metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2Dic7, D4⋊2Dic7, C28.56D4, C7⋊3C4≀C2, (C7×D4)⋊2C4, (C7×Q8)⋊2C4, C28.9(C2×C4), C4○D4.1D7, (C2×C14).3D4, (C4×Dic7)⋊2C2, (C2×C4).41D14, C4.Dic7⋊4C2, C4.3(C2×Dic7), C4.31(C7⋊D4), (C2×C28).20C22, C22.3(C7⋊D4), C2.8(C23.D7), C14.18(C22⋊C4), (C7×C4○D4).1C2, SmallGroup(224,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2Dic7
G = < a,b,c,d | a4=c14=1, b2=a2, d2=c7, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >
Smallest permutation representation of Q8⋊2Dic7
►On 56 points
Generators in S56
(1 14 24 18)(2 8 25 19)(3 9 26 20)(4 10 27 21)(5 11 28 15)(6 12 22 16)(7 13 23 17)(29 54 36 47)(30 55 37 48)(31 56 38 49)(32 43 39 50)(33 44 40 51)(34 45 41 52)(35 46 42 53)
(1 43 24 50)(2 51 25 44)(3 45 26 52)(4 53 27 46)(5 47 28 54)(6 55 22 48)(7 49 23 56)(8 40 19 33)(9 34 20 41)(10 42 21 35)(11 36 15 29)(12 30 16 37)(13 38 17 31)(14 32 18 39)
(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)
(1 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 19)(9 18)(10 17)(11 16)(12 15)(13 21)(14 20)(29 48 36 55)(30 47 37 54)(31 46 38 53)(32 45 39 52)(33 44 40 51)(34 43 41 50)(35 56 42 49)
G:=sub<Sym(56)| (1,14,24,18)(2,8,25,19)(3,9,26,20)(4,10,27,21)(5,11,28,15)(6,12,22,16)(7,13,23,17)(29,54,36,47)(30,55,37,48)(31,56,38,49)(32,43,39,50)(33,44,40,51)(34,45,41,52)(35,46,42,53), (1,43,24,50)(2,51,25,44)(3,45,26,52)(4,53,27,46)(5,47,28,54)(6,55,22,48)(7,49,23,56)(8,40,19,33)(9,34,20,41)(10,42,21,35)(11,36,15,29)(12,30,16,37)(13,38,17,31)(14,32,18,39), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,19)(9,18)(10,17)(11,16)(12,15)(13,21)(14,20)(29,48,36,55)(30,47,37,54)(31,46,38,53)(32,45,39,52)(33,44,40,51)(34,43,41,50)(35,56,42,49)>;
G:=Group( (1,14,24,18)(2,8,25,19)(3,9,26,20)(4,10,27,21)(5,11,28,15)(6,12,22,16)(7,13,23,17)(29,54,36,47)(30,55,37,48)(31,56,38,49)(32,43,39,50)(33,44,40,51)(34,45,41,52)(35,46,42,53), (1,43,24,50)(2,51,25,44)(3,45,26,52)(4,53,27,46)(5,47,28,54)(6,55,22,48)(7,49,23,56)(8,40,19,33)(9,34,20,41)(10,42,21,35)(11,36,15,29)(12,30,16,37)(13,38,17,31)(14,32,18,39), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,19)(9,18)(10,17)(11,16)(12,15)(13,21)(14,20)(29,48,36,55)(30,47,37,54)(31,46,38,53)(32,45,39,52)(33,44,40,51)(34,43,41,50)(35,56,42,49) );
G=PermutationGroup([(1,14,24,18),(2,8,25,19),(3,9,26,20),(4,10,27,21),(5,11,28,15),(6,12,22,16),(7,13,23,17),(29,54,36,47),(30,55,37,48),(31,56,38,49),(32,43,39,50),(33,44,40,51),(34,45,41,52),(35,46,42,53)], [(1,43,24,50),(2,51,25,44),(3,45,26,52),(4,53,27,46),(5,47,28,54),(6,55,22,48),(7,49,23,56),(8,40,19,33),(9,34,20,41),(10,42,21,35),(11,36,15,29),(12,30,16,37),(13,38,17,31),(14,32,18,39)], [(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)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,28),(7,27),(8,19),(9,18),(10,17),(11,16),(12,15),(13,21),(14,20),(29,48,36,55),(30,47,37,54),(31,46,38,53),(32,45,39,52),(33,44,40,51),(34,43,41,50),(35,56,42,49)])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | ··· | 14L | 28A | ··· | 28F | 28G | ··· | 28O |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D7 | D14 | Dic7 | Dic7 | C4≀C2 | C7⋊D4 | C7⋊D4 | Q8⋊2Dic7 |
| kernel | Q8⋊2Dic7 | C4.Dic7 | C4×Dic7 | C7×C4○D4 | C7×D4 | C7×Q8 | C28 | C2×C14 | C4○D4 | C2×C4 | D4 | Q8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 |
Matrix representation of Q8⋊2Dic7 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 15 |
| 34 | 5 | 0 | 0 |
| 108 | 79 | 0 | 0 |
| 0 | 0 | 0 | 98 |
| 0 | 0 | 98 | 0 |
| 0 | 112 | 0 | 0 |
| 1 | 104 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 1 |
| 98 | 0 | 0 | 0 |
| 91 | 15 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,98,0,0,0,0,15],[34,108,0,0,5,79,0,0,0,0,0,98,0,0,98,0],[0,1,0,0,112,104,0,0,0,0,112,0,0,0,0,1],[98,91,0,0,0,15,0,0,0,0,98,0,0,0,0,112] >;
Q8⋊2Dic7 in GAP, Magma, Sage, TeX
Q_8\rtimes_2{\rm Dic}_7 % in TeX
G:=Group("Q8:2Dic7"); // GroupNames label
G:=SmallGroup(224,43);
// by ID
G=gap.SmallGroup(224,43);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,86,579,297,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^14=1,b^2=a^2,d^2=c^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations