direct product, metabelian, supersoluble, monomial
Aliases: C7×C32⋊C6, C32⋊C42, He3⋊1C14, C3⋊S3⋊C21, (C3×C21)⋊3S3, (C3×C21)⋊8C6, (C7×He3)⋊4C2, C3.2(S3×C21), C32⋊1(S3×C7), C21.16(C3×S3), (C7×C3⋊S3)⋊1C3, SmallGroup(378,34)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C7×C32⋊C6 |
Generators and relations for C7×C32⋊C6
G = < a,b,c,d | a7=b3=c3=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >
Smallest permutation representation of C7×C32⋊C6
►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 16 40)(2 17 41)(3 18 42)(4 19 36)(5 20 37)(6 21 38)(7 15 39)(8 57 29)(9 58 30)(10 59 31)(11 60 32)(12 61 33)(13 62 34)(14 63 35)(22 46 56)(23 47 50)(24 48 51)(25 49 52)(26 43 53)(27 44 54)(28 45 55)
(1 58 27)(2 59 28)(3 60 22)(4 61 23)(5 62 24)(6 63 25)(7 57 26)(8 53 39)(9 54 40)(10 55 41)(11 56 42)(12 50 36)(13 51 37)(14 52 38)(15 29 43)(16 30 44)(17 31 45)(18 32 46)(19 33 47)(20 34 48)(21 35 49)
(8 29 39 43 53 15)(9 30 40 44 54 16)(10 31 41 45 55 17)(11 32 42 46 56 18)(12 33 36 47 50 19)(13 34 37 48 51 20)(14 35 38 49 52 21)(22 60)(23 61)(24 62)(25 63)(26 57)(27 58)(28 59)
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), (1,16,40)(2,17,41)(3,18,42)(4,19,36)(5,20,37)(6,21,38)(7,15,39)(8,57,29)(9,58,30)(10,59,31)(11,60,32)(12,61,33)(13,62,34)(14,63,35)(22,46,56)(23,47,50)(24,48,51)(25,49,52)(26,43,53)(27,44,54)(28,45,55), (1,58,27)(2,59,28)(3,60,22)(4,61,23)(5,62,24)(6,63,25)(7,57,26)(8,53,39)(9,54,40)(10,55,41)(11,56,42)(12,50,36)(13,51,37)(14,52,38)(15,29,43)(16,30,44)(17,31,45)(18,32,46)(19,33,47)(20,34,48)(21,35,49), (8,29,39,43,53,15)(9,30,40,44,54,16)(10,31,41,45,55,17)(11,32,42,46,56,18)(12,33,36,47,50,19)(13,34,37,48,51,20)(14,35,38,49,52,21)(22,60)(23,61)(24,62)(25,63)(26,57)(27,58)(28,59)>;
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), (1,16,40)(2,17,41)(3,18,42)(4,19,36)(5,20,37)(6,21,38)(7,15,39)(8,57,29)(9,58,30)(10,59,31)(11,60,32)(12,61,33)(13,62,34)(14,63,35)(22,46,56)(23,47,50)(24,48,51)(25,49,52)(26,43,53)(27,44,54)(28,45,55), (1,58,27)(2,59,28)(3,60,22)(4,61,23)(5,62,24)(6,63,25)(7,57,26)(8,53,39)(9,54,40)(10,55,41)(11,56,42)(12,50,36)(13,51,37)(14,52,38)(15,29,43)(16,30,44)(17,31,45)(18,32,46)(19,33,47)(20,34,48)(21,35,49), (8,29,39,43,53,15)(9,30,40,44,54,16)(10,31,41,45,55,17)(11,32,42,46,56,18)(12,33,36,47,50,19)(13,34,37,48,51,20)(14,35,38,49,52,21)(22,60)(23,61)(24,62)(25,63)(26,57)(27,58)(28,59) );
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)], [(1,16,40),(2,17,41),(3,18,42),(4,19,36),(5,20,37),(6,21,38),(7,15,39),(8,57,29),(9,58,30),(10,59,31),(11,60,32),(12,61,33),(13,62,34),(14,63,35),(22,46,56),(23,47,50),(24,48,51),(25,49,52),(26,43,53),(27,44,54),(28,45,55)], [(1,58,27),(2,59,28),(3,60,22),(4,61,23),(5,62,24),(6,63,25),(7,57,26),(8,53,39),(9,54,40),(10,55,41),(11,56,42),(12,50,36),(13,51,37),(14,52,38),(15,29,43),(16,30,44),(17,31,45),(18,32,46),(19,33,47),(20,34,48),(21,35,49)], [(8,29,39,43,53,15),(9,30,40,44,54,16),(10,31,41,45,55,17),(11,32,42,46,56,18),(12,33,36,47,50,19),(13,34,37,48,51,20),(14,35,38,49,52,21),(22,60),(23,61),(24,62),(25,63),(26,57),(27,58),(28,59)]])
70 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 7A | ··· | 7F | 14A | ··· | 14F | 21A | ··· | 21F | 21G | ··· | 21R | 21S | ··· | 21AJ | 42A | ··· | 42L |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 21 | ··· | 21 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 9 | 2 | 3 | 3 | 6 | 6 | 6 | 9 | 9 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 | S3 | C3×S3 | S3×C7 | S3×C21 | C32⋊C6 | C7×C32⋊C6 |
| kernel | C7×C32⋊C6 | C7×He3 | C7×C3⋊S3 | C3×C21 | C32⋊C6 | He3 | C3⋊S3 | C32 | C3×C21 | C21 | C32 | C3 | C7 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 1 | 2 | 6 | 12 | 1 | 6 |
Matrix representation of C7×C32⋊C6 ►in GL6(𝔽43)
| 41 | 0 | 0 | 0 | 0 | 0 |
| 0 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 0 | 0 | 0 |
| 0 | 0 | 0 | 41 | 0 | 0 |
| 0 | 0 | 0 | 0 | 41 | 0 |
| 0 | 0 | 0 | 0 | 0 | 41 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 42 | 42 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 42 | 42 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 42 | 42 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 42 | 42 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 42 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 42 | 42 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 42 | 42 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 42 | 42 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(43))| [41,0,0,0,0,0,0,41,0,0,0,0,0,0,41,0,0,0,0,0,0,41,0,0,0,0,0,0,41,0,0,0,0,0,0,41],[0,0,0,42,0,0,0,0,1,42,0,0,0,0,0,0,0,42,0,0,0,0,1,42,0,42,0,0,0,0,1,42,0,0,0,0],[0,42,0,0,0,0,1,42,0,0,0,0,0,0,0,42,0,0,0,0,1,42,0,0,0,0,0,0,0,42,0,0,0,0,1,42],[1,42,0,0,0,0,0,42,0,0,0,0,0,0,0,0,42,0,0,0,0,0,42,1,0,0,0,1,0,0,0,0,1,0,0,0] >;
C7×C32⋊C6 in GAP, Magma, Sage, TeX
C_7\times C_3^2\rtimes C_6
% in TeX
G:=Group("C7xC3^2:C6"); // GroupNames label
G:=SmallGroup(378,34);
// by ID
G=gap.SmallGroup(378,34);
# by ID
G:=PCGroup([5,-2,-3,-7,-3,-3,1683,1688,6304]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^3=c^3=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=c^-1>;
// generators/relations
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