metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.Dic7, Q8.Dic7, C28.42C23, (C7×D4).C4, (C7×Q8).C4, C7⋊3(C8○D4), C4○D4.3D7, C28.15(C2×C4), (C2×C4).58D14, C7⋊C8.13C22, C4.Dic7⋊8C2, C4.5(C2×Dic7), C4.42(C22×D7), (C2×C28).41C22, C14.27(C22×C4), C2.8(C22×Dic7), C22.1(C2×Dic7), (C2×C7⋊C8)⋊7C2, (C2×C14).7(C2×C4), (C7×C4○D4).2C2, SmallGroup(224,143)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.Dic7
G = < a,b,c,d | a4=b2=1, c14=a2, d2=a2c7, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c13 >
Subgroups: 134 in 62 conjugacy classes, 45 normal (12 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C7, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C14, C14 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C28, C28 [×3], C2×C14 [×3], C8○D4, C7⋊C8, C7⋊C8 [×3], C2×C28 [×3], C7×D4 [×3], C7×Q8, C2×C7⋊C8 [×3], C4.Dic7 [×3], C7×C4○D4, D4.Dic7
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D7, C22×C4, Dic7 [×4], D14 [×3], C8○D4, C2×Dic7 [×6], C22×D7, C22×Dic7, D4.Dic7
►On 112 points
Generators in S112
(1 22 15 8)(2 23 16 9)(3 24 17 10)(4 25 18 11)(5 26 19 12)(6 27 20 13)(7 28 21 14)(29 36 43 50)(30 37 44 51)(31 38 45 52)(32 39 46 53)(33 40 47 54)(34 41 48 55)(35 42 49 56)(57 78 71 64)(58 79 72 65)(59 80 73 66)(60 81 74 67)(61 82 75 68)(62 83 76 69)(63 84 77 70)(85 92 99 106)(86 93 100 107)(87 94 101 108)(88 95 102 109)(89 96 103 110)(90 97 104 111)(91 98 105 112)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)
(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 64 22 57 15 78 8 71)(2 77 23 70 16 63 9 84)(3 62 24 83 17 76 10 69)(4 75 25 68 18 61 11 82)(5 60 26 81 19 74 12 67)(6 73 27 66 20 59 13 80)(7 58 28 79 21 72 14 65)(29 88 50 109 43 102 36 95)(30 101 51 94 44 87 37 108)(31 86 52 107 45 100 38 93)(32 99 53 92 46 85 39 106)(33 112 54 105 47 98 40 91)(34 97 55 90 48 111 41 104)(35 110 56 103 49 96 42 89)
G:=sub<Sym(112)| (1,22,15,8)(2,23,16,9)(3,24,17,10)(4,25,18,11)(5,26,19,12)(6,27,20,13)(7,28,21,14)(29,36,43,50)(30,37,44,51)(31,38,45,52)(32,39,46,53)(33,40,47,54)(34,41,48,55)(35,42,49,56)(57,78,71,64)(58,79,72,65)(59,80,73,66)(60,81,74,67)(61,82,75,68)(62,83,76,69)(63,84,77,70)(85,92,99,106)(86,93,100,107)(87,94,101,108)(88,95,102,109)(89,96,103,110)(90,97,104,111)(91,98,105,112), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98), (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), (1,64,22,57,15,78,8,71)(2,77,23,70,16,63,9,84)(3,62,24,83,17,76,10,69)(4,75,25,68,18,61,11,82)(5,60,26,81,19,74,12,67)(6,73,27,66,20,59,13,80)(7,58,28,79,21,72,14,65)(29,88,50,109,43,102,36,95)(30,101,51,94,44,87,37,108)(31,86,52,107,45,100,38,93)(32,99,53,92,46,85,39,106)(33,112,54,105,47,98,40,91)(34,97,55,90,48,111,41,104)(35,110,56,103,49,96,42,89)>;
G:=Group( (1,22,15,8)(2,23,16,9)(3,24,17,10)(4,25,18,11)(5,26,19,12)(6,27,20,13)(7,28,21,14)(29,36,43,50)(30,37,44,51)(31,38,45,52)(32,39,46,53)(33,40,47,54)(34,41,48,55)(35,42,49,56)(57,78,71,64)(58,79,72,65)(59,80,73,66)(60,81,74,67)(61,82,75,68)(62,83,76,69)(63,84,77,70)(85,92,99,106)(86,93,100,107)(87,94,101,108)(88,95,102,109)(89,96,103,110)(90,97,104,111)(91,98,105,112), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98), (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), (1,64,22,57,15,78,8,71)(2,77,23,70,16,63,9,84)(3,62,24,83,17,76,10,69)(4,75,25,68,18,61,11,82)(5,60,26,81,19,74,12,67)(6,73,27,66,20,59,13,80)(7,58,28,79,21,72,14,65)(29,88,50,109,43,102,36,95)(30,101,51,94,44,87,37,108)(31,86,52,107,45,100,38,93)(32,99,53,92,46,85,39,106)(33,112,54,105,47,98,40,91)(34,97,55,90,48,111,41,104)(35,110,56,103,49,96,42,89) );
G=PermutationGroup([(1,22,15,8),(2,23,16,9),(3,24,17,10),(4,25,18,11),(5,26,19,12),(6,27,20,13),(7,28,21,14),(29,36,43,50),(30,37,44,51),(31,38,45,52),(32,39,46,53),(33,40,47,54),(34,41,48,55),(35,42,49,56),(57,78,71,64),(58,79,72,65),(59,80,73,66),(60,81,74,67),(61,82,75,68),(62,83,76,69),(63,84,77,70),(85,92,99,106),(86,93,100,107),(87,94,101,108),(88,95,102,109),(89,96,103,110),(90,97,104,111),(91,98,105,112)], [(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,29),(20,30),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,105),(64,106),(65,107),(66,108),(67,109),(68,110),(69,111),(70,112),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98)], [(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)], [(1,64,22,57,15,78,8,71),(2,77,23,70,16,63,9,84),(3,62,24,83,17,76,10,69),(4,75,25,68,18,61,11,82),(5,60,26,81,19,74,12,67),(6,73,27,66,20,59,13,80),(7,58,28,79,21,72,14,65),(29,88,50,109,43,102,36,95),(30,101,51,94,44,87,37,108),(31,86,52,107,45,100,38,93),(32,99,53,92,46,85,39,106),(33,112,54,105,47,98,40,91),(34,97,55,90,48,111,41,104),(35,110,56,103,49,96,42,89)])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 14A | 14B | 14C | 14D | ··· | 14L | 28A | ··· | 28F | 28G | ··· | 28O |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 7 | 7 | 7 | 7 | 14 | ··· | 14 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D7 | D14 | Dic7 | Dic7 | C8○D4 | D4.Dic7 |
| kernel | D4.Dic7 | C2×C7⋊C8 | C4.Dic7 | C7×C4○D4 | C7×D4 | C7×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C7 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 3 | 9 | 9 | 3 | 4 | 6 |
Matrix representation of D4.Dic7 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 98 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 98 |
| 0 | 0 | 15 | 0 |
| 103 | 1 | 0 | 0 |
| 13 | 89 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 33 | 12 | 0 | 0 |
| 60 | 80 | 0 | 0 |
| 0 | 0 | 44 | 0 |
| 0 | 0 | 0 | 44 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,15,0,0,0,0,98],[1,0,0,0,0,1,0,0,0,0,0,15,0,0,98,0],[103,13,0,0,1,89,0,0,0,0,15,0,0,0,0,15],[33,60,0,0,12,80,0,0,0,0,44,0,0,0,0,44] >;
D4.Dic7 in GAP, Magma, Sage, TeX
D_4.{\rm Dic}_7 % in TeX
G:=Group("D4.Dic7"); // GroupNames label
G:=SmallGroup(224,143);
// by ID
G=gap.SmallGroup(224,143);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,188,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^14=a^2,d^2=a^2*c^7,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^13>;
// generators/relations