metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊2Dic7, C56.40D4, Q16⋊2Dic7, C28.37SD16, (C7×D8)⋊4C4, (C7×Q16)⋊4C4, C4○D8.2D7, (C2×C14).4D8, C7⋊3(D8⋊2C4), C56.27(C2×C4), C8⋊Dic7⋊23C2, (C2×C8).51D14, C28.C8⋊8C2, C8.3(C2×Dic7), (C2×C28).118D4, C8.30(C7⋊D4), C4.12(D4.D7), C22.3(D4⋊D7), C4.5(C23.D7), C28.17(C22⋊C4), (C2×C56).154C22, C14.30(D4⋊C4), C2.10(D4⋊Dic7), (C7×C4○D8).5C2, (C2×C4).26(C7⋊D4), SmallGroup(448,123)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊2Dic7
G = < a,b,c,d | a8=b2=c14=1, d2=c7, bab=a-1, ac=ca, dad-1=a3, cbc-1=a4b, dbd-1=a5b, dcd-1=c-1 >
Subgroups: 244 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C14, C14, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C4.Q8, M5(2), C4○D8, C56, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×Q8, D8⋊2C4, C7⋊C16, C4⋊Dic7, C2×C56, C7×D8, C7×SD16, C7×Q16, C7×C4○D4, C28.C8, C8⋊Dic7, C7×C4○D8, D8⋊2Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, Dic7, D14, D4⋊C4, C2×Dic7, C7⋊D4, D8⋊2C4, D4⋊D7, D4.D7, C23.D7, D4⋊Dic7, D8⋊2Dic7
►On 112 points
Generators in S112
(1 50 27 40 13 43 21 34)(2 51 28 41 14 44 15 35)(3 52 22 42 8 45 16 29)(4 53 23 36 9 46 17 30)(5 54 24 37 10 47 18 31)(6 55 25 38 11 48 19 32)(7 56 26 39 12 49 20 33)(57 98 84 103 64 91 77 110)(58 85 71 104 65 92 78 111)(59 86 72 105 66 93 79 112)(60 87 73 106 67 94 80 99)(61 88 74 107 68 95 81 100)(62 89 75 108 69 96 82 101)(63 90 76 109 70 97 83 102)
(1 100)(2 108)(3 102)(4 110)(5 104)(6 112)(7 106)(8 109)(9 103)(10 111)(11 105)(12 99)(13 107)(14 101)(15 96)(16 90)(17 98)(18 92)(19 86)(20 94)(21 88)(22 97)(23 91)(24 85)(25 93)(26 87)(27 95)(28 89)(29 63)(30 57)(31 65)(32 59)(33 67)(34 61)(35 69)(36 64)(37 58)(38 66)(39 60)(40 68)(41 62)(42 70)(43 74)(44 82)(45 76)(46 84)(47 78)(48 72)(49 80)(50 81)(51 75)(52 83)(53 77)(54 71)(55 79)(56 73)
(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)
(2 7)(3 6)(4 5)(8 11)(9 10)(12 14)(15 26)(16 25)(17 24)(18 23)(19 22)(20 28)(21 27)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 49)(36 54)(37 53)(38 52)(39 51)(40 50)(41 56)(42 55)(57 92 64 85)(58 91 65 98)(59 90 66 97)(60 89 67 96)(61 88 68 95)(62 87 69 94)(63 86 70 93)(71 103 78 110)(72 102 79 109)(73 101 80 108)(74 100 81 107)(75 99 82 106)(76 112 83 105)(77 111 84 104)
G:=sub<Sym(112)| (1,50,27,40,13,43,21,34)(2,51,28,41,14,44,15,35)(3,52,22,42,8,45,16,29)(4,53,23,36,9,46,17,30)(5,54,24,37,10,47,18,31)(6,55,25,38,11,48,19,32)(7,56,26,39,12,49,20,33)(57,98,84,103,64,91,77,110)(58,85,71,104,65,92,78,111)(59,86,72,105,66,93,79,112)(60,87,73,106,67,94,80,99)(61,88,74,107,68,95,81,100)(62,89,75,108,69,96,82,101)(63,90,76,109,70,97,83,102), (1,100)(2,108)(3,102)(4,110)(5,104)(6,112)(7,106)(8,109)(9,103)(10,111)(11,105)(12,99)(13,107)(14,101)(15,96)(16,90)(17,98)(18,92)(19,86)(20,94)(21,88)(22,97)(23,91)(24,85)(25,93)(26,87)(27,95)(28,89)(29,63)(30,57)(31,65)(32,59)(33,67)(34,61)(35,69)(36,64)(37,58)(38,66)(39,60)(40,68)(41,62)(42,70)(43,74)(44,82)(45,76)(46,84)(47,78)(48,72)(49,80)(50,81)(51,75)(52,83)(53,77)(54,71)(55,79)(56,73), (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), (2,7)(3,6)(4,5)(8,11)(9,10)(12,14)(15,26)(16,25)(17,24)(18,23)(19,22)(20,28)(21,27)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,49)(36,54)(37,53)(38,52)(39,51)(40,50)(41,56)(42,55)(57,92,64,85)(58,91,65,98)(59,90,66,97)(60,89,67,96)(61,88,68,95)(62,87,69,94)(63,86,70,93)(71,103,78,110)(72,102,79,109)(73,101,80,108)(74,100,81,107)(75,99,82,106)(76,112,83,105)(77,111,84,104)>;
G:=Group( (1,50,27,40,13,43,21,34)(2,51,28,41,14,44,15,35)(3,52,22,42,8,45,16,29)(4,53,23,36,9,46,17,30)(5,54,24,37,10,47,18,31)(6,55,25,38,11,48,19,32)(7,56,26,39,12,49,20,33)(57,98,84,103,64,91,77,110)(58,85,71,104,65,92,78,111)(59,86,72,105,66,93,79,112)(60,87,73,106,67,94,80,99)(61,88,74,107,68,95,81,100)(62,89,75,108,69,96,82,101)(63,90,76,109,70,97,83,102), (1,100)(2,108)(3,102)(4,110)(5,104)(6,112)(7,106)(8,109)(9,103)(10,111)(11,105)(12,99)(13,107)(14,101)(15,96)(16,90)(17,98)(18,92)(19,86)(20,94)(21,88)(22,97)(23,91)(24,85)(25,93)(26,87)(27,95)(28,89)(29,63)(30,57)(31,65)(32,59)(33,67)(34,61)(35,69)(36,64)(37,58)(38,66)(39,60)(40,68)(41,62)(42,70)(43,74)(44,82)(45,76)(46,84)(47,78)(48,72)(49,80)(50,81)(51,75)(52,83)(53,77)(54,71)(55,79)(56,73), (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), (2,7)(3,6)(4,5)(8,11)(9,10)(12,14)(15,26)(16,25)(17,24)(18,23)(19,22)(20,28)(21,27)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,49)(36,54)(37,53)(38,52)(39,51)(40,50)(41,56)(42,55)(57,92,64,85)(58,91,65,98)(59,90,66,97)(60,89,67,96)(61,88,68,95)(62,87,69,94)(63,86,70,93)(71,103,78,110)(72,102,79,109)(73,101,80,108)(74,100,81,107)(75,99,82,106)(76,112,83,105)(77,111,84,104) );
G=PermutationGroup([[(1,50,27,40,13,43,21,34),(2,51,28,41,14,44,15,35),(3,52,22,42,8,45,16,29),(4,53,23,36,9,46,17,30),(5,54,24,37,10,47,18,31),(6,55,25,38,11,48,19,32),(7,56,26,39,12,49,20,33),(57,98,84,103,64,91,77,110),(58,85,71,104,65,92,78,111),(59,86,72,105,66,93,79,112),(60,87,73,106,67,94,80,99),(61,88,74,107,68,95,81,100),(62,89,75,108,69,96,82,101),(63,90,76,109,70,97,83,102)], [(1,100),(2,108),(3,102),(4,110),(5,104),(6,112),(7,106),(8,109),(9,103),(10,111),(11,105),(12,99),(13,107),(14,101),(15,96),(16,90),(17,98),(18,92),(19,86),(20,94),(21,88),(22,97),(23,91),(24,85),(25,93),(26,87),(27,95),(28,89),(29,63),(30,57),(31,65),(32,59),(33,67),(34,61),(35,69),(36,64),(37,58),(38,66),(39,60),(40,68),(41,62),(42,70),(43,74),(44,82),(45,76),(46,84),(47,78),(48,72),(49,80),(50,81),(51,75),(52,83),(53,77),(54,71),(55,79),(56,73)], [(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)], [(2,7),(3,6),(4,5),(8,11),(9,10),(12,14),(15,26),(16,25),(17,24),(18,23),(19,22),(20,28),(21,27),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,49),(36,54),(37,53),(38,52),(39,51),(40,50),(41,56),(42,55),(57,92,64,85),(58,91,65,98),(59,90,66,97),(60,89,67,96),(61,88,68,95),(62,87,69,94),(63,86,70,93),(71,103,78,110),(72,102,79,109),(73,101,80,108),(74,100,81,107),(75,99,82,106),(76,112,83,105),(77,111,84,104)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14L | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | 28H | 28I | 28J | ··· | 28O | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D7 | SD16 | D8 | D14 | Dic7 | Dic7 | C7⋊D4 | C7⋊D4 | D8⋊2C4 | D4.D7 | D4⋊D7 | D8⋊2Dic7 |
| kernel | D8⋊2Dic7 | C28.C8 | C8⋊Dic7 | C7×C4○D8 | C7×D8 | C7×Q16 | C56 | C2×C28 | C4○D8 | C28 | C2×C14 | C2×C8 | D8 | Q16 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 2 | 2 | 3 | 3 | 3 | 6 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of D8⋊2Dic7 ►in GL4(𝔽113) generated by
| 104 | 71 | 0 | 0 |
| 34 | 96 | 0 | 0 |
| 23 | 19 | 89 | 71 |
| 94 | 0 | 42 | 50 |
| 96 | 42 | 111 | 3 |
| 79 | 104 | 65 | 36 |
| 23 | 19 | 89 | 71 |
| 94 | 0 | 42 | 50 |
| 34 | 1 | 0 | 0 |
| 53 | 88 | 0 | 0 |
| 71 | 0 | 0 | 112 |
| 41 | 71 | 1 | 104 |
| 0 | 104 | 0 | 0 |
| 25 | 0 | 0 | 0 |
| 17 | 101 | 50 | 42 |
| 98 | 27 | 40 | 63 |
G:=sub<GL(4,GF(113))| [104,34,23,94,71,96,19,0,0,0,89,42,0,0,71,50],[96,79,23,94,42,104,19,0,111,65,89,42,3,36,71,50],[34,53,71,41,1,88,0,71,0,0,0,1,0,0,112,104],[0,25,17,98,104,0,101,27,0,0,50,40,0,0,42,63] >;
D8⋊2Dic7 in GAP, Magma, Sage, TeX
D_8\rtimes_2{\rm Dic}_7 % in TeX
G:=Group("D8:2Dic7"); // GroupNames label
G:=SmallGroup(448,123);
// by ID
G=gap.SmallGroup(448,123);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,387,675,794,80,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=c^7,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,c*b*c^-1=a^4*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations