direct product, non-abelian, soluble
Aliases: D7×SL2(𝔽3), D14.2A4, Q8⋊(C3×D7), (Q8×D7)⋊1C3, (C7×Q8)⋊1C6, C2.3(A4×D7), C14.8(C2×A4), C7⋊3(C2×SL2(𝔽3)), (C7×SL2(𝔽3))⋊3C2, SmallGroup(336,132)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — C14 — C7×Q8 — C7×SL2(𝔽3) — D7×SL2(𝔽3) |
| C7×Q8 — D7×SL2(𝔽3) |
Generators and relations for D7×SL2(𝔽3)
G = < a,b,c,d,e | a7=b2=c4=e3=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >
Smallest permutation representation of D7×SL2(𝔽3)
►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)
(1 16)(2 15)(3 21)(4 20)(5 19)(6 18)(7 17)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 56)(22 32)(23 31)(24 30)(25 29)(26 35)(27 34)(28 33)(36 46)(37 45)(38 44)(39 43)(40 49)(41 48)(42 47)
(1 31 17 24)(2 32 18 25)(3 33 19 26)(4 34 20 27)(5 35 21 28)(6 29 15 22)(7 30 16 23)(8 41 55 48)(9 42 56 49)(10 36 50 43)(11 37 51 44)(12 38 52 45)(13 39 53 46)(14 40 54 47)
(1 12 17 52)(2 13 18 53)(3 14 19 54)(4 8 20 55)(5 9 21 56)(6 10 15 50)(7 11 16 51)(22 36 29 43)(23 37 30 44)(24 38 31 45)(25 39 32 46)(26 40 33 47)(27 41 34 48)(28 42 35 49)
(8 34 48)(9 35 49)(10 29 43)(11 30 44)(12 31 45)(13 32 46)(14 33 47)(22 36 50)(23 37 51)(24 38 52)(25 39 53)(26 40 54)(27 41 55)(28 42 56)
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), (1,16)(2,15)(3,21)(4,20)(5,19)(6,18)(7,17)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,56)(22,32)(23,31)(24,30)(25,29)(26,35)(27,34)(28,33)(36,46)(37,45)(38,44)(39,43)(40,49)(41,48)(42,47), (1,31,17,24)(2,32,18,25)(3,33,19,26)(4,34,20,27)(5,35,21,28)(6,29,15,22)(7,30,16,23)(8,41,55,48)(9,42,56,49)(10,36,50,43)(11,37,51,44)(12,38,52,45)(13,39,53,46)(14,40,54,47), (1,12,17,52)(2,13,18,53)(3,14,19,54)(4,8,20,55)(5,9,21,56)(6,10,15,50)(7,11,16,51)(22,36,29,43)(23,37,30,44)(24,38,31,45)(25,39,32,46)(26,40,33,47)(27,41,34,48)(28,42,35,49), (8,34,48)(9,35,49)(10,29,43)(11,30,44)(12,31,45)(13,32,46)(14,33,47)(22,36,50)(23,37,51)(24,38,52)(25,39,53)(26,40,54)(27,41,55)(28,42,56)>;
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), (1,16)(2,15)(3,21)(4,20)(5,19)(6,18)(7,17)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,56)(22,32)(23,31)(24,30)(25,29)(26,35)(27,34)(28,33)(36,46)(37,45)(38,44)(39,43)(40,49)(41,48)(42,47), (1,31,17,24)(2,32,18,25)(3,33,19,26)(4,34,20,27)(5,35,21,28)(6,29,15,22)(7,30,16,23)(8,41,55,48)(9,42,56,49)(10,36,50,43)(11,37,51,44)(12,38,52,45)(13,39,53,46)(14,40,54,47), (1,12,17,52)(2,13,18,53)(3,14,19,54)(4,8,20,55)(5,9,21,56)(6,10,15,50)(7,11,16,51)(22,36,29,43)(23,37,30,44)(24,38,31,45)(25,39,32,46)(26,40,33,47)(27,41,34,48)(28,42,35,49), (8,34,48)(9,35,49)(10,29,43)(11,30,44)(12,31,45)(13,32,46)(14,33,47)(22,36,50)(23,37,51)(24,38,52)(25,39,53)(26,40,54)(27,41,55)(28,42,56) );
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)], [(1,16),(2,15),(3,21),(4,20),(5,19),(6,18),(7,17),(8,55),(9,54),(10,53),(11,52),(12,51),(13,50),(14,56),(22,32),(23,31),(24,30),(25,29),(26,35),(27,34),(28,33),(36,46),(37,45),(38,44),(39,43),(40,49),(41,48),(42,47)], [(1,31,17,24),(2,32,18,25),(3,33,19,26),(4,34,20,27),(5,35,21,28),(6,29,15,22),(7,30,16,23),(8,41,55,48),(9,42,56,49),(10,36,50,43),(11,37,51,44),(12,38,52,45),(13,39,53,46),(14,40,54,47)], [(1,12,17,52),(2,13,18,53),(3,14,19,54),(4,8,20,55),(5,9,21,56),(6,10,15,50),(7,11,16,51),(22,36,29,43),(23,37,30,44),(24,38,31,45),(25,39,32,46),(26,40,33,47),(27,41,34,48),(28,42,35,49)], [(8,34,48),(9,35,49),(10,29,43),(11,30,44),(12,31,45),(13,32,46),(14,33,47),(22,36,50),(23,37,51),(24,38,52),(25,39,53),(26,40,54),(27,41,55),(28,42,56)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 7A | 7B | 7C | 14A | 14B | 14C | 21A | ··· | 21F | 28A | 28B | 28C | 42A | ··· | 42F |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 14 | 14 | 14 | 21 | ··· | 21 | 28 | 28 | 28 | 42 | ··· | 42 |
| size | 1 | 1 | 7 | 7 | 4 | 4 | 6 | 42 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 12 | 12 | 12 | 8 | ··· | 8 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 |
| type | + | + | + | - | + | + | - | + | |||||
| image | C1 | C2 | C3 | C6 | D7 | SL2(𝔽3) | SL2(𝔽3) | C3×D7 | A4 | C2×A4 | D7×SL2(𝔽3) | D7×SL2(𝔽3) | A4×D7 |
| kernel | D7×SL2(𝔽3) | C7×SL2(𝔽3) | Q8×D7 | C7×Q8 | SL2(𝔽3) | D7 | D7 | Q8 | D14 | C14 | C1 | C1 | C2 |
| # reps | 1 | 1 | 2 | 2 | 3 | 2 | 4 | 6 | 1 | 1 | 3 | 6 | 3 |
Matrix representation of D7×SL2(𝔽3) ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 319 | 302 |
| 336 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 336 |
| 0 | 1 | 0 | 0 |
| 336 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 208 | 128 | 0 | 0 |
| 128 | 129 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 208 | 128 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,1,319,0,0,2,302],[336,0,0,0,0,336,0,0,0,0,1,0,0,0,2,336],[0,336,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[208,128,0,0,128,129,0,0,0,0,1,0,0,0,0,1],[1,208,0,0,0,128,0,0,0,0,1,0,0,0,0,1] >;
D7×SL2(𝔽3) in GAP, Magma, Sage, TeX
D_7\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("D7xSL(2,3)"); // GroupNames label
G:=SmallGroup(336,132);
// by ID
G=gap.SmallGroup(336,132);
# by ID
G:=PCGroup([6,-2,-3,-2,2,-7,-2,170,518,81,735,357,4324]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^4=e^3=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations
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