metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7⋊4D4, C23.14D14, C7⋊D4⋊C4, C7⋊2(C4×D4), C2.2(D4×D7), D14⋊2(C2×C4), C22⋊C4⋊7D7, C22⋊1(C4×D7), D14⋊C4⋊10C2, Dic7⋊C4⋊9C2, Dic7⋊1(C2×C4), (C2×C4).28D14, C14.18(C2×D4), (C4×Dic7)⋊11C2, C14.7(C22×C4), C14.22(C4○D4), C2.2(D4⋊2D7), (C2×C28).51C22, (C2×C14).22C23, (C22×Dic7)⋊1C2, C22.14(C22×D7), (C22×C14).11C22, (C2×Dic7).47C22, (C22×D7).16C22, (C2×C4×D7)⋊9C2, C2.9(C2×C4×D7), (C2×C14)⋊2(C2×C4), (C7×C22⋊C4)⋊9C2, (C2×C7⋊D4).2C2, SmallGroup(224,76)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7⋊4D4
G = < a,b,c,d | a14=c4=d2=1, b2=a7, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 358 in 94 conjugacy classes, 43 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C4×D4, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×C14, C4×Dic7, Dic7⋊C4, D14⋊C4, C7×C22⋊C4, C2×C4×D7, C22×Dic7, C2×C7⋊D4, Dic7⋊4D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, C22×D7, C2×C4×D7, D4×D7, D4⋊2D7, Dic7⋊4D4
►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 40 8 33)(2 39 9 32)(3 38 10 31)(4 37 11 30)(5 36 12 29)(6 35 13 42)(7 34 14 41)(15 68 22 61)(16 67 23 60)(17 66 24 59)(18 65 25 58)(19 64 26 57)(20 63 27 70)(21 62 28 69)(43 99 50 106)(44 112 51 105)(45 111 52 104)(46 110 53 103)(47 109 54 102)(48 108 55 101)(49 107 56 100)(71 98 78 91)(72 97 79 90)(73 96 80 89)(74 95 81 88)(75 94 82 87)(76 93 83 86)(77 92 84 85)
(1 77 69 108)(2 76 70 107)(3 75 57 106)(4 74 58 105)(5 73 59 104)(6 72 60 103)(7 71 61 102)(8 84 62 101)(9 83 63 100)(10 82 64 99)(11 81 65 112)(12 80 66 111)(13 79 67 110)(14 78 68 109)(15 47 34 98)(16 46 35 97)(17 45 36 96)(18 44 37 95)(19 43 38 94)(20 56 39 93)(21 55 40 92)(22 54 41 91)(23 53 42 90)(24 52 29 89)(25 51 30 88)(26 50 31 87)(27 49 32 86)(28 48 33 85)
(1 101)(2 102)(3 103)(4 104)(5 105)(6 106)(7 107)(8 108)(9 109)(10 110)(11 111)(12 112)(13 99)(14 100)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(41 49)(42 50)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 71)
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), (1,40,8,33)(2,39,9,32)(3,38,10,31)(4,37,11,30)(5,36,12,29)(6,35,13,42)(7,34,14,41)(15,68,22,61)(16,67,23,60)(17,66,24,59)(18,65,25,58)(19,64,26,57)(20,63,27,70)(21,62,28,69)(43,99,50,106)(44,112,51,105)(45,111,52,104)(46,110,53,103)(47,109,54,102)(48,108,55,101)(49,107,56,100)(71,98,78,91)(72,97,79,90)(73,96,80,89)(74,95,81,88)(75,94,82,87)(76,93,83,86)(77,92,84,85), (1,77,69,108)(2,76,70,107)(3,75,57,106)(4,74,58,105)(5,73,59,104)(6,72,60,103)(7,71,61,102)(8,84,62,101)(9,83,63,100)(10,82,64,99)(11,81,65,112)(12,80,66,111)(13,79,67,110)(14,78,68,109)(15,47,34,98)(16,46,35,97)(17,45,36,96)(18,44,37,95)(19,43,38,94)(20,56,39,93)(21,55,40,92)(22,54,41,91)(23,53,42,90)(24,52,29,89)(25,51,30,88)(26,50,31,87)(27,49,32,86)(28,48,33,85), (1,101)(2,102)(3,103)(4,104)(5,105)(6,106)(7,107)(8,108)(9,109)(10,110)(11,111)(12,112)(13,99)(14,100)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(41,49)(42,50)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,71)>;
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), (1,40,8,33)(2,39,9,32)(3,38,10,31)(4,37,11,30)(5,36,12,29)(6,35,13,42)(7,34,14,41)(15,68,22,61)(16,67,23,60)(17,66,24,59)(18,65,25,58)(19,64,26,57)(20,63,27,70)(21,62,28,69)(43,99,50,106)(44,112,51,105)(45,111,52,104)(46,110,53,103)(47,109,54,102)(48,108,55,101)(49,107,56,100)(71,98,78,91)(72,97,79,90)(73,96,80,89)(74,95,81,88)(75,94,82,87)(76,93,83,86)(77,92,84,85), (1,77,69,108)(2,76,70,107)(3,75,57,106)(4,74,58,105)(5,73,59,104)(6,72,60,103)(7,71,61,102)(8,84,62,101)(9,83,63,100)(10,82,64,99)(11,81,65,112)(12,80,66,111)(13,79,67,110)(14,78,68,109)(15,47,34,98)(16,46,35,97)(17,45,36,96)(18,44,37,95)(19,43,38,94)(20,56,39,93)(21,55,40,92)(22,54,41,91)(23,53,42,90)(24,52,29,89)(25,51,30,88)(26,50,31,87)(27,49,32,86)(28,48,33,85), (1,101)(2,102)(3,103)(4,104)(5,105)(6,106)(7,107)(8,108)(9,109)(10,110)(11,111)(12,112)(13,99)(14,100)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(41,49)(42,50)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,71) );
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)], [(1,40,8,33),(2,39,9,32),(3,38,10,31),(4,37,11,30),(5,36,12,29),(6,35,13,42),(7,34,14,41),(15,68,22,61),(16,67,23,60),(17,66,24,59),(18,65,25,58),(19,64,26,57),(20,63,27,70),(21,62,28,69),(43,99,50,106),(44,112,51,105),(45,111,52,104),(46,110,53,103),(47,109,54,102),(48,108,55,101),(49,107,56,100),(71,98,78,91),(72,97,79,90),(73,96,80,89),(74,95,81,88),(75,94,82,87),(76,93,83,86),(77,92,84,85)], [(1,77,69,108),(2,76,70,107),(3,75,57,106),(4,74,58,105),(5,73,59,104),(6,72,60,103),(7,71,61,102),(8,84,62,101),(9,83,63,100),(10,82,64,99),(11,81,65,112),(12,80,66,111),(13,79,67,110),(14,78,68,109),(15,47,34,98),(16,46,35,97),(17,45,36,96),(18,44,37,95),(19,43,38,94),(20,56,39,93),(21,55,40,92),(22,54,41,91),(23,53,42,90),(24,52,29,89),(25,51,30,88),(26,50,31,87),(27,49,32,86),(28,48,33,85)], [(1,101),(2,102),(3,103),(4,104),(5,105),(6,106),(7,107),(8,108),(9,109),(10,110),(11,111),(12,112),(13,99),(14,100),(15,93),(16,94),(17,95),(18,96),(19,97),(20,98),(21,85),(22,86),(23,87),(24,88),(25,89),(26,90),(27,91),(28,92),(29,51),(30,52),(31,53),(32,54),(33,55),(34,56),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(41,49),(42,50),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,78),(64,79),(65,80),(66,81),(67,82),(68,83),(69,84),(70,71)]])
