direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×C4○D8, D8⋊14D14, Q16⋊12D14, SD16⋊14D14, D56⋊18C22, C56.45C23, C28.14C24, D28.9C23, Dic28⋊16C22, Dic14.9C23, (D7×D8)⋊8C2, C4○D4⋊7D14, (C2×C8)⋊27D14, (D7×Q16)⋊8C2, D8⋊3D7⋊8C2, (C2×C56)⋊4C22, (D7×SD16)⋊7C2, (C4×D7).54D4, C4.221(D4×D7), C7⋊C8.24C23, D56⋊7C2⋊6C2, Q8.D14⋊8C2, C22.4(D4×D7), D4⋊D7⋊12C22, D14.66(C2×D4), C28.380(C2×D4), C4○D28⋊5C22, (C7×D8)⋊12C22, (C8×D7)⋊16C22, Q8⋊D7⋊11C22, (C7×D4).8C23, D4.8(C22×D7), (D4×D7).6C22, C8.42(C22×D7), C4.14(C23×D7), SD16⋊3D7⋊7C2, D4.8D14⋊1C2, Q8.8(C22×D7), (C7×Q8).8C23, (Q8×D7).5C22, D4⋊2D7⋊8C22, C56⋊C2⋊20C22, D4.D7⋊11C22, Dic7.71(C2×D4), (C7×Q16)⋊10C22, Q8⋊2D7⋊8C22, C7⋊Q16⋊10C22, (C4×D7).29C23, (C22×D7).64D4, (C2×C28).531C23, (C2×Dic7).124D4, (C7×SD16)⋊15C22, C14.115(C22×D4), (D7×C2×C8)⋊1C2, C7⋊5(C2×C4○D8), C2.88(C2×D4×D7), (D7×C4○D4)⋊1C2, (C7×C4○D8)⋊2C2, (C2×C7⋊C8)⋊37C22, (C2×C14).11(C2×D4), (C7×C4○D4)⋊1C22, (C2×C4×D7).261C22, (C2×C4).618(C22×D7), SmallGroup(448,1220)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1332 in 266 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2 [×8], C4 [×2], C4 [×6], C22, C22 [×12], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23 [×3], D7 [×2], D7 [×3], C14, C14 [×3], C2×C8, C2×C8 [×5], D8, D8 [×3], SD16 [×2], SD16 [×6], Q16, Q16 [×3], C22×C4 [×3], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×2], C4○D4 [×10], Dic7 [×2], Dic7 [×2], C28 [×2], C28 [×2], D14 [×2], D14 [×8], C2×C14, C2×C14 [×2], C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8, C4○D8 [×7], C2×C4○D4 [×2], C7⋊C8 [×2], C56 [×2], Dic14 [×2], Dic14 [×2], C4×D7 [×4], C4×D7 [×6], D28 [×2], D28 [×2], C2×Dic7, C2×Dic7 [×2], C7⋊D4 [×6], C2×C28, C2×C28 [×2], C7×D4 [×2], C7×D4 [×2], C7×Q8 [×2], C22×D7, C22×D7 [×2], C2×C4○D8, C8×D7 [×4], C56⋊C2 [×2], D56, Dic28, C2×C7⋊C8, D4⋊D7 [×2], D4.D7 [×2], Q8⋊D7 [×2], C7⋊Q16 [×2], C2×C56, C7×D8, C7×SD16 [×2], C7×Q16, C2×C4×D7, C2×C4×D7 [×2], C4○D28 [×2], C4○D28 [×2], D4×D7 [×2], D4×D7 [×2], D4⋊2D7 [×2], D4⋊2D7 [×2], Q8×D7 [×2], Q8⋊2D7 [×2], C7×C4○D4 [×2], D7×C2×C8, D56⋊7C2, D7×D8, D8⋊3D7, D7×SD16 [×2], SD16⋊3D7 [×2], D7×Q16, Q8.D14, D4.8D14 [×2], C7×C4○D8, D7×C4○D4 [×2], D7×C4○D8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C4○D8 [×2], C22×D4, C22×D7 [×7], C2×C4○D8, D4×D7 [×2], C23×D7, C2×D4×D7, D7×C4○D8
Generators and relations
G = < a,b,c,d,e | a7=b2=c4=e2=1, d4=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d3 >
►On 112 points
(1 102 21 12 86 40 43)(2 103 22 13 87 33 44)(3 104 23 14 88 34 45)(4 97 24 15 81 35 46)(5 98 17 16 82 36 47)(6 99 18 9 83 37 48)(7 100 19 10 84 38 41)(8 101 20 11 85 39 42)(25 78 54 63 66 91 109)(26 79 55 64 67 92 110)(27 80 56 57 68 93 111)(28 73 49 58 69 94 112)(29 74 50 59 70 95 105)(30 75 51 60 71 96 106)(31 76 52 61 72 89 107)(32 77 53 62 65 90 108)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 81)(25 109)(26 110)(27 111)(28 112)(29 105)(30 106)(31 107)(32 108)(33 103)(34 104)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(49 69)(50 70)(51 71)(52 72)(53 65)(54 66)(55 67)(56 68)(73 94)(74 95)(75 96)(76 89)(77 90)(78 91)(79 92)(80 93)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 31 29 27)(26 32 30 28)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)(65 71 69 67)(66 72 70 68)(73 79 77 75)(74 80 78 76)(81 83 85 87)(82 84 86 88)(89 95 93 91)(90 96 94 92)(97 99 101 103)(98 100 102 104)(105 111 109 107)(106 112 110 108)
(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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 57)(10 64)(11 63)(12 62)(13 61)(14 60)(15 59)(16 58)(17 49)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(33 89)(34 96)(35 95)(36 94)(37 93)(38 92)(39 91)(40 90)(41 110)(42 109)(43 108)(44 107)(45 106)(46 105)(47 112)(48 111)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 88)(72 87)(73 98)(74 97)(75 104)(76 103)(77 102)(78 101)(79 100)(80 99)
G:=sub<Sym(112)| (1,102,21,12,86,40,43)(2,103,22,13,87,33,44)(3,104,23,14,88,34,45)(4,97,24,15,81,35,46)(5,98,17,16,82,36,47)(6,99,18,9,83,37,48)(7,100,19,10,84,38,41)(8,101,20,11,85,39,42)(25,78,54,63,66,91,109)(26,79,55,64,67,92,110)(27,80,56,57,68,93,111)(28,73,49,58,69,94,112)(29,74,50,59,70,95,105)(30,75,51,60,71,96,106)(31,76,52,61,72,89,107)(32,77,53,62,65,90,108), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,81)(25,109)(26,110)(27,111)(28,112)(29,105)(30,106)(31,107)(32,108)(33,103)(34,104)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(49,69)(50,70)(51,71)(52,72)(53,65)(54,66)(55,67)(56,68)(73,94)(74,95)(75,96)(76,89)(77,90)(78,91)(79,92)(80,93), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,79,77,75)(74,80,78,76)(81,83,85,87)(82,84,86,88)(89,95,93,91)(90,96,94,92)(97,99,101,103)(98,100,102,104)(105,111,109,107)(106,112,110,108), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,57)(10,64)(11,63)(12,62)(13,61)(14,60)(15,59)(16,58)(17,49)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(33,89)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,112)(48,111)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,88)(72,87)(73,98)(74,97)(75,104)(76,103)(77,102)(78,101)(79,100)(80,99)>;
G:=Group( (1,102,21,12,86,40,43)(2,103,22,13,87,33,44)(3,104,23,14,88,34,45)(4,97,24,15,81,35,46)(5,98,17,16,82,36,47)(6,99,18,9,83,37,48)(7,100,19,10,84,38,41)(8,101,20,11,85,39,42)(25,78,54,63,66,91,109)(26,79,55,64,67,92,110)(27,80,56,57,68,93,111)(28,73,49,58,69,94,112)(29,74,50,59,70,95,105)(30,75,51,60,71,96,106)(31,76,52,61,72,89,107)(32,77,53,62,65,90,108), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,81)(25,109)(26,110)(27,111)(28,112)(29,105)(30,106)(31,107)(32,108)(33,103)(34,104)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(49,69)(50,70)(51,71)(52,72)(53,65)(54,66)(55,67)(56,68)(73,94)(74,95)(75,96)(76,89)(77,90)(78,91)(79,92)(80,93), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,79,77,75)(74,80,78,76)(81,83,85,87)(82,84,86,88)(89,95,93,91)(90,96,94,92)(97,99,101,103)(98,100,102,104)(105,111,109,107)(106,112,110,108), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,57)(10,64)(11,63)(12,62)(13,61)(14,60)(15,59)(16,58)(17,49)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(33,89)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,112)(48,111)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,88)(72,87)(73,98)(74,97)(75,104)(76,103)(77,102)(78,101)(79,100)(80,99) );
G=PermutationGroup([(1,102,21,12,86,40,43),(2,103,22,13,87,33,44),(3,104,23,14,88,34,45),(4,97,24,15,81,35,46),(5,98,17,16,82,36,47),(6,99,18,9,83,37,48),(7,100,19,10,84,38,41),(8,101,20,11,85,39,42),(25,78,54,63,66,91,109),(26,79,55,64,67,92,110),(27,80,56,57,68,93,111),(28,73,49,58,69,94,112),(29,74,50,59,70,95,105),(30,75,51,60,71,96,106),(31,76,52,61,72,89,107),(32,77,53,62,65,90,108)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,41),(8,42),(17,82),(18,83),(19,84),(20,85),(21,86),(22,87),(23,88),(24,81),(25,109),(26,110),(27,111),(28,112),(29,105),(30,106),(31,107),(32,108),(33,103),(34,104),(35,97),(36,98),(37,99),(38,100),(39,101),(40,102),(49,69),(50,70),(51,71),(52,72),(53,65),(54,66),(55,67),(56,68),(73,94),(74,95),(75,96),(76,89),(77,90),(78,91),(79,92),(80,93)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,31,29,27),(26,32,30,28),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60),(65,71,69,67),(66,72,70,68),(73,79,77,75),(74,80,78,76),(81,83,85,87),(82,84,86,88),(89,95,93,91),(90,96,94,92),(97,99,101,103),(98,100,102,104),(105,111,109,107),(106,112,110,108)], [(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)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,57),(10,64),(11,63),(12,62),(13,61),(14,60),(15,59),(16,58),(17,49),(18,56),(19,55),(20,54),(21,53),(22,52),(23,51),(24,50),(33,89),(34,96),(35,95),(36,94),(37,93),(38,92),(39,91),(40,90),(41,110),(42,109),(43,108),(44,107),(45,106),(46,105),(47,112),(48,111),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,88),(72,87),(73,98),(74,97),(75,104),(76,103),(77,102),(78,101),(79,100),(80,99)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 0 | 1 | 0 | 0 |
| 112 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 95 | 0 |
| 0 | 0 | 100 | 69 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 44 | 88 |
| 0 | 0 | 100 | 69 |
G:=sub<GL(4,GF(113))| [0,112,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,98,0,0,0,0,98],[1,0,0,0,0,1,0,0,0,0,95,100,0,0,0,69],[1,0,0,0,0,1,0,0,0,0,44,100,0,0,88,69] >;
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14L | 28A | ··· | 28F | 28G | 28H | 28I | 28J | ··· | 28O | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 7 | 7 | 14 | 28 | 28 | 1 | 1 | 2 | 4 | 4 | 7 | 7 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | C4○D8 | D4×D7 | D4×D7 | D7×C4○D8 |
| kernel | D7×C4○D8 | D7×C2×C8 | D56⋊7C2 | D7×D8 | D8⋊3D7 | D7×SD16 | SD16⋊3D7 | D7×Q16 | Q8.D14 | D4.8D14 | C7×C4○D8 | D7×C4○D4 | C4×D7 | C2×Dic7 | C22×D7 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | D7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 3 | 6 | 3 | 6 | 8 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
D_7\times C_4\circ D_8
% in TeX
G:=Group("D7xC4oD8"); // GroupNames label
G:=SmallGroup(448,1220);
// by ID
G=gap.SmallGroup(448,1220);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,570,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^4=e^2=1,d^4=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d^3>;
// generators/relations