direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C14×C22≀C2, C25⋊3C14, C23⋊6(C7×D4), (C24×C14)⋊1C2, C24⋊7(C2×C14), C22⋊3(D4×C14), (C2×C28)⋊10C23, (C22×D4)⋊3C14, (C22×C14)⋊17D4, (D4×C14)⋊60C22, (C22×C14)⋊3C23, C23⋊2(C22×C14), (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), (C14×C22⋊C4)⋊28C2, C22⋊C4⋊10(C2×C14), (C7×C22⋊C4)⋊64C22, SmallGroup(448,1304)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×C22≀C2
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 >
Subgroups: 1138 in 662 conjugacy classes, 210 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C22⋊C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C22≀C2, C22×D4, C25, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C2×C22≀C2, C7×C22⋊C4, C22×C28, D4×C14, D4×C14, C23×C14, C23×C14, C23×C14, C14×C22⋊C4, C7×C22≀C2, D4×C2×C14, C24×C14, C14×C22≀C2
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C22≀C2, C22×D4, C7×D4, C22×C14, C2×C22≀C2, D4×C14, C23×C14, C7×C22≀C2, D4×C2×C14, C14×C22≀C2
►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 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 29)(28 30)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)
(1 27)(2 28)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 105)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)
(1 27)(2 28)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 84)(58 71)(59 72)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 95)(16 96)(17 97)(18 98)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 91)(26 92)(27 93)(28 94)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 57)(12 58)(13 59)(14 60)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 71)(25 72)(26 73)(27 74)(28 75)(29 53)(30 54)(31 55)(32 56)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(85 101)(86 102)(87 103)(88 104)(89 105)(90 106)(91 107)(92 108)(93 109)(94 110)(95 111)(96 112)(97 99)(98 100)
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), (1,93)(2,94)(3,95)(4,96)(5,97)(6,98)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,29)(28,30)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,27)(2,28)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104), (1,27)(2,28)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,84)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,95)(16,96)(17,97)(18,98)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,91)(26,92)(27,93)(28,94)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,57)(12,58)(13,59)(14,60)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,71)(25,72)(26,73)(27,74)(28,75)(29,53)(30,54)(31,55)(32,56)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106)(91,107)(92,108)(93,109)(94,110)(95,111)(96,112)(97,99)(98,100)>;
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), (1,93)(2,94)(3,95)(4,96)(5,97)(6,98)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,29)(28,30)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,27)(2,28)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104), (1,27)(2,28)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,84)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,95)(16,96)(17,97)(18,98)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,91)(26,92)(27,93)(28,94)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,57)(12,58)(13,59)(14,60)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,71)(25,72)(26,73)(27,74)(28,75)(29,53)(30,54)(31,55)(32,56)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106)(91,107)(92,108)(93,109)(94,110)(95,111)(96,112)(97,99)(98,100) );
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)], [(1,93),(2,94),(3,95),(4,96),(5,97),(6,98),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,29),(28,30),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64),(71,106),(72,107),(73,108),(74,109),(75,110),(76,111),(77,112),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105)], [(1,27),(2,28),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(29,93),(30,94),(31,95),(32,96),(33,97),(34,98),(35,85),(36,86),(37,87),(38,88),(39,89),(40,90),(41,91),(42,92),(43,78),(44,79),(45,80),(46,81),(47,82),(48,83),(49,84),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,105),(58,106),(59,107),(60,108),(61,109),(62,110),(63,111),(64,112),(65,99),(66,100),(67,101),(68,102),(69,103),(70,104)], [(1,27),(2,28),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(29,93),(30,94),(31,95),(32,96),(33,97),(34,98),(35,85),(36,86),(37,87),(38,88),(39,89),(40,90),(41,91),(42,92),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(57,84),(58,71),(59,72),(60,73),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,81),(69,82),(70,83)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,95),(16,96),(17,97),(18,98),(19,85),(20,86),(21,87),(22,88),(23,89),(24,90),(25,91),(26,92),(27,93),(28,94),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64),(71,106),(72,107),(73,108),(74,109),(75,110),(76,111),(77,112),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105)], [(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,57),(12,58),(13,59),(14,60),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,71),(25,72),(26,73),(27,74),(28,75),(29,53),(30,54),(31,55),(32,56),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,49),(40,50),(41,51),(42,52),(85,101),(86,102),(87,103),(88,104),(89,105),(90,106),(91,107),(92,108),(93,109),(94,110),(95,111),(96,112),(97,99),(98,100)]])
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 |
Matrix representation of C14×C22≀C2 ►in 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] >;
C14×C22≀C2 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