metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.56D14, C22≀C2⋊9D7, D14⋊C4⋊8C22, (C2×Dic7)⋊20D4, (D4×Dic7)⋊10C2, C24⋊D7⋊4C2, C22.39(D4×D7), Dic7⋊D4⋊1C2, Dic7⋊4D4⋊1C2, (C2×D4).148D14, (C2×C28).25C23, Dic7⋊C4⋊6C22, C4⋊Dic7⋊23C22, C22⋊C4.44D14, Dic7.44(C2×D4), (C23×Dic7)⋊5C2, C14.53(C22×D4), C22⋊3(D4⋊2D7), (C2×C14).130C24, C7⋊3(C22.19C24), (C4×Dic7)⋊12C22, C22.D28⋊8C2, (C22×C14).7C23, C23.D7⋊11C22, C22⋊Dic14⋊11C2, (C2×Dic14)⋊18C22, (D4×C14).109C22, C23.18D14⋊2C2, C23.11D14⋊1C2, (C23×C14).66C22, (C22×D7).52C23, C22.151(C23×D7), C23.175(C22×D7), (C2×Dic7).219C23, (C22×Dic7)⋊10C22, C2.26(C2×D4×D7), (C2×C4×D7)⋊4C22, (C7×C22≀C2)⋊2C2, (C2×D4⋊2D7)⋊5C2, (C2×C14)⋊9(C4○D4), C14.75(C2×C4○D4), (C2×C14).52(C2×D4), (C2×C7⋊D4)⋊6C22, C2.26(C2×D4⋊2D7), (C2×C4).25(C22×D7), (C7×C22⋊C4).1C22, SmallGroup(448,1039)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1356 in 330 conjugacy classes, 111 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×12], C22, C22 [×6], C22 [×20], C7, C2×C4, C2×C4 [×2], C2×C4 [×25], D4 [×14], Q8 [×2], C23 [×2], C23 [×2], C23 [×7], D7, C14, C14 [×2], C14 [×7], C42 [×2], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×6], C22×C4 [×12], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24, Dic7 [×4], Dic7 [×5], C28 [×3], D14 [×3], C2×C14, C2×C14 [×6], C2×C14 [×17], C42⋊C2, C4×D4 [×4], C22≀C2, C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, Dic14 [×2], C4×D7 [×2], C2×Dic7 [×3], C2×Dic7 [×8], C2×Dic7 [×12], C7⋊D4 [×8], C2×C28, C2×C28 [×2], C7×D4 [×6], C22×D7, C22×C14 [×2], C22×C14 [×2], C22×C14 [×6], C22.19C24, C4×Dic7 [×2], Dic7⋊C4 [×4], C4⋊Dic7 [×2], D14⋊C4 [×2], C23.D7, C23.D7 [×4], C7×C22⋊C4, C7×C22⋊C4 [×2], C2×Dic14, C2×C4×D7, D4⋊2D7 [×4], C22×Dic7 [×3], C22×Dic7 [×4], C22×Dic7 [×4], C2×C7⋊D4 [×2], C2×C7⋊D4 [×2], D4×C14, D4×C14 [×2], C23×C14, C23.11D14, C22⋊Dic14 [×2], Dic7⋊4D4 [×2], C22.D28, D4×Dic7 [×2], C23.18D14, Dic7⋊D4 [×2], C24⋊D7, C7×C22≀C2, C2×D4⋊2D7, C23×Dic7, C24.56D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×4], C24, D14 [×7], C22×D4, C2×C4○D4 [×2], C22×D7 [×7], C22.19C24, D4×D7 [×2], D4⋊2D7 [×4], C23×D7, C2×D4×D7, C2×D4⋊2D7 [×2], C24.56D14
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=1, f2=d, ab=ba, ac=ca, eae-1=faf-1=ad=da, ebe-1=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
►On 112 points
(2 74)(4 76)(6 78)(8 80)(10 82)(12 84)(14 72)(15 64)(17 66)(19 68)(21 70)(23 58)(25 60)(27 62)(29 109)(31 111)(33 99)(35 101)(37 103)(39 105)(41 107)(43 93)(45 95)(47 97)(49 85)(51 87)(53 89)(55 91)
(2 87)(4 89)(6 91)(8 93)(10 95)(12 97)(14 85)(15 111)(17 99)(19 101)(21 103)(23 105)(25 107)(27 109)(29 62)(31 64)(33 66)(35 68)(37 70)(39 58)(41 60)(43 80)(45 82)(47 84)(49 72)(51 74)(53 76)(55 78)
(1 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 85)(15 111)(16 112)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 109)(28 110)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 57)(39 58)(40 59)(41 60)(42 61)(43 80)(44 81)(45 82)(46 83)(47 84)(48 71)(49 72)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 71)(14 72)(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)(29 109)(30 110)(31 111)(32 112)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(41 107)(42 108)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)
(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 33 73 99)(2 32 74 112)(3 31 75 111)(4 30 76 110)(5 29 77 109)(6 42 78 108)(7 41 79 107)(8 40 80 106)(9 39 81 105)(10 38 82 104)(11 37 83 103)(12 36 84 102)(13 35 71 101)(14 34 72 100)(15 88 64 52)(16 87 65 51)(17 86 66 50)(18 85 67 49)(19 98 68 48)(20 97 69 47)(21 96 70 46)(22 95 57 45)(23 94 58 44)(24 93 59 43)(25 92 60 56)(26 91 61 55)(27 90 62 54)(28 89 63 53)
G:=sub<Sym(112)| (2,74)(4,76)(6,78)(8,80)(10,82)(12,84)(14,72)(15,64)(17,66)(19,68)(21,70)(23,58)(25,60)(27,62)(29,109)(31,111)(33,99)(35,101)(37,103)(39,105)(41,107)(43,93)(45,95)(47,97)(49,85)(51,87)(53,89)(55,91), (2,87)(4,89)(6,91)(8,93)(10,95)(12,97)(14,85)(15,111)(17,99)(19,101)(21,103)(23,105)(25,107)(27,109)(29,62)(31,64)(33,66)(35,68)(37,70)(39,58)(41,60)(43,80)(45,82)(47,84)(49,72)(51,74)(53,76)(55,78), (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,85)(15,111)(16,112)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,57)(39,58)(40,59)(41,60)(42,61)(43,80)(44,81)(45,82)(46,83)(47,84)(48,71)(49,72)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,71)(14,72)(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)(29,109)(30,110)(31,111)(32,112)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,105)(40,106)(41,107)(42,108)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92), (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), (1,33,73,99)(2,32,74,112)(3,31,75,111)(4,30,76,110)(5,29,77,109)(6,42,78,108)(7,41,79,107)(8,40,80,106)(9,39,81,105)(10,38,82,104)(11,37,83,103)(12,36,84,102)(13,35,71,101)(14,34,72,100)(15,88,64,52)(16,87,65,51)(17,86,66,50)(18,85,67,49)(19,98,68,48)(20,97,69,47)(21,96,70,46)(22,95,57,45)(23,94,58,44)(24,93,59,43)(25,92,60,56)(26,91,61,55)(27,90,62,54)(28,89,63,53)>;
G:=Group( (2,74)(4,76)(6,78)(8,80)(10,82)(12,84)(14,72)(15,64)(17,66)(19,68)(21,70)(23,58)(25,60)(27,62)(29,109)(31,111)(33,99)(35,101)(37,103)(39,105)(41,107)(43,93)(45,95)(47,97)(49,85)(51,87)(53,89)(55,91), (2,87)(4,89)(6,91)(8,93)(10,95)(12,97)(14,85)(15,111)(17,99)(19,101)(21,103)(23,105)(25,107)(27,109)(29,62)(31,64)(33,66)(35,68)(37,70)(39,58)(41,60)(43,80)(45,82)(47,84)(49,72)(51,74)(53,76)(55,78), (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,85)(15,111)(16,112)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,57)(39,58)(40,59)(41,60)(42,61)(43,80)(44,81)(45,82)(46,83)(47,84)(48,71)(49,72)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,71)(14,72)(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)(29,109)(30,110)(31,111)(32,112)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,105)(40,106)(41,107)(42,108)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92), (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), (1,33,73,99)(2,32,74,112)(3,31,75,111)(4,30,76,110)(5,29,77,109)(6,42,78,108)(7,41,79,107)(8,40,80,106)(9,39,81,105)(10,38,82,104)(11,37,83,103)(12,36,84,102)(13,35,71,101)(14,34,72,100)(15,88,64,52)(16,87,65,51)(17,86,66,50)(18,85,67,49)(19,98,68,48)(20,97,69,47)(21,96,70,46)(22,95,57,45)(23,94,58,44)(24,93,59,43)(25,92,60,56)(26,91,61,55)(27,90,62,54)(28,89,63,53) );
G=PermutationGroup([(2,74),(4,76),(6,78),(8,80),(10,82),(12,84),(14,72),(15,64),(17,66),(19,68),(21,70),(23,58),(25,60),(27,62),(29,109),(31,111),(33,99),(35,101),(37,103),(39,105),(41,107),(43,93),(45,95),(47,97),(49,85),(51,87),(53,89),(55,91)], [(2,87),(4,89),(6,91),(8,93),(10,95),(12,97),(14,85),(15,111),(17,99),(19,101),(21,103),(23,105),(25,107),(27,109),(29,62),(31,64),(33,66),(35,68),(37,70),(39,58),(41,60),(43,80),(45,82),(47,84),(49,72),(51,74),(53,76),(55,78)], [(1,86),(2,87),(3,88),(4,89),(5,90),(6,91),(7,92),(8,93),(9,94),(10,95),(11,96),(12,97),(13,98),(14,85),(15,111),(16,112),(17,99),(18,100),(19,101),(20,102),(21,103),(22,104),(23,105),(24,106),(25,107),(26,108),(27,109),(28,110),(29,62),(30,63),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69),(37,70),(38,57),(39,58),(40,59),(41,60),(42,61),(43,80),(44,81),(45,82),(46,83),(47,84),(48,71),(49,72),(50,73),(51,74),(52,75),(53,76),(54,77),(55,78),(56,79)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,71),(14,72),(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),(29,109),(30,110),(31,111),(32,112),(33,99),(34,100),(35,101),(36,102),(37,103),(38,104),(39,105),(40,106),(41,107),(42,108),(43,93),(44,94),(45,95),(46,96),(47,97),(48,98),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,91),(56,92)], [(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)], [(1,33,73,99),(2,32,74,112),(3,31,75,111),(4,30,76,110),(5,29,77,109),(6,42,78,108),(7,41,79,107),(8,40,80,106),(9,39,81,105),(10,38,82,104),(11,37,83,103),(12,36,84,102),(13,35,71,101),(14,34,72,100),(15,88,64,52),(16,87,65,51),(17,86,66,50),(18,85,67,49),(19,98,68,48),(20,97,69,47),(21,96,70,46),(22,95,57,45),(23,94,58,44),(24,93,59,43),(25,92,60,56),(26,91,61,55),(27,90,62,54),(28,89,63,53)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 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 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 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 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 18 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 11 | 21 | 0 | 0 | 0 | 0 |
| 15 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 0 | 0 | 0 | 17 | 0 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,18,0,0,0,0,8,11,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,15,0,0,0,0,21,18,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,17,0,0,0,0,17,0] >;
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 14AB | 14AC | 14AD | 28A | ··· | 28I |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 28 | 4 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D4×D7 | D4⋊2D7 |
| kernel | C24.56D14 | C23.11D14 | C22⋊Dic14 | Dic7⋊4D4 | C22.D28 | D4×Dic7 | C23.18D14 | Dic7⋊D4 | C24⋊D7 | C7×C22≀C2 | C2×D4⋊2D7 | C23×Dic7 | C2×Dic7 | C22≀C2 | C2×C14 | C22⋊C4 | C2×D4 | C24 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 3 | 8 | 9 | 9 | 3 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_2^4._{56}D_{14} % in TeX
G:=Group("C2^4.56D14"); // GroupNames label
G:=SmallGroup(448,1039);
// by ID
G=gap.SmallGroup(448,1039);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,570,185,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=1,f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,e*b*e^-1=f*b*f^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations