metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.Dic7, D4.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 Q8.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, C4, C4, C22, C7, C8, C2×C4, D4, Q8, C14, C14, C2×C8, M4(2), C4○D4, C28, C28, C2×C14, C8○D4, C7⋊C8, C7⋊C8, C2×C28, C7×D4, C7×Q8, C2×C7⋊C8, C4.Dic7, C7×C4○D4, Q8.Dic7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, Dic7, D14, C8○D4, C2×Dic7, C22×D7, C22×Dic7, Q8.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 56)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)
(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 57 22 78 15 71 8 64)(2 70 23 63 16 84 9 77)(3 83 24 76 17 69 10 62)(4 68 25 61 18 82 11 75)(5 81 26 74 19 67 12 60)(6 66 27 59 20 80 13 73)(7 79 28 72 21 65 14 58)(29 105 50 98 43 91 36 112)(30 90 51 111 44 104 37 97)(31 103 52 96 45 89 38 110)(32 88 53 109 46 102 39 95)(33 101 54 94 47 87 40 108)(34 86 55 107 48 100 41 93)(35 99 56 92 49 85 42 106)
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), (1,56)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,99)(65,100)(66,101)(67,102)(68,103)(69,104)(70,105)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91), (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), (1,57,22,78,15,71,8,64)(2,70,23,63,16,84,9,77)(3,83,24,76,17,69,10,62)(4,68,25,61,18,82,11,75)(5,81,26,74,19,67,12,60)(6,66,27,59,20,80,13,73)(7,79,28,72,21,65,14,58)(29,105,50,98,43,91,36,112)(30,90,51,111,44,104,37,97)(31,103,52,96,45,89,38,110)(32,88,53,109,46,102,39,95)(33,101,54,94,47,87,40,108)(34,86,55,107,48,100,41,93)(35,99,56,92,49,85,42,106)>;
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), (1,56)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,99)(65,100)(66,101)(67,102)(68,103)(69,104)(70,105)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91), (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), (1,57,22,78,15,71,8,64)(2,70,23,63,16,84,9,77)(3,83,24,76,17,69,10,62)(4,68,25,61,18,82,11,75)(5,81,26,74,19,67,12,60)(6,66,27,59,20,80,13,73)(7,79,28,72,21,65,14,58)(29,105,50,98,43,91,36,112)(30,90,51,111,44,104,37,97)(31,103,52,96,45,89,38,110)(32,88,53,109,46,102,39,95)(33,101,54,94,47,87,40,108)(34,86,55,107,48,100,41,93)(35,99,56,92,49,85,42,106) );
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)], [(1,56),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(57,92),(58,93),(59,94),(60,95),(61,96),(62,97),(63,98),(64,99),(65,100),(66,101),(67,102),(68,103),(69,104),(70,105),(71,106),(72,107),(73,108),(74,109),(75,110),(76,111),(77,112),(78,85),(79,86),(80,87),(81,88),(82,89),(83,90),(84,91)], [(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)], [(1,57,22,78,15,71,8,64),(2,70,23,63,16,84,9,77),(3,83,24,76,17,69,10,62),(4,68,25,61,18,82,11,75),(5,81,26,74,19,67,12,60),(6,66,27,59,20,80,13,73),(7,79,28,72,21,65,14,58),(29,105,50,98,43,91,36,112),(30,90,51,111,44,104,37,97),(31,103,52,96,45,89,38,110),(32,88,53,109,46,102,39,95),(33,101,54,94,47,87,40,108),(34,86,55,107,48,100,41,93),(35,99,56,92,49,85,42,106)]])
Q8.Dic7 is a maximal subgroup of
M4(2).22D14 C42.196D14 D8⋊5Dic7 D8⋊4Dic7 M4(2).D14 M4(2).13D14 M4(2).15D14 M4(2).16D14 D7×C8○D4 C56.49C23 C28.76C24 D28.32C23 D28.33C23 D28.34C23 D28.35C23
Q8.Dic7 is a maximal quotient of
C28.5C42 C42.43D14 C42.187D14 D4×C7⋊C8 C42.47D14 C28⋊3M4(2) Q8×C7⋊C8 C42.210D14 (D4×C14).11C4
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 | Q8.Dic7 |
| kernel | Q8.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 Q8.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] >;
Q8.Dic7 in GAP, Magma, Sage, TeX
Q_8.{\rm Dic}_7 % in TeX
G:=Group("Q8.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