direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×M4(2), C8⋊6D14, C56⋊6C22, C28.38C23, (C8×D7)⋊7C2, C8⋊D7⋊5C2, C7⋊C8⋊11C22, (C4×D7).1C4, C4.15(C4×D7), C7⋊2(C2×M4(2)), C28.12(C2×C4), D14.6(C2×C4), (C2×C4).45D14, C4.Dic7⋊5C2, C22.7(C4×D7), (C7×M4(2))⋊3C2, Dic7.7(C2×C4), (C2×Dic7).6C4, (C22×D7).4C4, C4.38(C22×D7), C14.15(C22×C4), (C2×C28).25C22, (C4×D7).18C22, (C2×C4×D7).4C2, C2.16(C2×C4×D7), (C2×C14).5(C2×C4), SmallGroup(224,101)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×M4(2)
G = < a,b,c,d | a7=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >
Subgroups: 238 in 68 conjugacy classes, 39 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, D7, C14, C14, C2×C8, M4(2), M4(2), C22×C4, Dic7, C28, D14, D14, C2×C14, C2×M4(2), C7⋊C8, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C8×D7, C8⋊D7, C4.Dic7, C7×M4(2), C2×C4×D7, D7×M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, M4(2), C22×C4, D14, C2×M4(2), C4×D7, C22×D7, C2×C4×D7, D7×M4(2)
►On 56 points
(1 46 14 55 37 24 27)(2 47 15 56 38 17 28)(3 48 16 49 39 18 29)(4 41 9 50 40 19 30)(5 42 10 51 33 20 31)(6 43 11 52 34 21 32)(7 44 12 53 35 22 25)(8 45 13 54 36 23 26)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(49 53)(50 54)(51 55)(52 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 6)(4 8)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)
G:=sub<Sym(56)| (1,46,14,55,37,24,27)(2,47,15,56,38,17,28)(3,48,16,49,39,18,29)(4,41,9,50,40,19,30)(5,42,10,51,33,20,31)(6,43,11,52,34,21,32)(7,44,12,53,35,22,25)(8,45,13,54,36,23,26), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(49,53)(50,54)(51,55)(52,56), (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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)>;
G:=Group( (1,46,14,55,37,24,27)(2,47,15,56,38,17,28)(3,48,16,49,39,18,29)(4,41,9,50,40,19,30)(5,42,10,51,33,20,31)(6,43,11,52,34,21,32)(7,44,12,53,35,22,25)(8,45,13,54,36,23,26), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(49,53)(50,54)(51,55)(52,56), (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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56) );
G=PermutationGroup([[(1,46,14,55,37,24,27),(2,47,15,56,38,17,28),(3,48,16,49,39,18,29),(4,41,9,50,40,19,30),(5,42,10,51,33,20,31),(6,43,11,52,34,21,32),(7,44,12,53,35,22,25),(8,45,13,54,36,23,26)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,41),(24,42),(49,53),(50,54),(51,55),(52,56)], [(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)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(26,30),(28,32),(34,38),(36,40),(41,45),(43,47),(50,54),(52,56)]])
D7×M4(2) is a maximal subgroup of
M4(2).19D14 M4(2).21D14 C42⋊D14 M4(2).25D14 C28.70C24 C56.49C23 SD16⋊D14 D56⋊C22
D7×M4(2) is a maximal quotient of
C56⋊Q8 C42.182D14 C8⋊9D28 Dic7.C42 Dic7.5M4(2) Dic7.M4(2) D14⋊M4(2) D14⋊2M4(2) Dic7⋊M4(2) C42.27D14 C42.200D14 C42.202D14 D14⋊3M4(2) C28⋊M4(2) Dic7⋊4M4(2) D14⋊6M4(2) C56⋊D4 C56⋊18D4
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 7 | 7 | 14 | 1 | 1 | 2 | 7 | 7 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D7 | M4(2) | D14 | D14 | C4×D7 | C4×D7 | D7×M4(2) |
| kernel | D7×M4(2) | C8×D7 | C8⋊D7 | C4.Dic7 | C7×M4(2) | C2×C4×D7 | C4×D7 | C2×Dic7 | C22×D7 | M4(2) | D7 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 3 | 4 | 6 | 3 | 6 | 6 | 6 |
Matrix representation of D7×M4(2) ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 9 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,0,112,0,0,1,9],[112,0,0,0,0,112,0,0,0,0,0,1,0,0,1,0],[0,15,0,0,1,0,0,0,0,0,98,0,0,0,0,98],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1] >;
D7×M4(2) in GAP, Magma, Sage, TeX
D_7\times M_4(2)
% in TeX
G:=Group("D7xM4(2)"); // GroupNames label
G:=SmallGroup(224,101);
// by ID
G=gap.SmallGroup(224,101);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,188,50,69,6917]);
// 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^5>;
// generators/relations