metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD16⋊3D7, D4.5D14, D14.2D4, C8.11D14, Q8.2D14, C28.7C23, C56.11C22, Dic7.13D4, D28.3C22, Dic14.3C22, D4⋊D7⋊4C2, (C8×D7)⋊5C2, C7⋊3(C4○D8), C56⋊C2⋊6C2, C7⋊Q16⋊2C2, C2.21(D4×D7), C7⋊C8.6C22, D4⋊2D7⋊3C2, Q8⋊2D7⋊2C2, (C7×SD16)⋊4C2, C14.33(C2×D4), C4.7(C22×D7), (C7×D4).5C22, (C7×Q8).2C22, (C4×D7).10C22, SmallGroup(224,111)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16⋊3D7
G = < a,b,c,d | a8=b2=c7=d2=1, bab=a3, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 278 in 62 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, D4, Q8, Q8, D7, C14, C14, C2×C8, D8, SD16, SD16, Q16, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C4○D8, C7⋊C8, C56, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C7×D4, C7×Q8, C8×D7, C56⋊C2, D4⋊D7, C7⋊Q16, C7×SD16, D4⋊2D7, Q8⋊2D7, SD16⋊3D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4○D8, C22×D7, D4×D7, SD16⋊3D7
►On 112 points
Generators in S112
(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 32)(2 27)(3 30)(4 25)(5 28)(6 31)(7 26)(8 29)(9 43)(10 46)(11 41)(12 44)(13 47)(14 42)(15 45)(16 48)(17 67)(18 70)(19 65)(20 68)(21 71)(22 66)(23 69)(24 72)(33 107)(34 110)(35 105)(36 108)(37 111)(38 106)(39 109)(40 112)(49 98)(50 101)(51 104)(52 99)(53 102)(54 97)(55 100)(56 103)(57 87)(58 82)(59 85)(60 88)(61 83)(62 86)(63 81)(64 84)(73 95)(74 90)(75 93)(76 96)(77 91)(78 94)(79 89)(80 92)
(1 89 107 14 17 61 50)(2 90 108 15 18 62 51)(3 91 109 16 19 63 52)(4 92 110 9 20 64 53)(5 93 111 10 21 57 54)(6 94 112 11 22 58 55)(7 95 105 12 23 59 56)(8 96 106 13 24 60 49)(25 80 34 43 68 84 102)(26 73 35 44 69 85 103)(27 74 36 45 70 86 104)(28 75 37 46 71 87 97)(29 76 38 47 72 88 98)(30 77 39 48 65 81 99)(31 78 40 41 66 82 100)(32 79 33 42 67 83 101)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 97)(7 98)(8 99)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 52)(26 53)(27 54)(28 55)(29 56)(30 49)(31 50)(32 51)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 73)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 105)(81 96)(82 89)(83 90)(84 91)(85 92)(86 93)(87 94)(88 95)
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), (1,32)(2,27)(3,30)(4,25)(5,28)(6,31)(7,26)(8,29)(9,43)(10,46)(11,41)(12,44)(13,47)(14,42)(15,45)(16,48)(17,67)(18,70)(19,65)(20,68)(21,71)(22,66)(23,69)(24,72)(33,107)(34,110)(35,105)(36,108)(37,111)(38,106)(39,109)(40,112)(49,98)(50,101)(51,104)(52,99)(53,102)(54,97)(55,100)(56,103)(57,87)(58,82)(59,85)(60,88)(61,83)(62,86)(63,81)(64,84)(73,95)(74,90)(75,93)(76,96)(77,91)(78,94)(79,89)(80,92), (1,89,107,14,17,61,50)(2,90,108,15,18,62,51)(3,91,109,16,19,63,52)(4,92,110,9,20,64,53)(5,93,111,10,21,57,54)(6,94,112,11,22,58,55)(7,95,105,12,23,59,56)(8,96,106,13,24,60,49)(25,80,34,43,68,84,102)(26,73,35,44,69,85,103)(27,74,36,45,70,86,104)(28,75,37,46,71,87,97)(29,76,38,47,72,88,98)(30,77,39,48,65,81,99)(31,78,40,41,66,82,100)(32,79,33,42,67,83,101), (1,100)(2,101)(3,102)(4,103)(5,104)(6,97)(7,98)(8,99)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,52)(26,53)(27,54)(28,55)(29,56)(30,49)(31,50)(32,51)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,73)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,105)(81,96)(82,89)(83,90)(84,91)(85,92)(86,93)(87,94)(88,95)>;
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), (1,32)(2,27)(3,30)(4,25)(5,28)(6,31)(7,26)(8,29)(9,43)(10,46)(11,41)(12,44)(13,47)(14,42)(15,45)(16,48)(17,67)(18,70)(19,65)(20,68)(21,71)(22,66)(23,69)(24,72)(33,107)(34,110)(35,105)(36,108)(37,111)(38,106)(39,109)(40,112)(49,98)(50,101)(51,104)(52,99)(53,102)(54,97)(55,100)(56,103)(57,87)(58,82)(59,85)(60,88)(61,83)(62,86)(63,81)(64,84)(73,95)(74,90)(75,93)(76,96)(77,91)(78,94)(79,89)(80,92), (1,89,107,14,17,61,50)(2,90,108,15,18,62,51)(3,91,109,16,19,63,52)(4,92,110,9,20,64,53)(5,93,111,10,21,57,54)(6,94,112,11,22,58,55)(7,95,105,12,23,59,56)(8,96,106,13,24,60,49)(25,80,34,43,68,84,102)(26,73,35,44,69,85,103)(27,74,36,45,70,86,104)(28,75,37,46,71,87,97)(29,76,38,47,72,88,98)(30,77,39,48,65,81,99)(31,78,40,41,66,82,100)(32,79,33,42,67,83,101), (1,100)(2,101)(3,102)(4,103)(5,104)(6,97)(7,98)(8,99)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,52)(26,53)(27,54)(28,55)(29,56)(30,49)(31,50)(32,51)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,73)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,105)(81,96)(82,89)(83,90)(84,91)(85,92)(86,93)(87,94)(88,95) );
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)], [(1,32),(2,27),(3,30),(4,25),(5,28),(6,31),(7,26),(8,29),(9,43),(10,46),(11,41),(12,44),(13,47),(14,42),(15,45),(16,48),(17,67),(18,70),(19,65),(20,68),(21,71),(22,66),(23,69),(24,72),(33,107),(34,110),(35,105),(36,108),(37,111),(38,106),(39,109),(40,112),(49,98),(50,101),(51,104),(52,99),(53,102),(54,97),(55,100),(56,103),(57,87),(58,82),(59,85),(60,88),(61,83),(62,86),(63,81),(64,84),(73,95),(74,90),(75,93),(76,96),(77,91),(78,94),(79,89),(80,92)], [(1,89,107,14,17,61,50),(2,90,108,15,18,62,51),(3,91,109,16,19,63,52),(4,92,110,9,20,64,53),(5,93,111,10,21,57,54),(6,94,112,11,22,58,55),(7,95,105,12,23,59,56),(8,96,106,13,24,60,49),(25,80,34,43,68,84,102),(26,73,35,44,69,85,103),(27,74,36,45,70,86,104),(28,75,37,46,71,87,97),(29,76,38,47,72,88,98),(30,77,39,48,65,81,99),(31,78,40,41,66,82,100),(32,79,33,42,67,83,101)], [(1,100),(2,101),(3,102),(4,103),(5,104),(6,97),(7,98),(8,99),(9,44),(10,45),(11,46),(12,47),(13,48),(14,41),(15,42),(16,43),(17,40),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(25,52),(26,53),(27,54),(28,55),(29,56),(30,49),(31,50),(32,51),(57,74),(58,75),(59,76),(60,77),(61,78),(62,79),(63,80),(64,73),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,105),(81,96),(82,89),(83,90),(84,91),(85,92),(86,93),(87,94),(88,95)]])
SD16⋊3D7 is a maximal subgroup of
D28.29D4 D7×C4○D8 D8⋊11D14 SD16⋊D14 D8⋊5D14 D56⋊C22 D28.44D4
SD16⋊3D7 is a maximal quotient of
Dic7⋊4D8 D4.Dic14 (C8×Dic7)⋊C2 D4⋊2D7⋊C4 D14⋊D8 C8⋊Dic7⋊C2 D4⋊3D28 D28.D4 Dic7⋊4Q16 Dic14.11D4 Q8.2Dic14 Q8⋊Dic7⋊C2 Q8⋊2D7⋊C4 Q8.D28 D14⋊Q16 (C2×C8).D14 Dic7⋊8SD16 Dic14.Q8 C56.8Q8 (C8×D7)⋊C4 C8⋊8D28 C4.Q8⋊D7 C28.(C4○D4) D28.Q8 SD16×Dic7 (C7×D4).D4 (C7×Q8).D4 C56.43D4 C56⋊14D4 D28⋊7D4 Dic14.16D4
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | 14E | 14F | 28A | 28B | 28C | 28D | 28E | 28F | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 4 | 14 | 28 | 2 | 4 | 7 | 7 | 28 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 8 | 8 | 8 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 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 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | C4○D8 | D4×D7 | SD16⋊3D7 |
| kernel | SD16⋊3D7 | C8×D7 | C56⋊C2 | D4⋊D7 | C7⋊Q16 | C7×SD16 | D4⋊2D7 | Q8⋊2D7 | Dic7 | D14 | SD16 | C8 | D4 | Q8 | C7 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 3 | 6 |
Matrix representation of SD16⋊3D7 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 81 |
| 0 | 0 | 60 | 87 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 85 |
| 0 | 0 | 4 | 0 |
| 103 | 1 | 0 | 0 |
| 111 | 34 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 24 | 89 | 0 | 0 |
| 104 | 89 | 0 | 0 |
| 0 | 0 | 15 | 50 |
| 0 | 0 | 104 | 98 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,0,60,0,0,81,87],[1,0,0,0,0,1,0,0,0,0,0,4,0,0,85,0],[103,111,0,0,1,34,0,0,0,0,1,0,0,0,0,1],[24,104,0,0,89,89,0,0,0,0,15,104,0,0,50,98] >;
SD16⋊3D7 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes_3D_7 % in TeX
G:=Group("SD16:3D7"); // GroupNames label
G:=SmallGroup(224,111);
// by ID
G=gap.SmallGroup(224,111);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,362,116,86,297,159,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^7=d^2=1,b*a*b=a^3,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations