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)
(1 82)(2 83)(3 84)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(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)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 85)(42 86)(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)(63 112)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)
(1 70)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 39)(16 40)(17 41)(18 42)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(51 94)(52 95)(53 96)(54 97)(55 98)(56 85)(71 108)(72 109)(73 110)(74 111)(75 112)(76 99)(77 100)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 29)(14 30)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(49 108)(50 109)(51 110)(52 111)(53 112)(54 99)(55 100)(56 101)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)
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), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,39)(16,40)(17,41)(18,42)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,85)(57,83)(58,84)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,81)(70,82), (1,82)(2,83)(3,84)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,97)(16,98)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,43)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,99)(65,100)(66,101)(67,102)(68,103)(69,104)(70,105), (1,82)(2,83)(3,84)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(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)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,85)(42,86)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,99)(65,100)(66,101)(67,102)(68,103)(69,104)(70,105), (1,70)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,39)(16,40)(17,41)(18,42)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,85)(71,108)(72,109)(73,110)(74,111)(75,112)(76,99)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,29)(14,30)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,112)(54,99)(55,100)(56,101)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91)>;
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), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,39)(16,40)(17,41)(18,42)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,85)(57,83)(58,84)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,81)(70,82), (1,82)(2,83)(3,84)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,97)(16,98)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,43)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,99)(65,100)(66,101)(67,102)(68,103)(69,104)(70,105), (1,82)(2,83)(3,84)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(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)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,85)(42,86)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,99)(65,100)(66,101)(67,102)(68,103)(69,104)(70,105), (1,70)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,39)(16,40)(17,41)(18,42)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,85)(71,108)(72,109)(73,110)(74,111)(75,112)(76,99)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,29)(14,30)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,112)(54,99)(55,100)(56,101)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91) );
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)], [(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,111),(8,112),(9,99),(10,100),(11,101),(12,102),(13,103),(14,104),(15,39),(16,40),(17,41),(18,42),(19,29),(20,30),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,92),(50,93),(51,94),(52,95),(53,96),(54,97),(55,98),(56,85),(57,83),(58,84),(59,71),(60,72),(61,73),(62,74),(63,75),(64,76),(65,77),(66,78),(67,79),(68,80),(69,81),(70,82)], [(1,82),(2,83),(3,84),(4,71),(5,72),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,80),(14,81),(15,97),(16,98),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,56),(42,43),(57,106),(58,107),(59,108),(60,109),(61,110),(62,111),(63,112),(64,99),(65,100),(66,101),(67,102),(68,103),(69,104),(70,105)], [(1,82),(2,83),(3,84),(4,71),(5,72),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,80),(14,81),(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),(29,87),(30,88),(31,89),(32,90),(33,91),(34,92),(35,93),(36,94),(37,95),(38,96),(39,97),(40,98),(41,85),(42,86),(57,106),(58,107),(59,108),(60,109),(61,110),(62,111),(63,112),(64,99),(65,100),(66,101),(67,102),(68,103),(69,104),(70,105)], [(1,70),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,68),(14,69),(15,39),(16,40),(17,41),(18,42),(19,29),(20,30),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,92),(50,93),(51,94),(52,95),(53,96),(54,97),(55,98),(56,85),(71,108),(72,109),(73,110),(74,111),(75,112),(76,99),(77,100),(78,101),(79,102),(80,103),(81,104),(82,105),(83,106),(84,107)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,29),(14,30),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(43,102),(44,103),(45,104),(46,105),(47,106),(48,107),(49,108),(50,109),(51,110),(52,111),(53,112),(54,99),(55,100),(56,101),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,85),(79,86),(80,87),(81,88),(82,89),(83,90),(84,91)])
Matrix representation ►G ⊆ GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 | 0 |
| 0 | 0 | 20 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 0 | 23 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 28 | 0 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,20,0,0,0,0,0,20,0,0,0,0,0,23,0,0,0,0,0,23],[28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,28,0] >;
196 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2S | 2T | 2U | 4A | ··· | 4F | 7A | ··· | 7F | 14A | ··· | 14AP | 14AQ | ··· | 14DJ | 14DK | ··· | 14DV | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | D4 | C7×D4 |
| kernel | C14×C22≀C2 | C14×C22⋊C4 | C7×C22≀C2 | D4×C2×C14 | C24×C14 | C2×C22≀C2 | C2×C22⋊C4 | C22≀C2 | C22×D4 | C25 | C22×C14 | C23 |
| # reps | 1 | 3 | 8 | 3 | 1 | 6 | 18 | 48 | 18 | 6 | 12 | 72 |
In GAP, Magma, Sage, TeX
C_{14}\times C_2^2\wr C_2 % in TeX
G:=Group("C14xC2^2wrC2"); // GroupNames label
G:=SmallGroup(448,1304);
// by ID
G=gap.SmallGroup(448,1304);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^14=b^2=c^2=d^2=e^2=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