direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×SD16, C8⋊5D14, Q8⋊1D14, C56⋊5C22, D4.2D14, D14.13D4, C28.4C23, Dic7.4D4, D28.2C22, Dic14⋊2C22, (D4×D7).C2, (C8×D7)⋊4C2, C7⋊C8⋊6C22, Q8⋊D7⋊1C2, (Q8×D7)⋊1C2, C7⋊2(C2×SD16), C56⋊C2⋊5C2, D4.D7⋊3C2, C2.18(D4×D7), (C7×SD16)⋊3C2, C14.30(C2×D4), (C7×Q8)⋊1C22, C4.4(C22×D7), (C4×D7).9C22, (C7×D4).2C22, SmallGroup(224,108)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×SD16
G = < a,b,c,d | a7=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >
Subgroups: 342 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, D7, D7, C14, C14, C2×C8, SD16, SD16, C2×D4, C2×Q8, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×SD16, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, C7⋊D4, C7×D4, C7×Q8, C22×D7, C8×D7, C56⋊C2, D4.D7, Q8⋊D7, C7×SD16, D4×D7, Q8×D7, D7×SD16
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C2×SD16, C22×D7, D4×D7, D7×SD16
►On 56 points
(1 49 39 12 23 45 27)(2 50 40 13 24 46 28)(3 51 33 14 17 47 29)(4 52 34 15 18 48 30)(5 53 35 16 19 41 31)(6 54 36 9 20 42 32)(7 55 37 10 21 43 25)(8 56 38 11 22 44 26)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 13)(10 14)(11 15)(12 16)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 21)(18 24)(20 22)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(42 44)(43 47)(46 48)(50 52)(51 55)(54 56)
G:=sub<Sym(56)| (1,49,39,12,23,45,27)(2,50,40,13,24,46,28)(3,51,33,14,17,47,29)(4,52,34,15,18,48,30)(5,53,35,16,19,41,31)(6,54,36,9,20,42,32)(7,55,37,10,21,43,25)(8,56,38,11,22,44,26), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,13)(10,14)(11,15)(12,16)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,21)(18,24)(20,22)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(42,44)(43,47)(46,48)(50,52)(51,55)(54,56)>;
G:=Group( (1,49,39,12,23,45,27)(2,50,40,13,24,46,28)(3,51,33,14,17,47,29)(4,52,34,15,18,48,30)(5,53,35,16,19,41,31)(6,54,36,9,20,42,32)(7,55,37,10,21,43,25)(8,56,38,11,22,44,26), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,13)(10,14)(11,15)(12,16)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,21)(18,24)(20,22)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(42,44)(43,47)(46,48)(50,52)(51,55)(54,56) );
G=PermutationGroup([[(1,49,39,12,23,45,27),(2,50,40,13,24,46,28),(3,51,33,14,17,47,29),(4,52,34,15,18,48,30),(5,53,35,16,19,41,31),(6,54,36,9,20,42,32),(7,55,37,10,21,43,25),(8,56,38,11,22,44,26)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,13),(10,14),(11,15),(12,16),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56)], [(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)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,21),(18,24),(20,22),(25,29),(26,32),(28,30),(33,37),(34,40),(36,38),(42,44),(43,47),(46,48),(50,52),(51,55),(54,56)]])
D7×SD16 is a maximal subgroup of
D28.29D4 D8⋊11D14 D8⋊6D14 C56.C23
D7×SD16 is a maximal quotient of
Dic7⋊6SD16 Dic7.SD16 D4⋊Dic14 Dic14⋊2D4 D4.6D28 D14.SD16 D14⋊SD16 Dic7⋊7SD16 Q8⋊Dic14 Dic7.1Q16 D14.1SD16 Q8⋊2D28 D14⋊2SD16 Dic7⋊SD16 Dic7⋊8SD16 Dic14⋊Q8 C56⋊5Q8 D14.2SD16 D14.4SD16 C8⋊8D28 D28⋊Q8 Dic7⋊3SD16 Dic7⋊5SD16 D14⋊6SD16 Dic14⋊7D4 C56⋊14D4 C56⋊15D4
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 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 | 2 | 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 | 7 | 7 | 28 | 2 | 4 | 14 | 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 | SD16 | D14 | D14 | D14 | D4×D7 | D7×SD16 |
| kernel | D7×SD16 | C8×D7 | C56⋊C2 | D4.D7 | Q8⋊D7 | C7×SD16 | D4×D7 | Q8×D7 | Dic7 | D14 | SD16 | D7 | C8 | D4 | Q8 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 3 | 3 | 3 | 3 | 6 |
Matrix representation of D7×SD16 ►in GL4(𝔽113) generated by
| 0 | 1 | 0 | 0 |
| 112 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 26 | 22 |
| 0 | 0 | 36 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 81 | 112 |
G:=sub<GL(4,GF(113))| [0,112,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,26,36,0,0,22,0],[1,0,0,0,0,1,0,0,0,0,1,81,0,0,0,112] >;
D7×SD16 in GAP, Magma, Sage, TeX
D_7\times {\rm SD}_{16} % in TeX
G:=Group("D7xSD16"); // GroupNames label
G:=SmallGroup(224,108);
// by ID
G=gap.SmallGroup(224,108);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,116,86,297,159,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations