direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×C8.C22, Q16⋊3D14, SD16⋊5D14, C56.4C23, C28.23C24, M4(2)⋊11D14, Dic28⋊3C22, D28.16C23, Dic14.16C23, (D7×Q16)⋊1C2, (C2×Q8)⋊22D14, (D7×SD16)⋊3C2, (C4×D7).44D4, C4.191(D4×D7), C7⋊C8.11C23, Q8⋊D7⋊5C22, C8.4(C22×D7), Q16⋊D7⋊1C2, C4○D4.29D14, D14.68(C2×D4), C8.D14⋊3C2, SD16⋊D7⋊3C2, C28.244(C2×D4), C56⋊C2⋊5C22, C8⋊D7⋊5C22, (D7×M4(2))⋊3C2, D4.D7⋊6C22, (Q8×D7)⋊11C22, (C7×Q16)⋊1C22, (C8×D7).1C22, (D4×D7).7C22, C7⋊Q16⋊4C22, C4.23(C23×D7), C22.48(D4×D7), C28.C23⋊9C2, Dic7.61(C2×D4), (C2×Dic7).82D4, (Q8×C14)⋊20C22, (C7×SD16)⋊5C22, (C4×D7).15C23, (C7×D4).16C23, D4.16(C22×D7), D4.9D14⋊10C2, (C7×Q8).16C23, Q8.16(C22×D7), (C2×C28).114C23, C4○D28.30C22, (C22×D7).102D4, D4⋊2D7.6C22, C14.124(C22×D4), (C7×M4(2))⋊5C22, C4.Dic7⋊14C22, Q8⋊2D7.6C22, (C2×Dic14)⋊41C22, (C2×Q8×D7)⋊17C2, C2.97(C2×D4×D7), C7⋊4(C2×C8.C22), (D7×C4○D4).4C2, (C2×C14).69(C2×D4), (C7×C8.C22)⋊1C2, (C2×C4×D7).162C22, (C2×C4).98(C22×D7), (C7×C4○D4).25C22, SmallGroup(448,1229)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×C8.C22
G = < a,b,c,d,e | a7=b2=c8=d2=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, ede=c4d >
Subgroups: 1228 in 258 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C23, D7, D7, C14, C14, C2×C8, M4(2), M4(2), SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C8.C22, C22×Q8, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C7×Q8, C22×D7, C22×D7, C2×C8.C22, C8×D7, C8⋊D7, C56⋊C2, Dic28, C4.Dic7, D4.D7, Q8⋊D7, C7⋊Q16, C7×M4(2), C7×SD16, C7×Q16, C2×Dic14, C2×Dic14, C2×C4×D7, C2×C4×D7, C4○D28, C4○D28, D4×D7, D4×D7, D4⋊2D7, D4⋊2D7, Q8×D7, Q8×D7, Q8×D7, Q8⋊2D7, Q8×C14, C7×C4○D4, D7×M4(2), C8.D14, D7×SD16, SD16⋊D7, D7×Q16, Q16⋊D7, C28.C23, D4.9D14, C7×C8.C22, C2×Q8×D7, D7×C4○D4, D7×C8.C22
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C8.C22, C22×D4, C22×D7, C2×C8.C22, D4×D7, C23×D7, C2×D4×D7, D7×C8.C22
►On 112 points
(1 105 88 16 19 61 43)(2 106 81 9 20 62 44)(3 107 82 10 21 63 45)(4 108 83 11 22 64 46)(5 109 84 12 23 57 47)(6 110 85 13 24 58 48)(7 111 86 14 17 59 41)(8 112 87 15 18 60 42)(25 78 54 34 66 98 95)(26 79 55 35 67 99 96)(27 80 56 36 68 100 89)(28 73 49 37 69 101 90)(29 74 50 38 70 102 91)(30 75 51 39 71 103 92)(31 76 52 40 72 104 93)(32 77 53 33 65 97 94)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(17 86)(18 87)(19 88)(20 81)(21 82)(22 83)(23 84)(24 85)(25 95)(26 96)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(49 69)(50 70)(51 71)(52 72)(53 65)(54 66)(55 67)(56 68)(57 109)(58 110)(59 111)(60 112)(61 105)(62 106)(63 107)(64 108)(73 101)(74 102)(75 103)(76 104)(77 97)(78 98)(79 99)(80 100)
(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 25)(2 28)(3 31)(4 26)(5 29)(6 32)(7 27)(8 30)(9 37)(10 40)(11 35)(12 38)(13 33)(14 36)(15 39)(16 34)(17 68)(18 71)(19 66)(20 69)(21 72)(22 67)(23 70)(24 65)(41 89)(42 92)(43 95)(44 90)(45 93)(46 96)(47 91)(48 94)(49 81)(50 84)(51 87)(52 82)(53 85)(54 88)(55 83)(56 86)(57 102)(58 97)(59 100)(60 103)(61 98)(62 101)(63 104)(64 99)(73 106)(74 109)(75 112)(76 107)(77 110)(78 105)(79 108)(80 111)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)(58 62)(60 64)(66 70)(68 72)(74 78)(76 80)(81 85)(83 87)(89 93)(91 95)(98 102)(100 104)(106 110)(108 112)
G:=sub<Sym(112)| (1,105,88,16,19,61,43)(2,106,81,9,20,62,44)(3,107,82,10,21,63,45)(4,108,83,11,22,64,46)(5,109,84,12,23,57,47)(6,110,85,13,24,58,48)(7,111,86,14,17,59,41)(8,112,87,15,18,60,42)(25,78,54,34,66,98,95)(26,79,55,35,67,99,96)(27,80,56,36,68,100,89)(28,73,49,37,69,101,90)(29,74,50,38,70,102,91)(30,75,51,39,71,103,92)(31,76,52,40,72,104,93)(32,77,53,33,65,97,94), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(17,86)(18,87)(19,88)(20,81)(21,82)(22,83)(23,84)(24,85)(25,95)(26,96)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(49,69)(50,70)(51,71)(52,72)(53,65)(54,66)(55,67)(56,68)(57,109)(58,110)(59,111)(60,112)(61,105)(62,106)(63,107)(64,108)(73,101)(74,102)(75,103)(76,104)(77,97)(78,98)(79,99)(80,100), (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), (1,25)(2,28)(3,31)(4,26)(5,29)(6,32)(7,27)(8,30)(9,37)(10,40)(11,35)(12,38)(13,33)(14,36)(15,39)(16,34)(17,68)(18,71)(19,66)(20,69)(21,72)(22,67)(23,70)(24,65)(41,89)(42,92)(43,95)(44,90)(45,93)(46,96)(47,91)(48,94)(49,81)(50,84)(51,87)(52,82)(53,85)(54,88)(55,83)(56,86)(57,102)(58,97)(59,100)(60,103)(61,98)(62,101)(63,104)(64,99)(73,106)(74,109)(75,112)(76,107)(77,110)(78,105)(79,108)(80,111), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80)(81,85)(83,87)(89,93)(91,95)(98,102)(100,104)(106,110)(108,112)>;
G:=Group( (1,105,88,16,19,61,43)(2,106,81,9,20,62,44)(3,107,82,10,21,63,45)(4,108,83,11,22,64,46)(5,109,84,12,23,57,47)(6,110,85,13,24,58,48)(7,111,86,14,17,59,41)(8,112,87,15,18,60,42)(25,78,54,34,66,98,95)(26,79,55,35,67,99,96)(27,80,56,36,68,100,89)(28,73,49,37,69,101,90)(29,74,50,38,70,102,91)(30,75,51,39,71,103,92)(31,76,52,40,72,104,93)(32,77,53,33,65,97,94), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(17,86)(18,87)(19,88)(20,81)(21,82)(22,83)(23,84)(24,85)(25,95)(26,96)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(49,69)(50,70)(51,71)(52,72)(53,65)(54,66)(55,67)(56,68)(57,109)(58,110)(59,111)(60,112)(61,105)(62,106)(63,107)(64,108)(73,101)(74,102)(75,103)(76,104)(77,97)(78,98)(79,99)(80,100), (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), (1,25)(2,28)(3,31)(4,26)(5,29)(6,32)(7,27)(8,30)(9,37)(10,40)(11,35)(12,38)(13,33)(14,36)(15,39)(16,34)(17,68)(18,71)(19,66)(20,69)(21,72)(22,67)(23,70)(24,65)(41,89)(42,92)(43,95)(44,90)(45,93)(46,96)(47,91)(48,94)(49,81)(50,84)(51,87)(52,82)(53,85)(54,88)(55,83)(56,86)(57,102)(58,97)(59,100)(60,103)(61,98)(62,101)(63,104)(64,99)(73,106)(74,109)(75,112)(76,107)(77,110)(78,105)(79,108)(80,111), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80)(81,85)(83,87)(89,93)(91,95)(98,102)(100,104)(106,110)(108,112) );
G=PermutationGroup([[(1,105,88,16,19,61,43),(2,106,81,9,20,62,44),(3,107,82,10,21,63,45),(4,108,83,11,22,64,46),(5,109,84,12,23,57,47),(6,110,85,13,24,58,48),(7,111,86,14,17,59,41),(8,112,87,15,18,60,42),(25,78,54,34,66,98,95),(26,79,55,35,67,99,96),(27,80,56,36,68,100,89),(28,73,49,37,69,101,90),(29,74,50,38,70,102,91),(30,75,51,39,71,103,92),(31,76,52,40,72,104,93),(32,77,53,33,65,97,94)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,41),(8,42),(17,86),(18,87),(19,88),(20,81),(21,82),(22,83),(23,84),(24,85),(25,95),(26,96),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(49,69),(50,70),(51,71),(52,72),(53,65),(54,66),(55,67),(56,68),(57,109),(58,110),(59,111),(60,112),(61,105),(62,106),(63,107),(64,108),(73,101),(74,102),(75,103),(76,104),(77,97),(78,98),(79,99),(80,100)], [(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)], [(1,25),(2,28),(3,31),(4,26),(5,29),(6,32),(7,27),(8,30),(9,37),(10,40),(11,35),(12,38),(13,33),(14,36),(15,39),(16,34),(17,68),(18,71),(19,66),(20,69),(21,72),(22,67),(23,70),(24,65),(41,89),(42,92),(43,95),(44,90),(45,93),(46,96),(47,91),(48,94),(49,81),(50,84),(51,87),(52,82),(53,85),(54,88),(55,83),(56,86),(57,102),(58,97),(59,100),(60,103),(61,98),(62,101),(63,104),(64,99),(73,106),(74,109),(75,112),(76,107),(77,110),(78,105),(79,108),(80,111)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31),(34,38),(36,40),(42,46),(44,48),(50,54),(52,56),(58,62),(60,64),(66,70),(68,72),(74,78),(76,80),(81,85),(83,87),(89,93),(91,95),(98,102),(100,104),(106,110),(108,112)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 7 | 7 | 14 | 28 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | C8.C22 | D4×D7 | D4×D7 | D7×C8.C22 |
| kernel | D7×C8.C22 | D7×M4(2) | C8.D14 | D7×SD16 | SD16⋊D7 | D7×Q16 | Q16⋊D7 | C28.C23 | D4.9D14 | C7×C8.C22 | C2×Q8×D7 | D7×C4○D4 | C4×D7 | C2×Dic7 | C22×D7 | C8.C22 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | D7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 6 | 6 | 3 | 3 | 2 | 3 | 3 | 3 |
Matrix representation of D7×C8.C22 ►in GL8(𝔽113)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 112 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 41 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 103 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 | 104 | 77 |
| 0 | 0 | 0 | 0 | 44 | 44 | 97 | 64 |
| 0 | 0 | 0 | 0 | 60 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 83 | 109 | 109 | 9 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 112 | 112 | 112 | 91 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 41 | 41 | 0 | 112 |
G:=sub<GL(8,GF(113))| [0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,9,0,0,0,0,0,0,1,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[41,103,0,0,0,0,0,0,10,72,0,0,0,0,0,0,0,0,41,103,0,0,0,0,0,0,10,72,0,0,0,0,0,0,0,0,60,44,60,83,0,0,0,0,60,44,44,109,0,0,0,0,104,97,0,109,0,0,0,0,77,64,0,9],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,112,1,0,0,0,0,0,0,112,0,0,0,0,0,0,1,112,0,0,0,0,0,0,0,91,0,1],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,41,0,0,0,0,0,1,0,41,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112] >;
D7×C8.C22 in GAP, Magma, Sage, TeX
D_7\times C_8.C_2^2
% in TeX
G:=Group("D7xC8.C2^2"); // GroupNames label
G:=SmallGroup(448,1229);
// by ID
G=gap.SmallGroup(448,1229);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,570,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^8=d^2=e^2=1,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,d*c*d=c^3,e*c*e=c^5,e*d*e=c^4*d>;
// generators/relations