metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.46D4, C4.11D28, M4(2)⋊3D7, (C2×C4).1D14, (C22×D7).C4, (C2×D28).6C2, C7⋊1(C4.D4), C4.Dic7⋊2C2, C22.4(C4×D7), C2.9(D14⋊C4), C4.21(C7⋊D4), (C7×M4(2))⋊7C2, C14.8(C22⋊C4), (C2×C28).13C22, (C2×C14).2(C2×C4), SmallGroup(224,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.46D4
G = < a,b,c | a28=c2=1, b4=a14, bab-1=cac=a-1, cbc=a7b3 >
Smallest permutation representation of C28.46D4
►On 56 points
(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 49 8 42 15 35 22 56)(2 48 9 41 16 34 23 55)(3 47 10 40 17 33 24 54)(4 46 11 39 18 32 25 53)(5 45 12 38 19 31 26 52)(6 44 13 37 20 30 27 51)(7 43 14 36 21 29 28 50)
(1 8)(2 7)(3 6)(4 5)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(49 56)(50 55)(51 54)(52 53)
G:=sub<Sym(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), (1,49,8,42,15,35,22,56)(2,48,9,41,16,34,23,55)(3,47,10,40,17,33,24,54)(4,46,11,39,18,32,25,53)(5,45,12,38,19,31,26,52)(6,44,13,37,20,30,27,51)(7,43,14,36,21,29,28,50), (1,8)(2,7)(3,6)(4,5)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(49,56)(50,55)(51,54)(52,53)>;
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), (1,49,8,42,15,35,22,56)(2,48,9,41,16,34,23,55)(3,47,10,40,17,33,24,54)(4,46,11,39,18,32,25,53)(5,45,12,38,19,31,26,52)(6,44,13,37,20,30,27,51)(7,43,14,36,21,29,28,50), (1,8)(2,7)(3,6)(4,5)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(49,56)(50,55)(51,54)(52,53) );
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)], [(1,49,8,42,15,35,22,56),(2,48,9,41,16,34,23,55),(3,47,10,40,17,33,24,54),(4,46,11,39,18,32,25,53),(5,45,12,38,19,31,26,52),(6,44,13,37,20,30,27,51),(7,43,14,36,21,29,28,50)], [(1,8),(2,7),(3,6),(4,5),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39),(49,56),(50,55),(51,54),(52,53)]])
C28.46D4 is a maximal subgroup of
D7×C4.D4 D28.3D4 M4(2).21D14 D28.6D4 D4⋊4D28 M4(2)⋊D14 C8.21D28 C8.24D28 M4(2).31D14 D4.3D28 D4.4D28 D28⋊18D4 M4(2).D14 D28.39D4 M4(2).15D14
C28.46D4 is a maximal quotient of
C42.D14 (C22×D7)⋊C8 C4.Dic28 C4.D56 M4(2)⋊Dic7
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D7 | D14 | D28 | C7⋊D4 | C4×D7 | C4.D4 | C28.46D4 |
| kernel | C28.46D4 | C4.Dic7 | C7×M4(2) | C2×D28 | C22×D7 | C28 | M4(2) | C2×C4 | C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 3 | 6 | 6 | 6 | 1 | 6 |
Matrix representation of C28.46D4 ►in GL4(𝔽113) generated by
| 58 | 81 | 0 | 0 |
| 32 | 109 | 0 | 0 |
| 17 | 78 | 35 | 81 |
| 74 | 47 | 64 | 19 |
| 106 | 15 | 107 | 45 |
| 67 | 62 | 5 | 56 |
| 15 | 82 | 39 | 4 |
| 25 | 6 | 93 | 19 |
| 58 | 75 | 0 | 0 |
| 32 | 55 | 0 | 0 |
| 77 | 109 | 94 | 81 |
| 51 | 10 | 96 | 19 |
G:=sub<GL(4,GF(113))| [58,32,17,74,81,109,78,47,0,0,35,64,0,0,81,19],[106,67,15,25,15,62,82,6,107,5,39,93,45,56,4,19],[58,32,77,51,75,55,109,10,0,0,94,96,0,0,81,19] >;
C28.46D4 in GAP, Magma, Sage, TeX
C_{28}._{46}D_4 % in TeX
G:=Group("C28.46D4"); // GroupNames label
G:=SmallGroup(224,29);
// by ID
G=gap.SmallGroup(224,29);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,121,31,362,86,297,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=c^2=1,b^4=a^14,b*a*b^-1=c*a*c=a^-1,c*b*c=a^7*b^3>;
// generators/relations
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