direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C7⋊C3×C21, C21⋊C21, C72⋊2C32, C7⋊(C3×C21), (C7×C21)⋊1C3, SmallGroup(441,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C3×C21 |
Generators and relations for C7⋊C3×C21
G = < a,b,c | a21=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C7⋊C3×C21
►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 4 7 10 13 16 19)(2 5 8 11 14 17 20)(3 6 9 12 15 18 21)(22 28 34 40 25 31 37)(23 29 35 41 26 32 38)(24 30 36 42 27 33 39)(43 55 46 58 49 61 52)(44 56 47 59 50 62 53)(45 57 48 60 51 63 54)
(1 63 24)(2 43 25)(3 44 26)(4 45 27)(5 46 28)(6 47 29)(7 48 30)(8 49 31)(9 50 32)(10 51 33)(11 52 34)(12 53 35)(13 54 36)(14 55 37)(15 56 38)(16 57 39)(17 58 40)(18 59 41)(19 60 42)(20 61 22)(21 62 23)
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), (1,4,7,10,13,16,19)(2,5,8,11,14,17,20)(3,6,9,12,15,18,21)(22,28,34,40,25,31,37)(23,29,35,41,26,32,38)(24,30,36,42,27,33,39)(43,55,46,58,49,61,52)(44,56,47,59,50,62,53)(45,57,48,60,51,63,54), (1,63,24)(2,43,25)(3,44,26)(4,45,27)(5,46,28)(6,47,29)(7,48,30)(8,49,31)(9,50,32)(10,51,33)(11,52,34)(12,53,35)(13,54,36)(14,55,37)(15,56,38)(16,57,39)(17,58,40)(18,59,41)(19,60,42)(20,61,22)(21,62,23)>;
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), (1,4,7,10,13,16,19)(2,5,8,11,14,17,20)(3,6,9,12,15,18,21)(22,28,34,40,25,31,37)(23,29,35,41,26,32,38)(24,30,36,42,27,33,39)(43,55,46,58,49,61,52)(44,56,47,59,50,62,53)(45,57,48,60,51,63,54), (1,63,24)(2,43,25)(3,44,26)(4,45,27)(5,46,28)(6,47,29)(7,48,30)(8,49,31)(9,50,32)(10,51,33)(11,52,34)(12,53,35)(13,54,36)(14,55,37)(15,56,38)(16,57,39)(17,58,40)(18,59,41)(19,60,42)(20,61,22)(21,62,23) );
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)], [(1,4,7,10,13,16,19),(2,5,8,11,14,17,20),(3,6,9,12,15,18,21),(22,28,34,40,25,31,37),(23,29,35,41,26,32,38),(24,30,36,42,27,33,39),(43,55,46,58,49,61,52),(44,56,47,59,50,62,53),(45,57,48,60,51,63,54)], [(1,63,24),(2,43,25),(3,44,26),(4,45,27),(5,46,28),(6,47,29),(7,48,30),(8,49,31),(9,50,32),(10,51,33),(11,52,34),(12,53,35),(13,54,36),(14,55,37),(15,56,38),(16,57,39),(17,58,40),(18,59,41),(19,60,42),(20,61,22),(21,62,23)]])
105 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3H | 7A | ··· | 7F | 7G | ··· | 7T | 21A | ··· | 21L | 21M | ··· | 21AN | 21AO | ··· | 21BX |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 7 | ··· | 7 | 7 | ··· | 7 | 21 | ··· | 21 | 21 | ··· | 21 | 21 | ··· | 21 |
| size | 1 | 1 | 1 | 7 | ··· | 7 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 7 | ··· | 7 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | |||||||||
| image | C1 | C3 | C3 | C7 | C21 | C21 | C7⋊C3 | C3×C7⋊C3 | C7×C7⋊C3 | C7⋊C3×C21 |
| kernel | C7⋊C3×C21 | C7×C7⋊C3 | C7×C21 | C3×C7⋊C3 | C7⋊C3 | C21 | C21 | C7 | C3 | C1 |
| # reps | 1 | 6 | 2 | 6 | 36 | 12 | 2 | 4 | 12 | 24 |
Matrix representation of C7⋊C3×C21 ►in GL3(𝔽43) generated by
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 41 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 16 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 6 | 0 | 0 |
G:=sub<GL(3,GF(43))| [9,0,0,0,9,0,0,0,9],[41,0,0,0,4,0,0,0,16],[0,0,6,6,0,0,0,6,0] >;
C7⋊C3×C21 in GAP, Magma, Sage, TeX
C_7\rtimes C_3\times C_{21} % in TeX
G:=Group("C7:C3xC21"); // GroupNames label
G:=SmallGroup(441,10);
// by ID
G=gap.SmallGroup(441,10);
# by ID
G:=PCGroup([4,-3,-3,-7,-7,2019]);
// Polycyclic
G:=Group<a,b,c|a^21=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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