metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: 2+ 1+4.2D7, (C2×C28).21D4, (C2×D4).84D14, C23⋊Dic7⋊9C2, C14.82C22≀C2, (C22×C14).26D4, C7⋊3(C23.7D4), C23.D7⋊6C22, C23.5(C22×D7), C23.14(C7⋊D4), (C22×C14).5C23, (D4×C14).178C22, C2.16(C24⋊D7), (C22×Dic7)⋊3C22, (C7×2+ 1+4).2C2, C23.18D14⋊17C2, (C2×C14).44(C2×D4), (C2×C4).12(C7⋊D4), C22.16(C2×C7⋊D4), SmallGroup(448,777)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for 2+ 1+4.2D7
G = < a,b,c,d,e,f | a4=b2=d2=e7=1, c2=f2=a2, bab=a-1, ac=ca, ad=da, ae=ea, faf-1=a-1cd, fcf-1=bc=cb, fdf-1=bd=db, be=eb, bf=fb, dcd=a2c, ce=ec, de=ed, fef-1=e-1 >
Subgroups: 628 in 160 conjugacy classes, 43 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C4○D4, Dic7, C28, C2×C14, C2×C14, C23⋊C4, C22.D4, 2+ 1+4, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C22×C14, C23.7D4, Dic7⋊C4, C23.D7, C23.D7, C22×Dic7, D4×C14, D4×C14, C7×C4○D4, C23⋊Dic7, C23.18D14, C7×2+ 1+4, 2+ 1+4.2D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, C23.7D4, C2×C7⋊D4, C24⋊D7, 2+ 1+4.2D7
►On 112 points
Generators in S112
(1 34 13 41)(2 35 14 42)(3 29 8 36)(4 30 9 37)(5 31 10 38)(6 32 11 39)(7 33 12 40)(15 43 22 50)(16 44 23 51)(17 45 24 52)(18 46 25 53)(19 47 26 54)(20 48 27 55)(21 49 28 56)(57 92 64 85)(58 93 65 86)(59 94 66 87)(60 95 67 88)(61 96 68 89)(62 97 69 90)(63 98 70 91)(71 106 78 99)(72 107 79 100)(73 108 80 101)(74 109 81 102)(75 110 82 103)(76 111 83 104)(77 112 84 105)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)
(1 62 13 69)(2 63 14 70)(3 57 8 64)(4 58 9 65)(5 59 10 66)(6 60 11 67)(7 61 12 68)(15 71 22 78)(16 72 23 79)(17 73 24 80)(18 74 25 81)(19 75 26 82)(20 76 27 83)(21 77 28 84)(29 92 36 85)(30 93 37 86)(31 94 38 87)(32 95 39 88)(33 96 40 89)(34 97 41 90)(35 98 42 91)(43 106 50 99)(44 107 51 100)(45 108 52 101)(46 109 53 102)(47 110 54 103)(48 111 55 104)(49 112 56 105)
(1 83)(2 84)(3 78)(4 79)(5 80)(6 81)(7 82)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(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)(29 99)(30 100)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 111)(42 112)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 97)(56 98)
(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 89 13 96)(2 88 14 95)(3 87 8 94)(4 86 9 93)(5 85 10 92)(6 91 11 98)(7 90 12 97)(15 108 22 101)(16 107 23 100)(17 106 24 99)(18 112 25 105)(19 111 26 104)(20 110 27 103)(21 109 28 102)(29 80 36 73)(30 79 37 72)(31 78 38 71)(32 84 39 77)(33 83 40 76)(34 82 41 75)(35 81 42 74)(43 59 50 66)(44 58 51 65)(45 57 52 64)(46 63 53 70)(47 62 54 69)(48 61 55 68)(49 60 56 67)
G:=sub<Sym(112)| (1,34,13,41)(2,35,14,42)(3,29,8,36)(4,30,9,37)(5,31,10,38)(6,32,11,39)(7,33,12,40)(15,43,22,50)(16,44,23,51)(17,45,24,52)(18,46,25,53)(19,47,26,54)(20,48,27,55)(21,49,28,56)(57,92,64,85)(58,93,65,86)(59,94,66,87)(60,95,67,88)(61,96,68,89)(62,97,69,90)(63,98,70,91)(71,106,78,99)(72,107,79,100)(73,108,80,101)(74,109,81,102)(75,110,82,103)(76,111,83,104)(77,112,84,105), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (1,62,13,69)(2,63,14,70)(3,57,8,64)(4,58,9,65)(5,59,10,66)(6,60,11,67)(7,61,12,68)(15,71,22,78)(16,72,23,79)(17,73,24,80)(18,74,25,81)(19,75,26,82)(20,76,27,83)(21,77,28,84)(29,92,36,85)(30,93,37,86)(31,94,38,87)(32,95,39,88)(33,96,40,89)(34,97,41,90)(35,98,42,91)(43,106,50,99)(44,107,51,100)(45,108,52,101)(46,109,53,102)(47,110,54,103)(48,111,55,104)(49,112,56,105), (1,83)(2,84)(3,78)(4,79)(5,80)(6,81)(7,82)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(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)(29,99)(30,100)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,111)(42,112)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98), (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), (1,89,13,96)(2,88,14,95)(3,87,8,94)(4,86,9,93)(5,85,10,92)(6,91,11,98)(7,90,12,97)(15,108,22,101)(16,107,23,100)(17,106,24,99)(18,112,25,105)(19,111,26,104)(20,110,27,103)(21,109,28,102)(29,80,36,73)(30,79,37,72)(31,78,38,71)(32,84,39,77)(33,83,40,76)(34,82,41,75)(35,81,42,74)(43,59,50,66)(44,58,51,65)(45,57,52,64)(46,63,53,70)(47,62,54,69)(48,61,55,68)(49,60,56,67)>;
G:=Group( (1,34,13,41)(2,35,14,42)(3,29,8,36)(4,30,9,37)(5,31,10,38)(6,32,11,39)(7,33,12,40)(15,43,22,50)(16,44,23,51)(17,45,24,52)(18,46,25,53)(19,47,26,54)(20,48,27,55)(21,49,28,56)(57,92,64,85)(58,93,65,86)(59,94,66,87)(60,95,67,88)(61,96,68,89)(62,97,69,90)(63,98,70,91)(71,106,78,99)(72,107,79,100)(73,108,80,101)(74,109,81,102)(75,110,82,103)(76,111,83,104)(77,112,84,105), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (1,62,13,69)(2,63,14,70)(3,57,8,64)(4,58,9,65)(5,59,10,66)(6,60,11,67)(7,61,12,68)(15,71,22,78)(16,72,23,79)(17,73,24,80)(18,74,25,81)(19,75,26,82)(20,76,27,83)(21,77,28,84)(29,92,36,85)(30,93,37,86)(31,94,38,87)(32,95,39,88)(33,96,40,89)(34,97,41,90)(35,98,42,91)(43,106,50,99)(44,107,51,100)(45,108,52,101)(46,109,53,102)(47,110,54,103)(48,111,55,104)(49,112,56,105), (1,83)(2,84)(3,78)(4,79)(5,80)(6,81)(7,82)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(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)(29,99)(30,100)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,111)(42,112)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98), (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), (1,89,13,96)(2,88,14,95)(3,87,8,94)(4,86,9,93)(5,85,10,92)(6,91,11,98)(7,90,12,97)(15,108,22,101)(16,107,23,100)(17,106,24,99)(18,112,25,105)(19,111,26,104)(20,110,27,103)(21,109,28,102)(29,80,36,73)(30,79,37,72)(31,78,38,71)(32,84,39,77)(33,83,40,76)(34,82,41,75)(35,81,42,74)(43,59,50,66)(44,58,51,65)(45,57,52,64)(46,63,53,70)(47,62,54,69)(48,61,55,68)(49,60,56,67) );
G=PermutationGroup([[(1,34,13,41),(2,35,14,42),(3,29,8,36),(4,30,9,37),(5,31,10,38),(6,32,11,39),(7,33,12,40),(15,43,22,50),(16,44,23,51),(17,45,24,52),(18,46,25,53),(19,47,26,54),(20,48,27,55),(21,49,28,56),(57,92,64,85),(58,93,65,86),(59,94,66,87),(60,95,67,88),(61,96,68,89),(62,97,69,90),(63,98,70,91),(71,106,78,99),(72,107,79,100),(73,108,80,101),(74,109,81,102),(75,110,82,103),(76,111,83,104),(77,112,84,105)], [(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,106),(86,107),(87,108),(88,109),(89,110),(90,111),(91,112),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105)], [(1,62,13,69),(2,63,14,70),(3,57,8,64),(4,58,9,65),(5,59,10,66),(6,60,11,67),(7,61,12,68),(15,71,22,78),(16,72,23,79),(17,73,24,80),(18,74,25,81),(19,75,26,82),(20,76,27,83),(21,77,28,84),(29,92,36,85),(30,93,37,86),(31,94,38,87),(32,95,39,88),(33,96,40,89),(34,97,41,90),(35,98,42,91),(43,106,50,99),(44,107,51,100),(45,108,52,101),(46,109,53,102),(47,110,54,103),(48,111,55,104),(49,112,56,105)], [(1,83),(2,84),(3,78),(4,79),(5,80),(6,81),(7,82),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(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),(29,99),(30,100),(31,101),(32,102),(33,103),(34,104),(35,105),(36,106),(37,107),(38,108),(39,109),(40,110),(41,111),(42,112),(43,85),(44,86),(45,87),(46,88),(47,89),(48,90),(49,91),(50,92),(51,93),(52,94),(53,95),(54,96),(55,97),(56,98)], [(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)], [(1,89,13,96),(2,88,14,95),(3,87,8,94),(4,86,9,93),(5,85,10,92),(6,91,11,98),(7,90,12,97),(15,108,22,101),(16,107,23,100),(17,106,24,99),(18,112,25,105),(19,111,26,104),(20,110,27,103),(21,109,28,102),(29,80,36,73),(30,79,37,72),(31,78,38,71),(32,84,39,77),(33,83,40,76),(34,82,41,75),(35,81,42,74),(43,59,50,66),(44,58,51,65),(45,57,52,64),(46,63,53,70),(47,62,54,69),(48,61,55,68),(49,60,56,67)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 14A | 14B | 14C | 14D | ··· | 14AD | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 28 | 28 | 56 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | C7⋊D4 | C7⋊D4 | C23.7D4 | 2+ 1+4.2D7 |
| kernel | 2+ 1+4.2D7 | C23⋊Dic7 | C23.18D14 | C7×2+ 1+4 | C2×C28 | C22×C14 | 2+ 1+4 | C2×D4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 3 | 3 | 1 | 3 | 3 | 3 | 9 | 18 | 18 | 2 | 3 |
Matrix representation of 2+ 1+4.2D7 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 27 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 24 |
| 0 | 0 | 0 | 0 | 17 | 17 |
| 0 | 0 | 17 | 5 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 17 | 17 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 24 | 0 | 0 | 0 | 0 | 0 |
| 0 | 23 | 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 | 0 | 1 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 24 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,0,28,0,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,27,1,0,0,0,0,0,0,1,0,0,0,0,0,2,28],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,17,12,0,0,0,0,5,12,0,0,12,17,0,0,0,0,24,17,0,0],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,17,12,0,0,0,0,0,12,0,0,12,17,0,0,0,0,0,17,0,0],[24,0,0,0,0,0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,6,0,0,0,0,0,0,0,12,0,0,0,0,0,24,17,0,0,0,0,0,0,17,0,0,0,0,0,0,17] >;
2+ 1+4.2D7 in GAP, Magma, Sage, TeX
2_+^{1+4}._2D_7 % in TeX
G:=Group("ES+(2,2).2D7"); // GroupNames label
G:=SmallGroup(448,777);
// by ID
G=gap.SmallGroup(448,777);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,254,570,438,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=b^2=d^2=e^7=1,c^2=f^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a^-1*c*d,f*c*f^-1=b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=a^2*c,c*e=e*c,d*e=e*d,f*e*f^-1=e^-1>;
// generators/relations