Dic7⋊4D4 is a maximal subgroup of
C24.24D14 C24.27D14 C24.31D14 C42.188D14 C42.91D14 C42⋊10D14 C42.96D14 C4×D4⋊2D7 C42.104D14 C4×D4×D7 C42⋊11D14 C42.108D14 Dic14⋊23D4 C42.119D14 C24.56D14 C24⋊3D14 C24.33D14 C24.35D14 C28⋊(C4○D4) Dic14⋊20D4 C14.342+ 1+4 C4⋊C4⋊21D14 C14.402+ 1+4 C14.422+ 1+4 C14.432+ 1+4 C14.442+ 1+4 C22⋊Q8⋊25D7 C4⋊C4⋊26D14 Dic14⋊21D4 C14.1182+ 1+4 C14.522+ 1+4 C14.562+ 1+4 C14.572+ 1+4 C4⋊C4.197D14 C14.1212+ 1+4 C14.822- 1+4 C4⋊C4⋊28D14 C14.1222+ 1+4 C14.832- 1+4 C14.642+ 1+4 C14.662+ 1+4 C14.672+ 1+4 C14.852- 1+4 C42.233D14 C42.137D14 C42.138D14 Dic14⋊10D4 C42⋊20D14 C42⋊21D14 C42.234D14 C42.143D14 C42.160D14 C42.189D14 C42.161D14 C42.162D14 C42.163D14 C42.164D14
Dic7⋊4D4 is a maximal quotient of
C14.(C4×Q8) Dic7⋊C42 Dic7⋊C4⋊C4 C14.(C4×D4) D14⋊C42 D14⋊C4⋊C4 D14⋊C4⋊5C4 C2.(C4×D28) C7⋊D4⋊C8 D14⋊2M4(2) Dic7⋊M4(2) C7⋊C8⋊26D4 Dic7⋊4D8 D4.D7⋊C4 Dic7⋊6SD16 D4⋊D7⋊C4 Dic7⋊7SD16 C7⋊Q16⋊C4 Dic7⋊4Q16 Q8⋊D7⋊C4 M4(2).22D14 C42.196D14 C22⋊C4×Dic7 C24.3D14 C24.4D14 C24.46D14 C24.12D14 C24.13D14 C23.45D28
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | 7 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | D14 | C4×D7 | D4×D7 | D4⋊2D7 |
| kernel | Dic7⋊4D4 | C4×Dic7 | Dic7⋊C4 | D14⋊C4 | C7×C22⋊C4 | C2×C4×D7 | C22×Dic7 | C2×C7⋊D4 | C7⋊D4 | Dic7 | C22⋊C4 | C14 | C2×C4 | C23 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 3 | 2 | 6 | 3 | 12 | 3 | 3 |
Matrix representation of Dic7⋊4D4 ►in GL4(𝔽29) generated by
| 1 | 20 | 0 | 0 |
| 1 | 21 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 7 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 26 | 0 | 0 |
| 19 | 0 | 0 | 0 |
| 0 | 0 | 1 | 28 |
| 0 | 0 | 2 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 2 | 28 |
G:=sub<GL(4,GF(29))| [1,1,0,0,20,21,0,0,0,0,28,0,0,0,0,28],[0,4,0,0,7,0,0,0,0,0,17,0,0,0,0,17],[0,19,0,0,26,0,0,0,0,0,1,2,0,0,28,28],[28,0,0,0,0,28,0,0,0,0,1,2,0,0,0,28] >;
Dic7⋊4D4 in GAP, Magma, Sage, TeX
{\rm Dic}_7\rtimes_4D_4 % in TeX
G:=Group("Dic7:4D4"); // GroupNames label
G:=SmallGroup(224,76);
// by ID
G=gap.SmallGroup(224,76);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,188,50,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^14=c^4=d^2=1,b^2=a^7,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations