metabelian, supersoluble, monomial
Aliases: C32⋊F7, C7⋊He3⋊1C2, C7⋊(C32⋊C6), (C3×C21)⋊3C6, C3⋊D21⋊2C3, C21.4(C3×S3), C3.4(C3⋊F7), (C3×C7⋊C3)⋊1S3, SmallGroup(378,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C21 — C32⋊F7 |
Generators and relations for C32⋊F7
G = < a,b,c,d | a3=b3=c7=d6=1, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, dbd-1=b-1, dcd-1=c5 >
Character table of C32⋊F7
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 7 | 21A | 21B | 21C | 21D | 21E | 21F | 21G | 21H | |
| size | 1 | 63 | 2 | 6 | 21 | 21 | 42 | 42 | 63 | 63 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 2 | 0 | 2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | 2 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ10 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ11 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C6 |
| ρ12 | 6 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1-√21/2 | -1 | -1 | 1-√21/2 | 1+√21/2 | 1+√21/2 | 1-√21/2 | 1+√21/2 | orthogonal lifted from C3⋊F7 |
| ρ13 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | ζ32ζ75+2ζ32ζ73+ζ32ζ72+2ζ32ζ7+ζ32+2ζ75+ζ73+2ζ72+ζ7+1 | 1+√21/2 | 1-√21/2 | ζ3ζ76+2ζ3ζ73+2ζ3ζ72+ζ3ζ7+ζ3+2ζ76+ζ73+ζ72+2ζ7+1 | ζ3ζ76+2ζ3ζ75+2ζ3ζ74+ζ3ζ7+ζ3+2ζ76+ζ75+ζ74+2ζ7+1 | ζ3ζ75+2ζ3ζ73+ζ3ζ72+2ζ3ζ7+ζ3+2ζ75+ζ73+2ζ72+ζ7+1 | 2ζ32ζ76+ζ32ζ74+ζ32ζ73+2ζ32ζ72+ζ32+ζ76+2ζ74+2ζ73+ζ72+1 | 2ζ32ζ75+ζ32ζ74+ζ32ζ73+2ζ32ζ7+ζ32+ζ75+2ζ74+2ζ73+ζ7+1 | orthogonal faithful |
| ρ14 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | ζ3ζ76+2ζ3ζ75+2ζ3ζ74+ζ3ζ7+ζ3+2ζ76+ζ75+ζ74+2ζ7+1 | 1-√21/2 | 1+√21/2 | 2ζ32ζ75+ζ32ζ74+ζ32ζ73+2ζ32ζ7+ζ32+ζ75+2ζ74+2ζ73+ζ7+1 | 2ζ32ζ76+ζ32ζ74+ζ32ζ73+2ζ32ζ72+ζ32+ζ76+2ζ74+2ζ73+ζ72+1 | ζ3ζ76+2ζ3ζ73+2ζ3ζ72+ζ3ζ7+ζ3+2ζ76+ζ73+ζ72+2ζ7+1 | ζ3ζ75+2ζ3ζ73+ζ3ζ72+2ζ3ζ7+ζ3+2ζ75+ζ73+2ζ72+ζ7+1 | ζ32ζ75+2ζ32ζ73+ζ32ζ72+2ζ32ζ7+ζ32+2ζ75+ζ73+2ζ72+ζ7+1 | orthogonal faithful |
| ρ15 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | ζ3ζ75+2ζ3ζ73+ζ3ζ72+2ζ3ζ7+ζ3+2ζ75+ζ73+2ζ72+ζ7+1 | 1-√21/2 | 1+√21/2 | ζ3ζ76+2ζ3ζ75+2ζ3ζ74+ζ3ζ7+ζ3+2ζ76+ζ75+ζ74+2ζ7+1 | ζ3ζ76+2ζ3ζ73+2ζ3ζ72+ζ3ζ7+ζ3+2ζ76+ζ73+ζ72+2ζ7+1 | ζ32ζ75+2ζ32ζ73+ζ32ζ72+2ζ32ζ7+ζ32+2ζ75+ζ73+2ζ72+ζ7+1 | 2ζ32ζ75+ζ32ζ74+ζ32ζ73+2ζ32ζ7+ζ32+ζ75+2ζ74+2ζ73+ζ7+1 | 2ζ32ζ76+ζ32ζ74+ζ32ζ73+2ζ32ζ72+ζ32+ζ76+2ζ74+2ζ73+ζ72+1 | orthogonal faithful |
| ρ16 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | ζ3ζ76+2ζ3ζ73+2ζ3ζ72+ζ3ζ7+ζ3+2ζ76+ζ73+ζ72+2ζ7+1 | 1+√21/2 | 1-√21/2 | 2ζ32ζ76+ζ32ζ74+ζ32ζ73+2ζ32ζ72+ζ32+ζ76+2ζ74+2ζ73+ζ72+1 | 2ζ32ζ75+ζ32ζ74+ζ32ζ73+2ζ32ζ7+ζ32+ζ75+2ζ74+2ζ73+ζ7+1 | ζ3ζ76+2ζ3ζ75+2ζ3ζ74+ζ3ζ7+ζ3+2ζ76+ζ75+ζ74+2ζ7+1 | ζ32ζ75+2ζ32ζ73+ζ32ζ72+2ζ32ζ7+ζ32+2ζ75+ζ73+2ζ72+ζ7+1 | ζ3ζ75+2ζ3ζ73+ζ3ζ72+2ζ3ζ7+ζ3+2ζ75+ζ73+2ζ72+ζ7+1 | orthogonal faithful |
| ρ17 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2ζ32ζ75+ζ32ζ74+ζ32ζ73+2ζ32ζ7+ζ32+ζ75+2ζ74+2ζ73+ζ7+1 | 1-√21/2 | 1+√21/2 | ζ3ζ75+2ζ3ζ73+ζ3ζ72+2ζ3ζ7+ζ3+2ζ75+ζ73+2ζ72+ζ7+1 | ζ32ζ75+2ζ32ζ73+ζ32ζ72+2ζ32ζ7+ζ32+2ζ75+ζ73+2ζ72+ζ7+1 | 2ζ32ζ76+ζ32ζ74+ζ32ζ73+2ζ32ζ72+ζ32+ζ76+2ζ74+2ζ73+ζ72+1 | ζ3ζ76+2ζ3ζ75+2ζ3ζ74+ζ3ζ7+ζ3+2ζ76+ζ75+ζ74+2ζ7+1 | ζ3ζ76+2ζ3ζ73+2ζ3ζ72+ζ3ζ7+ζ3+2ζ76+ζ73+ζ72+2ζ7+1 | orthogonal faithful |
| ρ18 | 6 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1+√21/2 | -1 | -1 | 1+√21/2 | 1-√21/2 | 1-√21/2 | 1+√21/2 | 1-√21/2 | orthogonal lifted from C3⋊F7 |
| ρ19 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2ζ32ζ76+ζ32ζ74+ζ32ζ73+2ζ32ζ72+ζ32+ζ76+2ζ74+2ζ73+ζ72+1 | 1+√21/2 | 1-√21/2 | ζ32ζ75+2ζ32ζ73+ζ32ζ72+2ζ32ζ7+ζ32+2ζ75+ζ73+2ζ72+ζ7+1 | ζ3ζ75+2ζ3ζ73+ζ3ζ72+2ζ3ζ7+ζ3+2ζ75+ζ73+2ζ72+ζ7+1 | 2ζ32ζ75+ζ32ζ74+ζ32ζ73+2ζ32ζ7+ζ32+ζ75+2ζ74+2ζ73+ζ7+1 | ζ3ζ76+2ζ3ζ73+2ζ3ζ72+ζ3ζ7+ζ3+2ζ76+ζ73+ζ72+2ζ7+1 | ζ3ζ76+2ζ3ζ75+2ζ3ζ74+ζ3ζ7+ζ3+2ζ76+ζ75+ζ74+2ζ7+1 | orthogonal faithful |
►On 63 points
Generators in S63
(1 43 22)(2 44 23)(3 45 24)(4 46 25)(5 47 26)(6 48 27)(7 49 28)(8 50 29)(9 51 30)(10 52 31)(11 53 32)(12 54 33)(13 55 34)(14 56 35)(15 57 36)(16 58 37)(17 59 38)(18 60 39)(19 61 40)(20 62 41)(21 63 42)
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)(43 57 50)(44 58 51)(45 59 52)(46 60 53)(47 61 54)(48 62 55)(49 63 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)(57 58 59 60 61 62 63)
(2 4 3 7 5 6)(8 15)(9 18 10 21 12 20)(11 17 14 19 13 16)(22 50 29 43 36 57)(23 53 31 49 40 62)(24 56 33 48 37 60)(25 52 35 47 41 58)(26 55 30 46 38 63)(27 51 32 45 42 61)(28 54 34 44 39 59)
G:=sub<Sym(63)| (1,43,22)(2,44,23)(3,45,24)(4,46,25)(5,47,26)(6,48,27)(7,49,28)(8,50,29)(9,51,30)(10,52,31)(11,53,32)(12,54,33)(13,55,34)(14,56,35)(15,57,36)(16,58,37)(17,59,38)(18,60,39)(19,61,40)(20,62,41)(21,63,42), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56), (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)(57,58,59,60,61,62,63), (2,4,3,7,5,6)(8,15)(9,18,10,21,12,20)(11,17,14,19,13,16)(22,50,29,43,36,57)(23,53,31,49,40,62)(24,56,33,48,37,60)(25,52,35,47,41,58)(26,55,30,46,38,63)(27,51,32,45,42,61)(28,54,34,44,39,59)>;
G:=Group( (1,43,22)(2,44,23)(3,45,24)(4,46,25)(5,47,26)(6,48,27)(7,49,28)(8,50,29)(9,51,30)(10,52,31)(11,53,32)(12,54,33)(13,55,34)(14,56,35)(15,57,36)(16,58,37)(17,59,38)(18,60,39)(19,61,40)(20,62,41)(21,63,42), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56), (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)(57,58,59,60,61,62,63), (2,4,3,7,5,6)(8,15)(9,18,10,21,12,20)(11,17,14,19,13,16)(22,50,29,43,36,57)(23,53,31,49,40,62)(24,56,33,48,37,60)(25,52,35,47,41,58)(26,55,30,46,38,63)(27,51,32,45,42,61)(28,54,34,44,39,59) );
G=PermutationGroup([[(1,43,22),(2,44,23),(3,45,24),(4,46,25),(5,47,26),(6,48,27),(7,49,28),(8,50,29),(9,51,30),(10,52,31),(11,53,32),(12,54,33),(13,55,34),(14,56,35),(15,57,36),(16,58,37),(17,59,38),(18,60,39),(19,61,40),(20,62,41),(21,63,42)], [(1,15,8),(2,16,9),(3,17,10),(4,18,11),(5,19,12),(6,20,13),(7,21,14),(22,36,29),(23,37,30),(24,38,31),(25,39,32),(26,40,33),(27,41,34),(28,42,35),(43,57,50),(44,58,51),(45,59,52),(46,60,53),(47,61,54),(48,62,55),(49,63,56)], [(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),(57,58,59,60,61,62,63)], [(2,4,3,7,5,6),(8,15),(9,18,10,21,12,20),(11,17,14,19,13,16),(22,50,29,43,36,57),(23,53,31,49,40,62),(24,56,33,48,37,60),(25,52,35,47,41,58),(26,55,30,46,38,63),(27,51,32,45,42,61),(28,54,34,44,39,59)]])
Matrix representation of C32⋊F7 ►in GL6(𝔽43)
| 40 | 33 | 15 | 30 | 29 | 4 |
| 39 | 36 | 29 | 11 | 26 | 25 |
| 18 | 14 | 11 | 4 | 29 | 1 |
| 42 | 17 | 13 | 10 | 3 | 28 |
| 15 | 14 | 32 | 28 | 25 | 18 |
| 25 | 40 | 39 | 14 | 10 | 7 |
| 2 | 5 | 5 | 0 | 5 | 0 |
| 0 | 2 | 5 | 5 | 0 | 5 |
| 38 | 38 | 40 | 0 | 0 | 38 |
| 5 | 0 | 0 | 2 | 5 | 5 |
| 38 | 0 | 38 | 38 | 40 | 0 |
| 0 | 38 | 0 | 38 | 38 | 40 |
| 42 | 42 | 42 | 42 | 42 | 42 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 42 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 42 |
| 0 | 0 | 0 | 42 | 0 | 0 |
| 0 | 42 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 42 | 0 |
G:=sub<GL(6,GF(43))| [40,39,18,42,15,25,33,36,14,17,14,40,15,29,11,13,32,39,30,11,4,10,28,14,29,26,29,3,25,10,4,25,1,28,18,7],[2,0,38,5,38,0,5,2,38,0,0,38,5,5,40,0,38,0,0,5,0,2,38,38,5,0,0,5,40,38,0,5,38,5,0,40],[42,1,0,0,0,0,42,0,1,0,0,0,42,0,0,1,0,0,42,0,0,0,1,0,42,0,0,0,0,1,42,0,0,0,0,0],[42,0,0,0,1,0,0,0,0,42,1,0,0,0,0,0,1,0,0,0,42,0,1,0,0,0,0,0,1,42,0,42,0,0,1,0] >;
C32⋊F7 in GAP, Magma, Sage, TeX
C_3^2\rtimes F_7
% in TeX
G:=Group("C3^2:F7"); // GroupNames label
G:=SmallGroup(378,22);
// by ID
G=gap.SmallGroup(378,22);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,182,187,723,8104,1359]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^7=d^6=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations
Export
Subgroup lattice of C32⋊F7 in TeX
Character table of C32⋊F7 in TeX