metabelian, supersoluble, monomial
Aliases: C7⋊He3⋊3C2, (C3×C21)⋊9C6, C21.9(C3×S3), C7⋊2(C32⋊C6), C3⋊S3⋊(C7⋊C3), C32⋊(C2×C7⋊C3), (C3×C7⋊C3)⋊2S3, C3.4(S3×C7⋊C3), (C7×C3⋊S3)⋊2C3, SmallGroup(378,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C21 — C7⋊He3⋊C2 |
Generators and relations for C7⋊He3⋊C2
G = < a,b,c,d,e | a7=b3=c3=d3=e2=1, ab=ba, ac=ca, dad-1=a4, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1, cd=dc, ece=c-1, de=ed >
Character table of C7⋊He3⋊C2
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 7A | 7B | 14A | 14B | 21A | 21B | 21C | 21D | 21E | 21F | 21G | 21H | |
| size | 1 | 9 | 2 | 6 | 21 | 21 | 42 | 42 | 63 | 63 | 3 | 3 | 27 | 27 | 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 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 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 | 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 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 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 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | 2 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | complex lifted from C3×S3 |
| ρ10 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C7⋊C3 |
| ρ11 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C7⋊C3 |
| ρ12 | 3 | -3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 1+√-7/2 | 1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ13 | 3 | -3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 1-√-7/2 | 1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ14 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 6 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | -3 | 0 | 0 | orthogonal lifted from C32⋊C6 |
| ρ15 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 0 | 0 | -ζ74+2ζ72-ζ7 | 1+√-7/2 | -ζ76-ζ75+2ζ73 | 2ζ74-ζ72-ζ7 | 2ζ76-ζ75-ζ73 | 1-√-7/2 | -ζ74-ζ72+2ζ7 | -ζ76+2ζ75-ζ73 | complex faithful |
| ρ16 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 0 | 0 | -ζ76-ζ75+2ζ73 | 1-√-7/2 | -ζ74-ζ72+2ζ7 | 2ζ76-ζ75-ζ73 | -ζ74+2ζ72-ζ7 | 1+√-7/2 | -ζ76+2ζ75-ζ73 | 2ζ74-ζ72-ζ7 | complex faithful |
| ρ17 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 0 | 0 | 2ζ76-ζ75-ζ73 | 1-√-7/2 | -ζ74+2ζ72-ζ7 | -ζ76+2ζ75-ζ73 | 2ζ74-ζ72-ζ7 | 1+√-7/2 | -ζ76-ζ75+2ζ73 | -ζ74-ζ72+2ζ7 | complex faithful |
| ρ18 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 0 | 0 | 2ζ74-ζ72-ζ7 | 1+√-7/2 | 2ζ76-ζ75-ζ73 | -ζ74-ζ72+2ζ7 | -ζ76+2ζ75-ζ73 | 1-√-7/2 | -ζ74+2ζ72-ζ7 | -ζ76-ζ75+2ζ73 | complex faithful |
| ρ19 | 6 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 0 | 0 | 1-√-7/2 | -1-√-7 | 1+√-7/2 | 1-√-7/2 | 1+√-7/2 | -1+√-7 | 1-√-7/2 | 1+√-7/2 | complex lifted from S3×C7⋊C3 |
| ρ20 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 0 | 0 | -ζ74-ζ72+2ζ7 | 1+√-7/2 | -ζ76+2ζ75-ζ73 | -ζ74+2ζ72-ζ7 | -ζ76-ζ75+2ζ73 | 1-√-7/2 | 2ζ74-ζ72-ζ7 | 2ζ76-ζ75-ζ73 | complex faithful |
| ρ21 | 6 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 0 | 0 | 1+√-7/2 | -1+√-7 | 1-√-7/2 | 1+√-7/2 | 1-√-7/2 | -1-√-7 | 1+√-7/2 | 1-√-7/2 | complex lifted from S3×C7⋊C3 |
| ρ22 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 0 | 0 | -ζ76+2ζ75-ζ73 | 1-√-7/2 | 2ζ74-ζ72-ζ7 | -ζ76-ζ75+2ζ73 | -ζ74-ζ72+2ζ7 | 1+√-7/2 | 2ζ76-ζ75-ζ73 | -ζ74+2ζ72-ζ7 | complex faithful |
►On 63 points
Generators in S63
(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)
(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)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(22 29 36)(23 31 40)(24 33 37)(25 35 41)(26 30 38)(27 32 42)(28 34 39)(43 57 50)(44 59 54)(45 61 51)(46 63 55)(47 58 52)(48 60 56)(49 62 53)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)
G:=sub<Sym(63)| (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), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(22,29,36)(23,31,40)(24,33,37)(25,35,41)(26,30,38)(27,32,42)(28,34,39)(43,57,50)(44,59,54)(45,61,51)(46,63,55)(47,58,52)(48,60,56)(49,62,53), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(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)(57,58,59,60,61,62,63), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(22,29,36)(23,31,40)(24,33,37)(25,35,41)(26,30,38)(27,32,42)(28,34,39)(43,57,50)(44,59,54)(45,61,51)(46,63,55)(47,58,52)(48,60,56)(49,62,53), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(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),(57,58,59,60,61,62,63)], [(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)], [(2,3,5),(4,7,6),(9,10,12),(11,14,13),(16,17,19),(18,21,20),(22,29,36),(23,31,40),(24,33,37),(25,35,41),(26,30,38),(27,32,42),(28,34,39),(43,57,50),(44,59,54),(45,61,51),(46,63,55),(47,58,52),(48,60,56),(49,62,53)], [(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,57),(30,58),(31,59),(32,60),(33,61),(34,62),(35,63),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56)]])
Matrix representation of C7⋊He3⋊C2 ►in GL6(𝔽43)
| 18 | 19 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 19 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 30 | 15 | 35 | 33 | 35 | 20 |
| 35 | 2 | 38 | 20 | 17 | 42 |
| 38 | 39 | 11 | 42 | 38 | 36 |
| 10 | 8 | 23 | 40 | 23 | 15 |
| 23 | 26 | 1 | 15 | 28 | 39 |
| 1 | 5 | 7 | 39 | 1 | 18 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 42 | 0 | 0 | 42 | 0 | 0 |
| 0 | 42 | 0 | 0 | 42 | 0 |
| 0 | 0 | 42 | 0 | 0 | 42 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 24 | 42 | 42 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 24 | 42 | 42 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 42 | 0 | 0 | 42 | 0 | 0 |
| 0 | 42 | 0 | 0 | 42 | 0 |
| 0 | 0 | 42 | 0 | 0 | 42 |
G:=sub<GL(6,GF(43))| [18,1,0,0,0,0,19,0,1,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,19,0,1,0,0,0,1,0,0],[30,35,38,10,23,1,15,2,39,8,26,5,35,38,11,23,1,7,33,20,42,40,15,39,35,17,38,23,28,1,20,42,36,15,39,18],[0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,1,0,0,42,0,0,0,1,0,0,42,0,0,0,1,0,0,42],[1,24,0,0,0,0,0,42,1,0,0,0,0,42,0,0,0,0,0,0,0,1,24,0,0,0,0,0,42,1,0,0,0,0,42,0],[1,0,0,42,0,0,0,1,0,0,42,0,0,0,1,0,0,42,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42] >;
C7⋊He3⋊C2 in GAP, Magma, Sage, TeX
C_7\rtimes {\rm He}_3\rtimes C_2 % in TeX
G:=Group("C7:He3:C2"); // GroupNames label
G:=SmallGroup(378,17);
// by ID
G=gap.SmallGroup(378,17);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,182,187,723,1359]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^4,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,d*e=e*d>;
// generators/relations
Export
Subgroup lattice of C7⋊He3⋊C2 in TeX
Character table of C7⋊He3⋊C2 in TeX