metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.2D14, SD16⋊2D7, D14.8D4, D4.4D14, Q8.1D14, Dic28⋊6C2, C28.6C23, C56.9C22, Dic7.10D4, Dic14.2C22, (Q8×D7)⋊2C2, C8⋊D7⋊2C2, D4.D7⋊4C2, D4⋊2D7.C2, C7⋊Q16⋊1C2, C2.20(D4×D7), C7⋊C8.1C22, (C7×SD16)⋊2C2, C14.32(C2×D4), C7⋊2(C8.C22), C4.6(C22×D7), (C4×D7).3C22, (C7×D4).4C22, (C7×Q8).1C22, SmallGroup(224,110)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16⋊D7
G = < a,b,c,d | a8=b2=c7=d2=1, bab=a3, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 246 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, D4, Q8, Q8, D7, C14, C14, M4(2), SD16, SD16, Q16, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, C2×C14, C8.C22, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, C2×Dic7, C7⋊D4, C7×D4, C7×Q8, C8⋊D7, Dic28, D4.D7, C7⋊Q16, C7×SD16, D4⋊2D7, Q8×D7, SD16⋊D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C8.C22, C22×D7, D4×D7, SD16⋊D7
►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 25)(2 28)(3 31)(4 26)(5 29)(6 32)(7 27)(8 30)(9 69)(10 72)(11 67)(12 70)(13 65)(14 68)(15 71)(16 66)(17 50)(18 53)(19 56)(20 51)(21 54)(22 49)(23 52)(24 55)(33 87)(34 82)(35 85)(36 88)(37 83)(38 86)(39 81)(40 84)(41 89)(42 92)(43 95)(44 90)(45 93)(46 96)(47 91)(48 94)(57 102)(58 97)(59 100)(60 103)(61 98)(62 101)(63 104)(64 99)(73 110)(74 105)(75 108)(76 111)(77 106)(78 109)(79 112)(80 107)
(1 105 17 104 93 72 40)(2 106 18 97 94 65 33)(3 107 19 98 95 66 34)(4 108 20 99 96 67 35)(5 109 21 100 89 68 36)(6 110 22 101 90 69 37)(7 111 23 102 91 70 38)(8 112 24 103 92 71 39)(9 83 32 73 49 62 44)(10 84 25 74 50 63 45)(11 85 26 75 51 64 46)(12 86 27 76 52 57 47)(13 87 28 77 53 58 48)(14 88 29 78 54 59 41)(15 81 30 79 55 60 42)(16 82 31 80 56 61 43)
(1 40)(2 37)(3 34)(4 39)(5 36)(6 33)(7 38)(8 35)(9 73)(10 78)(11 75)(12 80)(13 77)(14 74)(15 79)(16 76)(17 93)(18 90)(19 95)(20 92)(21 89)(22 94)(23 91)(24 96)(25 88)(26 85)(27 82)(28 87)(29 84)(30 81)(31 86)(32 83)(41 50)(42 55)(43 52)(44 49)(45 54)(46 51)(47 56)(48 53)(57 61)(59 63)(65 110)(66 107)(67 112)(68 109)(69 106)(70 111)(71 108)(72 105)(97 101)(99 103)
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), (1,25)(2,28)(3,31)(4,26)(5,29)(6,32)(7,27)(8,30)(9,69)(10,72)(11,67)(12,70)(13,65)(14,68)(15,71)(16,66)(17,50)(18,53)(19,56)(20,51)(21,54)(22,49)(23,52)(24,55)(33,87)(34,82)(35,85)(36,88)(37,83)(38,86)(39,81)(40,84)(41,89)(42,92)(43,95)(44,90)(45,93)(46,96)(47,91)(48,94)(57,102)(58,97)(59,100)(60,103)(61,98)(62,101)(63,104)(64,99)(73,110)(74,105)(75,108)(76,111)(77,106)(78,109)(79,112)(80,107), (1,105,17,104,93,72,40)(2,106,18,97,94,65,33)(3,107,19,98,95,66,34)(4,108,20,99,96,67,35)(5,109,21,100,89,68,36)(6,110,22,101,90,69,37)(7,111,23,102,91,70,38)(8,112,24,103,92,71,39)(9,83,32,73,49,62,44)(10,84,25,74,50,63,45)(11,85,26,75,51,64,46)(12,86,27,76,52,57,47)(13,87,28,77,53,58,48)(14,88,29,78,54,59,41)(15,81,30,79,55,60,42)(16,82,31,80,56,61,43), (1,40)(2,37)(3,34)(4,39)(5,36)(6,33)(7,38)(8,35)(9,73)(10,78)(11,75)(12,80)(13,77)(14,74)(15,79)(16,76)(17,93)(18,90)(19,95)(20,92)(21,89)(22,94)(23,91)(24,96)(25,88)(26,85)(27,82)(28,87)(29,84)(30,81)(31,86)(32,83)(41,50)(42,55)(43,52)(44,49)(45,54)(46,51)(47,56)(48,53)(57,61)(59,63)(65,110)(66,107)(67,112)(68,109)(69,106)(70,111)(71,108)(72,105)(97,101)(99,103)>;
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), (1,25)(2,28)(3,31)(4,26)(5,29)(6,32)(7,27)(8,30)(9,69)(10,72)(11,67)(12,70)(13,65)(14,68)(15,71)(16,66)(17,50)(18,53)(19,56)(20,51)(21,54)(22,49)(23,52)(24,55)(33,87)(34,82)(35,85)(36,88)(37,83)(38,86)(39,81)(40,84)(41,89)(42,92)(43,95)(44,90)(45,93)(46,96)(47,91)(48,94)(57,102)(58,97)(59,100)(60,103)(61,98)(62,101)(63,104)(64,99)(73,110)(74,105)(75,108)(76,111)(77,106)(78,109)(79,112)(80,107), (1,105,17,104,93,72,40)(2,106,18,97,94,65,33)(3,107,19,98,95,66,34)(4,108,20,99,96,67,35)(5,109,21,100,89,68,36)(6,110,22,101,90,69,37)(7,111,23,102,91,70,38)(8,112,24,103,92,71,39)(9,83,32,73,49,62,44)(10,84,25,74,50,63,45)(11,85,26,75,51,64,46)(12,86,27,76,52,57,47)(13,87,28,77,53,58,48)(14,88,29,78,54,59,41)(15,81,30,79,55,60,42)(16,82,31,80,56,61,43), (1,40)(2,37)(3,34)(4,39)(5,36)(6,33)(7,38)(8,35)(9,73)(10,78)(11,75)(12,80)(13,77)(14,74)(15,79)(16,76)(17,93)(18,90)(19,95)(20,92)(21,89)(22,94)(23,91)(24,96)(25,88)(26,85)(27,82)(28,87)(29,84)(30,81)(31,86)(32,83)(41,50)(42,55)(43,52)(44,49)(45,54)(46,51)(47,56)(48,53)(57,61)(59,63)(65,110)(66,107)(67,112)(68,109)(69,106)(70,111)(71,108)(72,105)(97,101)(99,103) );
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)], [(1,25),(2,28),(3,31),(4,26),(5,29),(6,32),(7,27),(8,30),(9,69),(10,72),(11,67),(12,70),(13,65),(14,68),(15,71),(16,66),(17,50),(18,53),(19,56),(20,51),(21,54),(22,49),(23,52),(24,55),(33,87),(34,82),(35,85),(36,88),(37,83),(38,86),(39,81),(40,84),(41,89),(42,92),(43,95),(44,90),(45,93),(46,96),(47,91),(48,94),(57,102),(58,97),(59,100),(60,103),(61,98),(62,101),(63,104),(64,99),(73,110),(74,105),(75,108),(76,111),(77,106),(78,109),(79,112),(80,107)], [(1,105,17,104,93,72,40),(2,106,18,97,94,65,33),(3,107,19,98,95,66,34),(4,108,20,99,96,67,35),(5,109,21,100,89,68,36),(6,110,22,101,90,69,37),(7,111,23,102,91,70,38),(8,112,24,103,92,71,39),(9,83,32,73,49,62,44),(10,84,25,74,50,63,45),(11,85,26,75,51,64,46),(12,86,27,76,52,57,47),(13,87,28,77,53,58,48),(14,88,29,78,54,59,41),(15,81,30,79,55,60,42),(16,82,31,80,56,61,43)], [(1,40),(2,37),(3,34),(4,39),(5,36),(6,33),(7,38),(8,35),(9,73),(10,78),(11,75),(12,80),(13,77),(14,74),(15,79),(16,76),(17,93),(18,90),(19,95),(20,92),(21,89),(22,94),(23,91),(24,96),(25,88),(26,85),(27,82),(28,87),(29,84),(30,81),(31,86),(32,83),(41,50),(42,55),(43,52),(44,49),(45,54),(46,51),(47,56),(48,53),(57,61),(59,63),(65,110),(66,107),(67,112),(68,109),(69,106),(70,111),(71,108),(72,105),(97,101),(99,103)]])
SD16⋊D7 is a maximal subgroup of
D28.29D4 D8⋊10D14 D8.10D14 SD16⋊D14 D8⋊6D14 D7×C8.C22 D28.44D4
SD16⋊D7 is a maximal quotient of
D4.D7⋊C4 Dic7.D8 C28⋊Q8⋊C2 Dic14.D4 D4⋊(C4×D7) D14.D8 C7⋊C8⋊1D4 D4.D28 C7⋊Q16⋊C4 Dic7⋊Q16 Q8.Dic14 C56⋊C4.C2 (Q8×D7)⋊C4 D14⋊4Q16 D14⋊C8.C2 C7⋊C8.D4 Dic28⋊9C4 Dic14⋊Q8 C56⋊3Q8 Dic14.Q8 C8⋊(C4×D7) D14.2SD16 C28.(C4○D4) C8.2D28 Dic7⋊3SD16 SD16⋊Dic7 (C7×Q8).D4 C56.31D4 Dic14⋊7D4 Dic14.16D4 C56⋊8D4
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | 14E | 14F | 28A | 28B | 28C | 28D | 28E | 28F | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 4 | 14 | 2 | 4 | 14 | 28 | 28 | 2 | 2 | 2 | 4 | 28 | 2 | 2 | 2 | 8 | 8 | 8 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | C8.C22 | D4×D7 | SD16⋊D7 |
| kernel | SD16⋊D7 | C8⋊D7 | Dic28 | D4.D7 | C7⋊Q16 | C7×SD16 | D4⋊2D7 | Q8×D7 | Dic7 | D14 | SD16 | C8 | D4 | Q8 | C7 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 3 | 6 |
Matrix representation of SD16⋊D7 ►in GL4(𝔽113) generated by
| 0 | 0 | 79 | 1 |
| 0 | 0 | 53 | 0 |
| 0 | 32 | 20 | 96 |
| 1 | 71 | 17 | 93 |
| 11 | 59 | 42 | 32 |
| 76 | 102 | 1 | 0 |
| 0 | 29 | 96 | 54 |
| 68 | 82 | 59 | 17 |
| 88 | 1 | 0 | 0 |
| 53 | 34 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 9 |
| 34 | 9 | 0 | 0 |
| 60 | 79 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(113))| [0,0,0,1,0,0,32,71,79,53,20,17,1,0,96,93],[11,76,0,68,59,102,29,82,42,1,96,59,32,0,54,17],[88,53,0,0,1,34,0,0,0,0,0,112,0,0,1,9],[34,60,0,0,9,79,0,0,0,0,0,1,0,0,1,0] >;
SD16⋊D7 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes D_7 % in TeX
G:=Group("SD16:D7"); // GroupNames label
G:=SmallGroup(224,110);
// by ID
G=gap.SmallGroup(224,110);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,362,116,297,159,69,6917]);
// 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,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations