direct product, non-abelian, soluble
Aliases: C7×SL2(𝔽3), Q8⋊C21, C14.2A4, C2.(C7×A4), (C7×Q8)⋊1C3, SmallGroup(168,22)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C7×SL2(𝔽3) |
Generators and relations for C7×SL2(𝔽3)
G = < a,b,c,d | a7=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >
Smallest permutation representation of C7×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 8 39 48)(2 9 40 49)(3 10 41 43)(4 11 42 44)(5 12 36 45)(6 13 37 46)(7 14 38 47)(15 51 30 28)(16 52 31 22)(17 53 32 23)(18 54 33 24)(19 55 34 25)(20 56 35 26)(21 50 29 27)
(1 21 39 29)(2 15 40 30)(3 16 41 31)(4 17 42 32)(5 18 36 33)(6 19 37 34)(7 20 38 35)(8 27 48 50)(9 28 49 51)(10 22 43 52)(11 23 44 53)(12 24 45 54)(13 25 46 55)(14 26 47 56)
(8 27 21)(9 28 15)(10 22 16)(11 23 17)(12 24 18)(13 25 19)(14 26 20)(29 48 50)(30 49 51)(31 43 52)(32 44 53)(33 45 54)(34 46 55)(35 47 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), (1,8,39,48)(2,9,40,49)(3,10,41,43)(4,11,42,44)(5,12,36,45)(6,13,37,46)(7,14,38,47)(15,51,30,28)(16,52,31,22)(17,53,32,23)(18,54,33,24)(19,55,34,25)(20,56,35,26)(21,50,29,27), (1,21,39,29)(2,15,40,30)(3,16,41,31)(4,17,42,32)(5,18,36,33)(6,19,37,34)(7,20,38,35)(8,27,48,50)(9,28,49,51)(10,22,43,52)(11,23,44,53)(12,24,45,54)(13,25,46,55)(14,26,47,56), (8,27,21)(9,28,15)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(29,48,50)(30,49,51)(31,43,52)(32,44,53)(33,45,54)(34,46,55)(35,47,56)>;
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), (1,8,39,48)(2,9,40,49)(3,10,41,43)(4,11,42,44)(5,12,36,45)(6,13,37,46)(7,14,38,47)(15,51,30,28)(16,52,31,22)(17,53,32,23)(18,54,33,24)(19,55,34,25)(20,56,35,26)(21,50,29,27), (1,21,39,29)(2,15,40,30)(3,16,41,31)(4,17,42,32)(5,18,36,33)(6,19,37,34)(7,20,38,35)(8,27,48,50)(9,28,49,51)(10,22,43,52)(11,23,44,53)(12,24,45,54)(13,25,46,55)(14,26,47,56), (8,27,21)(9,28,15)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(29,48,50)(30,49,51)(31,43,52)(32,44,53)(33,45,54)(34,46,55)(35,47,56) );
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)], [(1,8,39,48),(2,9,40,49),(3,10,41,43),(4,11,42,44),(5,12,36,45),(6,13,37,46),(7,14,38,47),(15,51,30,28),(16,52,31,22),(17,53,32,23),(18,54,33,24),(19,55,34,25),(20,56,35,26),(21,50,29,27)], [(1,21,39,29),(2,15,40,30),(3,16,41,31),(4,17,42,32),(5,18,36,33),(6,19,37,34),(7,20,38,35),(8,27,48,50),(9,28,49,51),(10,22,43,52),(11,23,44,53),(12,24,45,54),(13,25,46,55),(14,26,47,56)], [(8,27,21),(9,28,15),(10,22,16),(11,23,17),(12,24,18),(13,25,19),(14,26,20),(29,48,50),(30,49,51),(31,43,52),(32,44,53),(33,45,54),(34,46,55),(35,47,56)]])
C7×SL2(𝔽3) is a maximal subgroup of
Q8.D21 Q8⋊D21 Dic7.2A4
49 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4 | 6A | 6B | 7A | ··· | 7F | 14A | ··· | 14F | 21A | ··· | 21L | 28A | ··· | 28F | 42A | ··· | 42L |
| order | 1 | 2 | 3 | 3 | 4 | 6 | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 |
| size | 1 | 1 | 4 | 4 | 6 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 |
| type | + | - | + | ||||||
| image | C1 | C3 | C7 | C21 | SL2(𝔽3) | SL2(𝔽3) | C7×SL2(𝔽3) | A4 | C7×A4 |
| kernel | C7×SL2(𝔽3) | C7×Q8 | SL2(𝔽3) | Q8 | C7 | C7 | C1 | C14 | C2 |
| # reps | 1 | 2 | 6 | 12 | 1 | 2 | 18 | 1 | 6 |
Matrix representation of C7×SL2(𝔽3) ►in GL2(𝔽29) generated by
| 25 | 0 |
| 0 | 25 |
| 3 | 12 |
| 4 | 26 |
| 5 | 23 |
| 14 | 24 |
| 0 | 16 |
| 9 | 28 |
G:=sub<GL(2,GF(29))| [25,0,0,25],[3,4,12,26],[5,14,23,24],[0,9,16,28] >;
C7×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_7\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C7xSL(2,3)"); // GroupNames label
G:=SmallGroup(168,22);
// by ID
G=gap.SmallGroup(168,22);
# by ID
G:=PCGroup([5,-3,-7,-2,2,-2,632,72,1263,133,58]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=d^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations
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