direct product, metacyclic, supersoluble, monomial
Aliases: M4(2)×C7⋊C3, C56⋊7C6, C28.4C12, (C7×M4(2))⋊C3, (C2×C28).6C6, C7⋊3(C3×M4(2)), (C2×C14).3C12, C28.30(C2×C6), C14.14(C2×C12), C4.(C4×C7⋊C3), C8⋊3(C2×C7⋊C3), (C8×C7⋊C3)⋊7C2, C22.(C4×C7⋊C3), (C4×C7⋊C3).4C4, C4.7(C22×C7⋊C3), (C22×C7⋊C3).3C4, (C4×C7⋊C3).23C22, C2.5(C2×C4×C7⋊C3), (C2×C4×C7⋊C3).7C2, (C2×C4).2(C2×C7⋊C3), (C2×C7⋊C3).14(C2×C4), SmallGroup(336,52)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×C7⋊C3
G = < a,b,c,d | a8=b2=c7=d3=1, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Smallest permutation representation of M4(2)×C7⋊C3
►On 56 points
Generators in S56
(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)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)
(1 56 38 30 12 21 46)(2 49 39 31 13 22 47)(3 50 40 32 14 23 48)(4 51 33 25 15 24 41)(5 52 34 26 16 17 42)(6 53 35 27 9 18 43)(7 54 36 28 10 19 44)(8 55 37 29 11 20 45)
(9 53 35)(10 54 36)(11 55 37)(12 56 38)(13 49 39)(14 50 40)(15 51 33)(16 52 34)(17 26 42)(18 27 43)(19 28 44)(20 29 45)(21 30 46)(22 31 47)(23 32 48)(24 25 41)
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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55), (1,56,38,30,12,21,46)(2,49,39,31,13,22,47)(3,50,40,32,14,23,48)(4,51,33,25,15,24,41)(5,52,34,26,16,17,42)(6,53,35,27,9,18,43)(7,54,36,28,10,19,44)(8,55,37,29,11,20,45), (9,53,35)(10,54,36)(11,55,37)(12,56,38)(13,49,39)(14,50,40)(15,51,33)(16,52,34)(17,26,42)(18,27,43)(19,28,44)(20,29,45)(21,30,46)(22,31,47)(23,32,48)(24,25,41)>;
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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55), (1,56,38,30,12,21,46)(2,49,39,31,13,22,47)(3,50,40,32,14,23,48)(4,51,33,25,15,24,41)(5,52,34,26,16,17,42)(6,53,35,27,9,18,43)(7,54,36,28,10,19,44)(8,55,37,29,11,20,45), (9,53,35)(10,54,36)(11,55,37)(12,56,38)(13,49,39)(14,50,40)(15,51,33)(16,52,34)(17,26,42)(18,27,43)(19,28,44)(20,29,45)(21,30,46)(22,31,47)(23,32,48)(24,25,41) );
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)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(49,53),(51,55)], [(1,56,38,30,12,21,46),(2,49,39,31,13,22,47),(3,50,40,32,14,23,48),(4,51,33,25,15,24,41),(5,52,34,26,16,17,42),(6,53,35,27,9,18,43),(7,54,36,28,10,19,44),(8,55,37,29,11,20,45)], [(9,53,35),(10,54,36),(11,55,37),(12,56,38),(13,49,39),(14,50,40),(15,51,33),(16,52,34),(17,26,42),(18,27,43),(19,28,44),(20,29,45),(21,30,46),(22,31,47),(23,32,48),(24,25,41)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 7A | 7B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 14A | 14B | 14C | 14D | 24A | ··· | 24H | 28A | 28B | 28C | 28D | 28E | 28F | 56A | ··· | 56H |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 7 | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 14 | 24 | ··· | 24 | 28 | 28 | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 7 | 7 | 1 | 1 | 2 | 7 | 7 | 14 | 14 | 3 | 3 | 2 | 2 | 2 | 2 | 7 | 7 | 7 | 7 | 14 | 14 | 3 | 3 | 6 | 6 | 14 | ··· | 14 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | ··· | 6 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | M4(2) | C3×M4(2) | C7⋊C3 | C2×C7⋊C3 | C2×C7⋊C3 | C4×C7⋊C3 | C4×C7⋊C3 | M4(2)×C7⋊C3 |
| kernel | M4(2)×C7⋊C3 | C8×C7⋊C3 | C2×C4×C7⋊C3 | C7×M4(2) | C4×C7⋊C3 | C22×C7⋊C3 | C56 | C2×C28 | C28 | C2×C14 | C7⋊C3 | C7 | M4(2) | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of M4(2)×C7⋊C3 ►in GL5(𝔽337)
| 149 | 335 | 0 | 0 | 0 |
| 222 | 188 | 0 | 0 | 0 |
| 0 | 0 | 189 | 0 | 0 |
| 0 | 0 | 0 | 189 | 0 |
| 0 | 0 | 0 | 0 | 189 |
| 1 | 0 | 0 | 0 | 0 |
| 149 | 336 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 213 | 212 | 212 |
| 0 | 0 | 336 | 0 | 336 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 124 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 336 | 0 | 336 |
G:=sub<GL(5,GF(337))| [149,222,0,0,0,335,188,0,0,0,0,0,189,0,0,0,0,0,189,0,0,0,0,0,189],[1,149,0,0,0,0,336,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,213,336,0,0,1,212,0,0,0,1,212,336],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,336,0,0,124,1,0,0,0,1,0,336] >;
M4(2)×C7⋊C3 in GAP, Magma, Sage, TeX
M_4(2)\times C_7\rtimes C_3
% in TeX
G:=Group("M4(2)xC7:C3"); // GroupNames label
G:=SmallGroup(336,52);
// by ID
G=gap.SmallGroup(336,52);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,313,79,69,881]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^7=d^3=1,b*a*b=a^5,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations
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