metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊4Dic7, Q16⋊4Dic7, SD16⋊2Dic7, (C7×D8)⋊6C4, C7⋊C8.36D4, (C7×Q16)⋊6C4, C4○D8.4D7, C56⋊C4⋊2C2, C7⋊5(C8.26D4), C56.31(C2×C4), (C7×SD16)⋊2C4, C4.218(D4×D7), C56.C4⋊9C2, C8.6(C2×Dic7), C4○D4.23D14, (C2×C8).100D14, C28.377(C2×D4), C14.100(C4×D4), Q8.Dic7⋊4C2, Q8.4(C2×Dic7), D4.4(C2×Dic7), C2.17(D4×Dic7), D4⋊2Dic7⋊5C2, C28.78(C22×C4), (C2×C56).45C22, C4.8(C22×Dic7), (C2×C28).468C23, C22.4(D4⋊2D7), (C4×Dic7).57C22, C4.Dic7.23C22, (C7×C4○D8).3C2, (C7×D4).11(C2×C4), (C7×Q8).11(C2×C4), (C2×C7⋊C8).169C22, (C2×C14).12(C4○D4), (C7×C4○D4).10C22, (C2×C4).555(C22×D7), SmallGroup(448,731)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊4Dic7
G = < a,b,c,d | a8=b2=c14=1, d2=c7, bab=a-1, ac=ca, dad-1=a5, cbc-1=a4b, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 340 in 104 conjugacy classes, 53 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C14, C14, C42, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C8⋊C4, C4≀C2, C8.C4, C8○D4, C4○D8, C7⋊C8, C7⋊C8, C56, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C8.26D4, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, C4×Dic7, C2×C56, C7×D8, C7×SD16, C7×Q16, C7×C4○D4, C56⋊C4, C56.C4, D4⋊2Dic7, Q8.Dic7, C7×C4○D8, D8⋊4Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, Dic7, D14, C4×D4, C2×Dic7, C22×D7, C8.26D4, D4×D7, D4⋊2D7, C22×Dic7, D4×Dic7, D8⋊4Dic7
►On 112 points
Generators in S112
(1 31 15 52 26 38 9 45)(2 32 16 53 27 39 10 46)(3 33 17 54 28 40 11 47)(4 34 18 55 22 41 12 48)(5 35 19 56 23 42 13 49)(6 29 20 50 24 36 14 43)(7 30 21 51 25 37 8 44)(57 75 87 101 64 82 94 108)(58 76 88 102 65 83 95 109)(59 77 89 103 66 84 96 110)(60 78 90 104 67 71 97 111)(61 79 91 105 68 72 98 112)(62 80 92 106 69 73 85 99)(63 81 93 107 70 74 86 100)
(1 61)(2 69)(3 63)(4 57)(5 65)(6 59)(7 67)(8 97)(9 91)(10 85)(11 93)(12 87)(13 95)(14 89)(15 98)(16 92)(17 86)(18 94)(19 88)(20 96)(21 90)(22 64)(23 58)(24 66)(25 60)(26 68)(27 62)(28 70)(29 110)(30 104)(31 112)(32 106)(33 100)(34 108)(35 102)(36 103)(37 111)(38 105)(39 99)(40 107)(41 101)(42 109)(43 77)(44 71)(45 79)(46 73)(47 81)(48 75)(49 83)(50 84)(51 78)(52 72)(53 80)(54 74)(55 82)(56 76)
(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 7)(2 6)(3 5)(8 9)(10 14)(11 13)(15 21)(16 20)(17 19)(23 28)(24 27)(25 26)(29 39)(30 38)(31 37)(32 36)(33 42)(34 41)(35 40)(43 53)(44 52)(45 51)(46 50)(47 56)(48 55)(49 54)(57 87 64 94)(58 86 65 93)(59 85 66 92)(60 98 67 91)(61 97 68 90)(62 96 69 89)(63 95 70 88)(71 112 78 105)(72 111 79 104)(73 110 80 103)(74 109 81 102)(75 108 82 101)(76 107 83 100)(77 106 84 99)
G:=sub<Sym(112)| (1,31,15,52,26,38,9,45)(2,32,16,53,27,39,10,46)(3,33,17,54,28,40,11,47)(4,34,18,55,22,41,12,48)(5,35,19,56,23,42,13,49)(6,29,20,50,24,36,14,43)(7,30,21,51,25,37,8,44)(57,75,87,101,64,82,94,108)(58,76,88,102,65,83,95,109)(59,77,89,103,66,84,96,110)(60,78,90,104,67,71,97,111)(61,79,91,105,68,72,98,112)(62,80,92,106,69,73,85,99)(63,81,93,107,70,74,86,100), (1,61)(2,69)(3,63)(4,57)(5,65)(6,59)(7,67)(8,97)(9,91)(10,85)(11,93)(12,87)(13,95)(14,89)(15,98)(16,92)(17,86)(18,94)(19,88)(20,96)(21,90)(22,64)(23,58)(24,66)(25,60)(26,68)(27,62)(28,70)(29,110)(30,104)(31,112)(32,106)(33,100)(34,108)(35,102)(36,103)(37,111)(38,105)(39,99)(40,107)(41,101)(42,109)(43,77)(44,71)(45,79)(46,73)(47,81)(48,75)(49,83)(50,84)(51,78)(52,72)(53,80)(54,74)(55,82)(56,76), (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), (1,7)(2,6)(3,5)(8,9)(10,14)(11,13)(15,21)(16,20)(17,19)(23,28)(24,27)(25,26)(29,39)(30,38)(31,37)(32,36)(33,42)(34,41)(35,40)(43,53)(44,52)(45,51)(46,50)(47,56)(48,55)(49,54)(57,87,64,94)(58,86,65,93)(59,85,66,92)(60,98,67,91)(61,97,68,90)(62,96,69,89)(63,95,70,88)(71,112,78,105)(72,111,79,104)(73,110,80,103)(74,109,81,102)(75,108,82,101)(76,107,83,100)(77,106,84,99)>;
G:=Group( (1,31,15,52,26,38,9,45)(2,32,16,53,27,39,10,46)(3,33,17,54,28,40,11,47)(4,34,18,55,22,41,12,48)(5,35,19,56,23,42,13,49)(6,29,20,50,24,36,14,43)(7,30,21,51,25,37,8,44)(57,75,87,101,64,82,94,108)(58,76,88,102,65,83,95,109)(59,77,89,103,66,84,96,110)(60,78,90,104,67,71,97,111)(61,79,91,105,68,72,98,112)(62,80,92,106,69,73,85,99)(63,81,93,107,70,74,86,100), (1,61)(2,69)(3,63)(4,57)(5,65)(6,59)(7,67)(8,97)(9,91)(10,85)(11,93)(12,87)(13,95)(14,89)(15,98)(16,92)(17,86)(18,94)(19,88)(20,96)(21,90)(22,64)(23,58)(24,66)(25,60)(26,68)(27,62)(28,70)(29,110)(30,104)(31,112)(32,106)(33,100)(34,108)(35,102)(36,103)(37,111)(38,105)(39,99)(40,107)(41,101)(42,109)(43,77)(44,71)(45,79)(46,73)(47,81)(48,75)(49,83)(50,84)(51,78)(52,72)(53,80)(54,74)(55,82)(56,76), (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), (1,7)(2,6)(3,5)(8,9)(10,14)(11,13)(15,21)(16,20)(17,19)(23,28)(24,27)(25,26)(29,39)(30,38)(31,37)(32,36)(33,42)(34,41)(35,40)(43,53)(44,52)(45,51)(46,50)(47,56)(48,55)(49,54)(57,87,64,94)(58,86,65,93)(59,85,66,92)(60,98,67,91)(61,97,68,90)(62,96,69,89)(63,95,70,88)(71,112,78,105)(72,111,79,104)(73,110,80,103)(74,109,81,102)(75,108,82,101)(76,107,83,100)(77,106,84,99) );
G=PermutationGroup([[(1,31,15,52,26,38,9,45),(2,32,16,53,27,39,10,46),(3,33,17,54,28,40,11,47),(4,34,18,55,22,41,12,48),(5,35,19,56,23,42,13,49),(6,29,20,50,24,36,14,43),(7,30,21,51,25,37,8,44),(57,75,87,101,64,82,94,108),(58,76,88,102,65,83,95,109),(59,77,89,103,66,84,96,110),(60,78,90,104,67,71,97,111),(61,79,91,105,68,72,98,112),(62,80,92,106,69,73,85,99),(63,81,93,107,70,74,86,100)], [(1,61),(2,69),(3,63),(4,57),(5,65),(6,59),(7,67),(8,97),(9,91),(10,85),(11,93),(12,87),(13,95),(14,89),(15,98),(16,92),(17,86),(18,94),(19,88),(20,96),(21,90),(22,64),(23,58),(24,66),(25,60),(26,68),(27,62),(28,70),(29,110),(30,104),(31,112),(32,106),(33,100),(34,108),(35,102),(36,103),(37,111),(38,105),(39,99),(40,107),(41,101),(42,109),(43,77),(44,71),(45,79),(46,73),(47,81),(48,75),(49,83),(50,84),(51,78),(52,72),(53,80),(54,74),(55,82),(56,76)], [(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)], [(1,7),(2,6),(3,5),(8,9),(10,14),(11,13),(15,21),(16,20),(17,19),(23,28),(24,27),(25,26),(29,39),(30,38),(31,37),(32,36),(33,42),(34,41),(35,40),(43,53),(44,52),(45,51),(46,50),(47,56),(48,55),(49,54),(57,87,64,94),(58,86,65,93),(59,85,66,92),(60,98,67,91),(61,97,68,90),(62,96,69,89),(63,95,70,88),(71,112,78,105),(72,111,79,104),(73,110,80,103),(74,109,81,102),(75,108,82,101),(76,107,83,100),(77,106,84,99)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14L | 28A | ··· | 28F | 28G | 28H | 28I | 28J | ··· | 28O | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 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 | 1 | 1 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | C4○D4 | D14 | Dic7 | Dic7 | Dic7 | D14 | C8.26D4 | D4×D7 | D4⋊2D7 | D8⋊4Dic7 |
| kernel | D8⋊4Dic7 | C56⋊C4 | C56.C4 | D4⋊2Dic7 | Q8.Dic7 | C7×C4○D8 | C7×D8 | C7×SD16 | C7×Q16 | C7⋊C8 | C4○D8 | C2×C14 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 2 | 3 | 2 | 3 | 3 | 6 | 3 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of D8⋊4Dic7 ►in GL4(𝔽113) generated by
| 47 | 10 | 0 | 0 |
| 94 | 66 | 0 | 0 |
| 17 | 70 | 64 | 37 |
| 2 | 33 | 76 | 49 |
| 98 | 76 | 12 | 12 |
| 14 | 27 | 0 | 68 |
| 17 | 70 | 64 | 37 |
| 64 | 80 | 64 | 37 |
| 1 | 112 | 0 | 0 |
| 81 | 33 | 0 | 0 |
| 26 | 44 | 0 | 1 |
| 105 | 43 | 112 | 79 |
| 70 | 98 | 0 | 0 |
| 48 | 43 | 0 | 0 |
| 76 | 82 | 112 | 0 |
| 110 | 83 | 34 | 1 |
G:=sub<GL(4,GF(113))| [47,94,17,2,10,66,70,33,0,0,64,76,0,0,37,49],[98,14,17,64,76,27,70,80,12,0,64,64,12,68,37,37],[1,81,26,105,112,33,44,43,0,0,0,112,0,0,1,79],[70,48,76,110,98,43,82,83,0,0,112,34,0,0,0,1] >;
D8⋊4Dic7 in GAP, Magma, Sage, TeX
D_8\rtimes_4{\rm Dic}_7 % in TeX
G:=Group("D8:4Dic7"); // GroupNames label
G:=SmallGroup(448,731);
// by ID
G=gap.SmallGroup(448,731);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,758,219,136,851,438,102,18822]);
// 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^5,c*b*c^-1=a^4*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations