direct product, non-abelian, supersoluble, monomial
Aliases: C7×He3⋊C2, He3⋊2C14, (C3×C21)⋊4S3, (C7×He3)⋊5C2, C32⋊2(S3×C7), C21.4(C3⋊S3), C3.2(C7×C3⋊S3), SmallGroup(378,41)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — 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, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1, cd=dc, ce=ec, ede=d-1 >
Smallest permutation representation of C7×He3⋊C2
►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 57 44)(2 58 45)(3 59 46)(4 60 47)(5 61 48)(6 62 49)(7 63 43)(8 39 29)(9 40 30)(10 41 31)(11 42 32)(12 36 33)(13 37 34)(14 38 35)(15 26 53)(16 27 54)(17 28 55)(18 22 56)(19 23 50)(20 24 51)(21 25 52)
(1 27 9)(2 28 10)(3 22 11)(4 23 12)(5 24 13)(6 25 14)(7 26 8)(15 29 43)(16 30 44)(17 31 45)(18 32 46)(19 33 47)(20 34 48)(21 35 49)(36 60 50)(37 61 51)(38 62 52)(39 63 53)(40 57 54)(41 58 55)(42 59 56)
(1 40 30)(2 41 31)(3 42 32)(4 36 33)(5 37 34)(6 38 35)(7 39 29)(8 53 15)(9 54 16)(10 55 17)(11 56 18)(12 50 19)(13 51 20)(14 52 21)(22 59 46)(23 60 47)(24 61 48)(25 62 49)(26 63 43)(27 57 44)(28 58 45)
(15 53)(16 54)(17 55)(18 56)(19 50)(20 51)(21 52)(29 39)(30 40)(31 41)(32 42)(33 36)(34 37)(35 38)(43 63)(44 57)(45 58)(46 59)(47 60)(48 61)(49 62)
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), (1,57,44)(2,58,45)(3,59,46)(4,60,47)(5,61,48)(6,62,49)(7,63,43)(8,39,29)(9,40,30)(10,41,31)(11,42,32)(12,36,33)(13,37,34)(14,38,35)(15,26,53)(16,27,54)(17,28,55)(18,22,56)(19,23,50)(20,24,51)(21,25,52), (1,27,9)(2,28,10)(3,22,11)(4,23,12)(5,24,13)(6,25,14)(7,26,8)(15,29,43)(16,30,44)(17,31,45)(18,32,46)(19,33,47)(20,34,48)(21,35,49)(36,60,50)(37,61,51)(38,62,52)(39,63,53)(40,57,54)(41,58,55)(42,59,56), (1,40,30)(2,41,31)(3,42,32)(4,36,33)(5,37,34)(6,38,35)(7,39,29)(8,53,15)(9,54,16)(10,55,17)(11,56,18)(12,50,19)(13,51,20)(14,52,21)(22,59,46)(23,60,47)(24,61,48)(25,62,49)(26,63,43)(27,57,44)(28,58,45), (15,53)(16,54)(17,55)(18,56)(19,50)(20,51)(21,52)(29,39)(30,40)(31,41)(32,42)(33,36)(34,37)(35,38)(43,63)(44,57)(45,58)(46,59)(47,60)(48,61)(49,62)>;
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), (1,57,44)(2,58,45)(3,59,46)(4,60,47)(5,61,48)(6,62,49)(7,63,43)(8,39,29)(9,40,30)(10,41,31)(11,42,32)(12,36,33)(13,37,34)(14,38,35)(15,26,53)(16,27,54)(17,28,55)(18,22,56)(19,23,50)(20,24,51)(21,25,52), (1,27,9)(2,28,10)(3,22,11)(4,23,12)(5,24,13)(6,25,14)(7,26,8)(15,29,43)(16,30,44)(17,31,45)(18,32,46)(19,33,47)(20,34,48)(21,35,49)(36,60,50)(37,61,51)(38,62,52)(39,63,53)(40,57,54)(41,58,55)(42,59,56), (1,40,30)(2,41,31)(3,42,32)(4,36,33)(5,37,34)(6,38,35)(7,39,29)(8,53,15)(9,54,16)(10,55,17)(11,56,18)(12,50,19)(13,51,20)(14,52,21)(22,59,46)(23,60,47)(24,61,48)(25,62,49)(26,63,43)(27,57,44)(28,58,45), (15,53)(16,54)(17,55)(18,56)(19,50)(20,51)(21,52)(29,39)(30,40)(31,41)(32,42)(33,36)(34,37)(35,38)(43,63)(44,57)(45,58)(46,59)(47,60)(48,61)(49,62) );
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)], [(1,57,44),(2,58,45),(3,59,46),(4,60,47),(5,61,48),(6,62,49),(7,63,43),(8,39,29),(9,40,30),(10,41,31),(11,42,32),(12,36,33),(13,37,34),(14,38,35),(15,26,53),(16,27,54),(17,28,55),(18,22,56),(19,23,50),(20,24,51),(21,25,52)], [(1,27,9),(2,28,10),(3,22,11),(4,23,12),(5,24,13),(6,25,14),(7,26,8),(15,29,43),(16,30,44),(17,31,45),(18,32,46),(19,33,47),(20,34,48),(21,35,49),(36,60,50),(37,61,51),(38,62,52),(39,63,53),(40,57,54),(41,58,55),(42,59,56)], [(1,40,30),(2,41,31),(3,42,32),(4,36,33),(5,37,34),(6,38,35),(7,39,29),(8,53,15),(9,54,16),(10,55,17),(11,56,18),(12,50,19),(13,51,20),(14,52,21),(22,59,46),(23,60,47),(24,61,48),(25,62,49),(26,63,43),(27,57,44),(28,58,45)], [(15,53),(16,54),(17,55),(18,56),(19,50),(20,51),(21,52),(29,39),(30,40),(31,41),(32,42),(33,36),(34,37),(35,38),(43,63),(44,57),(45,58),(46,59),(47,60),(48,61),(49,62)]])
70 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 7A | ··· | 7F | 14A | ··· | 14F | 21A | ··· | 21L | 21M | ··· | 21AJ | 42A | ··· | 42L |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 9 | 1 | 1 | 6 | 6 | 6 | 6 | 9 | 9 | 1 | ··· | 1 | 9 | ··· | 9 | 1 | ··· | 1 | 6 | ··· | 6 | 9 | ··· | 9 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 |
| type | + | + | + | |||||
| image | C1 | C2 | C7 | C14 | S3 | S3×C7 | He3⋊C2 | C7×He3⋊C2 |
| kernel | C7×He3⋊C2 | C7×He3 | He3⋊C2 | He3 | C3×C21 | C32 | C7 | C1 |
| # reps | 1 | 1 | 6 | 6 | 4 | 24 | 4 | 24 |
Matrix representation of C7×He3⋊C2 ►in GL3(𝔽43) generated by
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 0 | 1 | 0 |
| 42 | 42 | 35 |
| 0 | 0 | 1 |
| 36 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 0 | 36 | 0 |
| 42 | 42 | 35 |
| 37 | 1 | 1 |
| 1 | 0 | 0 |
| 42 | 42 | 35 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(43))| [4,0,0,0,4,0,0,0,4],[0,42,0,1,42,0,0,35,1],[36,0,0,0,36,0,0,0,36],[0,42,37,36,42,1,0,35,1],[1,42,0,0,42,0,0,35,1] >;
C7×He3⋊C2 in GAP, Magma, Sage, TeX
C_7\times {\rm He}_3\rtimes C_2 % in TeX
G:=Group("C7xHe3:C2"); // GroupNames label
G:=SmallGroup(378,41);
// by ID
G=gap.SmallGroup(378,41);
# by ID
G:=PCGroup([5,-2,-7,-3,-3,-3,422,1683,253]);
// 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,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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