metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7⋊3D4, C23.9D14, (C2×D4)⋊5D7, (C2×C14)⋊3D4, (D4×C14)⋊9C2, C7⋊5(C4⋊D4), C2.27(D4×D7), D14⋊C4⋊15C2, (C2×C4).19D14, C14.51(C2×D4), Dic7⋊C4⋊15C2, C22⋊1(C7⋊D4), C23.D7⋊12C2, C14.32(C4○D4), (C2×C14).54C23, (C2×C28).62C22, (C22×Dic7)⋊6C2, C2.18(D4⋊2D7), C22.61(C22×D7), (C22×C14).21C22, (C2×Dic7).38C22, (C22×D7).11C22, (C2×C7⋊D4)⋊6C2, C2.15(C2×C7⋊D4), SmallGroup(224,134)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7⋊D4
G = < a,b,c,d | a14=c4=d2=1, b2=a7, bab-1=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd=c-1 >
Subgroups: 382 in 94 conjugacy classes, 35 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, Dic7, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C4⋊D4, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, Dic7⋊C4, D14⋊C4, C23.D7, C22×Dic7, C2×C7⋊D4, D4×C14, Dic7⋊D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C7⋊D4, C22×D7, D4×D7, D4⋊2D7, C2×C7⋊D4, Dic7⋊D4
►On 112 points
(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 68 8 61)(2 67 9 60)(3 66 10 59)(4 65 11 58)(5 64 12 57)(6 63 13 70)(7 62 14 69)(15 39 22 32)(16 38 23 31)(17 37 24 30)(18 36 25 29)(19 35 26 42)(20 34 27 41)(21 33 28 40)(43 110 50 103)(44 109 51 102)(45 108 52 101)(46 107 53 100)(47 106 54 99)(48 105 55 112)(49 104 56 111)(71 86 78 93)(72 85 79 92)(73 98 80 91)(74 97 81 90)(75 96 82 89)(76 95 83 88)(77 94 84 87)
(1 95 50 30)(2 96 51 31)(3 97 52 32)(4 98 53 33)(5 85 54 34)(6 86 55 35)(7 87 56 36)(8 88 43 37)(9 89 44 38)(10 90 45 39)(11 91 46 40)(12 92 47 41)(13 93 48 42)(14 94 49 29)(15 59 81 108)(16 60 82 109)(17 61 83 110)(18 62 84 111)(19 63 71 112)(20 64 72 99)(21 65 73 100)(22 66 74 101)(23 67 75 102)(24 68 76 103)(25 69 77 104)(26 70 78 105)(27 57 79 106)(28 58 80 107)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 71)(11 72)(12 73)(13 74)(14 75)(15 55)(16 56)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 109)(37 110)(38 111)(39 112)(40 99)(41 100)(42 101)(57 98)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)
G:=sub<Sym(112)| (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), (1,68,8,61)(2,67,9,60)(3,66,10,59)(4,65,11,58)(5,64,12,57)(6,63,13,70)(7,62,14,69)(15,39,22,32)(16,38,23,31)(17,37,24,30)(18,36,25,29)(19,35,26,42)(20,34,27,41)(21,33,28,40)(43,110,50,103)(44,109,51,102)(45,108,52,101)(46,107,53,100)(47,106,54,99)(48,105,55,112)(49,104,56,111)(71,86,78,93)(72,85,79,92)(73,98,80,91)(74,97,81,90)(75,96,82,89)(76,95,83,88)(77,94,84,87), (1,95,50,30)(2,96,51,31)(3,97,52,32)(4,98,53,33)(5,85,54,34)(6,86,55,35)(7,87,56,36)(8,88,43,37)(9,89,44,38)(10,90,45,39)(11,91,46,40)(12,92,47,41)(13,93,48,42)(14,94,49,29)(15,59,81,108)(16,60,82,109)(17,61,83,110)(18,62,84,111)(19,63,71,112)(20,64,72,99)(21,65,73,100)(22,66,74,101)(23,67,75,102)(24,68,76,103)(25,69,77,104)(26,70,78,105)(27,57,79,106)(28,58,80,107), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,82)(8,83)(9,84)(10,71)(11,72)(12,73)(13,74)(14,75)(15,55)(16,56)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,99)(41,100)(42,101)(57,98)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)>;
G:=Group( (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), (1,68,8,61)(2,67,9,60)(3,66,10,59)(4,65,11,58)(5,64,12,57)(6,63,13,70)(7,62,14,69)(15,39,22,32)(16,38,23,31)(17,37,24,30)(18,36,25,29)(19,35,26,42)(20,34,27,41)(21,33,28,40)(43,110,50,103)(44,109,51,102)(45,108,52,101)(46,107,53,100)(47,106,54,99)(48,105,55,112)(49,104,56,111)(71,86,78,93)(72,85,79,92)(73,98,80,91)(74,97,81,90)(75,96,82,89)(76,95,83,88)(77,94,84,87), (1,95,50,30)(2,96,51,31)(3,97,52,32)(4,98,53,33)(5,85,54,34)(6,86,55,35)(7,87,56,36)(8,88,43,37)(9,89,44,38)(10,90,45,39)(11,91,46,40)(12,92,47,41)(13,93,48,42)(14,94,49,29)(15,59,81,108)(16,60,82,109)(17,61,83,110)(18,62,84,111)(19,63,71,112)(20,64,72,99)(21,65,73,100)(22,66,74,101)(23,67,75,102)(24,68,76,103)(25,69,77,104)(26,70,78,105)(27,57,79,106)(28,58,80,107), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,82)(8,83)(9,84)(10,71)(11,72)(12,73)(13,74)(14,75)(15,55)(16,56)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,99)(41,100)(42,101)(57,98)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97) );
G=PermutationGroup([[(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)], [(1,68,8,61),(2,67,9,60),(3,66,10,59),(4,65,11,58),(5,64,12,57),(6,63,13,70),(7,62,14,69),(15,39,22,32),(16,38,23,31),(17,37,24,30),(18,36,25,29),(19,35,26,42),(20,34,27,41),(21,33,28,40),(43,110,50,103),(44,109,51,102),(45,108,52,101),(46,107,53,100),(47,106,54,99),(48,105,55,112),(49,104,56,111),(71,86,78,93),(72,85,79,92),(73,98,80,91),(74,97,81,90),(75,96,82,89),(76,95,83,88),(77,94,84,87)], [(1,95,50,30),(2,96,51,31),(3,97,52,32),(4,98,53,33),(5,85,54,34),(6,86,55,35),(7,87,56,36),(8,88,43,37),(9,89,44,38),(10,90,45,39),(11,91,46,40),(12,92,47,41),(13,93,48,42),(14,94,49,29),(15,59,81,108),(16,60,82,109),(17,61,83,110),(18,62,84,111),(19,63,71,112),(20,64,72,99),(21,65,73,100),(22,66,74,101),(23,67,75,102),(24,68,76,103),(25,69,77,104),(26,70,78,105),(27,57,79,106),(28,58,80,107)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,81),(7,82),(8,83),(9,84),(10,71),(11,72),(12,73),(13,74),(14,75),(15,55),(16,56),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(29,102),(30,103),(31,104),(32,105),(33,106),(34,107),(35,108),(36,109),(37,110),(38,111),(39,112),(40,99),(41,100),(42,101),(57,98),(58,85),(59,86),(60,87),(61,88),(62,89),(63,90),(64,91),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97)]])
Dic7⋊D4 is a maximal subgroup of
C42.102D14 C42.104D14 C42⋊12D14 Dic14⋊23D4 C42⋊17D14 C42.118D14 C42.119D14 C24.56D14 C24⋊3D14 C24.33D14 C24.34D14 C24.35D14 C24.36D14 C28⋊(C4○D4) C14.682- 1+4 Dic14⋊19D4 Dic14⋊20D4 C14.342+ 1+4 D7×C4⋊D4 C14.372+ 1+4 C14.722- 1+4 C14.402+ 1+4 C14.422+ 1+4 C14.442+ 1+4 C14.452+ 1+4 C14.462+ 1+4 C14.1152+ 1+4 C14.472+ 1+4 C14.482+ 1+4 C14.492+ 1+4 C4⋊C4.197D14 C14.1212+ 1+4 C14.822- 1+4 C14.642+ 1+4 C14.842- 1+4 C14.662+ 1+4 C14.852- 1+4 C14.862- 1+4 C42.137D14 C42.138D14 C42⋊20D14 C42.145D14 C42⋊26D14 Dic14⋊11D4 C42.168D14 C42⋊28D14 D4×C7⋊D4 C24⋊7D14 C24.42D14 C14.1042- 1+4 C14.1452+ 1+4 (C2×C28)⋊17D4 C14.1082- 1+4
Dic7⋊D4 is a maximal quotient of
C24.44D14 C24.4D14 C24.6D14 C24.9D14 C24.13D14 C23.45D28 C24.14D14 C23.16D28 Dic7⋊(C4⋊C4) (C2×C28).54D4 D14⋊C4⋊7C4 (C2×C4).45D28 C7⋊C8⋊22D4 C4⋊D4⋊D7 C7⋊C8⋊23D4 C7⋊C8⋊5D4 C7⋊C8⋊24D4 C7⋊C8⋊6D4 C7⋊C8.29D4 C7⋊C8.6D4 Dic7⋊D8 (C2×D8).D7 Dic7⋊3SD16 Dic7⋊5SD16 (C7×D4).D4 (C7×Q8).D4 Dic7⋊3Q16 (C2×Q16)⋊D7 M4(2).D14 M4(2).13D14 M4(2).15D14 M4(2).16D14 C24.18D14 C24.20D14 C24.21D14
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 28 | 4 | 14 | 14 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | D4×D7 | D4⋊2D7 |
| kernel | Dic7⋊D4 | Dic7⋊C4 | D14⋊C4 | C23.D7 | C22×Dic7 | C2×C7⋊D4 | D4×C14 | Dic7 | C2×C14 | C2×D4 | C14 | C2×C4 | C23 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 3 | 2 | 3 | 6 | 12 | 3 | 3 |
Matrix representation of Dic7⋊D4 ►in GL4(𝔽29) generated by
| 0 | 28 | 0 | 0 |
| 1 | 22 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 14 | 0 | 0 |
| 19 | 20 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 14 | 0 | 0 |
| 15 | 20 | 0 | 0 |
| 0 | 0 | 11 | 5 |
| 0 | 0 | 22 | 18 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 18 | 24 |
| 0 | 0 | 24 | 11 |
G:=sub<GL(4,GF(29))| [0,1,0,0,28,22,0,0,0,0,1,0,0,0,0,1],[9,19,0,0,14,20,0,0,0,0,1,0,0,0,0,1],[9,15,0,0,14,20,0,0,0,0,11,22,0,0,5,18],[28,0,0,0,0,28,0,0,0,0,18,24,0,0,24,11] >;
Dic7⋊D4 in GAP, Magma, Sage, TeX
{\rm Dic}_7\rtimes D_4 % in TeX
G:=Group("Dic7:D4"); // GroupNames label
G:=SmallGroup(224,134);
// by ID
G=gap.SmallGroup(224,134);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,218,188,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^14=c^4=d^2=1,b^2=a^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations