metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊2Dic7, Q8⋊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 D4⋊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 D4⋊2Dic7
►On 56 points
(1 15 14 28)(2 16 8 22)(3 17 9 23)(4 18 10 24)(5 19 11 25)(6 20 12 26)(7 21 13 27)(29 56 36 49)(30 43 37 50)(31 44 38 51)(32 45 39 52)(33 46 40 53)(34 47 41 54)(35 48 42 55)
(1 49 14 56)(2 43 8 50)(3 51 9 44)(4 45 10 52)(5 53 11 46)(6 47 12 54)(7 55 13 48)(15 36 28 29)(16 30 22 37)(17 38 23 31)(18 32 24 39)(19 40 25 33)(20 34 26 41)(21 42 27 35)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(7 14)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(21 28)(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)| (1,15,14,28)(2,16,8,22)(3,17,9,23)(4,18,10,24)(5,19,11,25)(6,20,12,26)(7,21,13,27)(29,56,36,49)(30,43,37,50)(31,44,38,51)(32,45,39,52)(33,46,40,53)(34,47,41,54)(35,48,42,55), (1,49,14,56)(2,43,8,50)(3,51,9,44)(4,45,10,52)(5,53,11,46)(6,47,12,54)(7,55,13,48)(15,36,28,29)(16,30,22,37)(17,38,23,31)(18,32,24,39)(19,40,25,33)(20,34,26,41)(21,42,27,35), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(7,14)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(21,28)(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( (1,15,14,28)(2,16,8,22)(3,17,9,23)(4,18,10,24)(5,19,11,25)(6,20,12,26)(7,21,13,27)(29,56,36,49)(30,43,37,50)(31,44,38,51)(32,45,39,52)(33,46,40,53)(34,47,41,54)(35,48,42,55), (1,49,14,56)(2,43,8,50)(3,51,9,44)(4,45,10,52)(5,53,11,46)(6,47,12,54)(7,55,13,48)(15,36,28,29)(16,30,22,37)(17,38,23,31)(18,32,24,39)(19,40,25,33)(20,34,26,41)(21,42,27,35), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(7,14)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(21,28)(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([[(1,15,14,28),(2,16,8,22),(3,17,9,23),(4,18,10,24),(5,19,11,25),(6,20,12,26),(7,21,13,27),(29,56,36,49),(30,43,37,50),(31,44,38,51),(32,45,39,52),(33,46,40,53),(34,47,41,54),(35,48,42,55)], [(1,49,14,56),(2,43,8,50),(3,51,9,44),(4,45,10,52),(5,53,11,46),(6,47,12,54),(7,55,13,48),(15,36,28,29),(16,30,22,37),(17,38,23,31),(18,32,24,39),(19,40,25,33),(20,34,26,41),(21,42,27,35)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(7,14),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22),(21,28),(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)]])
D4⋊2Dic7 is a maximal subgroup of
D7×C4≀C2 C42⋊D14 C56.93D4 C56.50D4 D8⋊5Dic7 D8⋊4Dic7 D28⋊18D4 D28.38D4 D28.39D4 D28.40D4 (D4×C14)⋊9C4 2+ 1+4⋊D7 2+ 1+4.D7 2- 1+4⋊D7 2- 1+4.D7
D4⋊2Dic7 is a maximal quotient of
C28.2C42 C28.57D8 C28.26Q16 (D4×C14)⋊C4 C4⋊C4⋊Dic7 C42.7D14 C42.8D14
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 | D4⋊2Dic7 |
| kernel | D4⋊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 D4⋊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] >;
D4⋊2Dic7 in GAP, Magma, Sage, TeX
D_4\rtimes_2{\rm Dic}_7 % in TeX
G:=Group("D4:2Dic7"); // 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
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