direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C32×D7, C21⋊6C6, C7⋊3(C3×C6), (C3×C21)⋊3C2, SmallGroup(126,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C32×D7 |
Generators and relations for C32×D7
G = < a,b,c,d | a3=b3=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of C32×D7
►On 63 points
Generators in S63
(1 62 34)(2 63 35)(3 57 29)(4 58 30)(5 59 31)(6 60 32)(7 61 33)(8 43 36)(9 44 37)(10 45 38)(11 46 39)(12 47 40)(13 48 41)(14 49 42)(15 50 22)(16 51 23)(17 52 24)(18 53 25)(19 54 26)(20 55 27)(21 56 28)
(1 20 13)(2 21 14)(3 15 8)(4 16 9)(5 17 10)(6 18 11)(7 19 12)(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)
(1 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)
G:=sub<Sym(63)| (1,62,34)(2,63,35)(3,57,29)(4,58,30)(5,59,31)(6,60,32)(7,61,33)(8,43,36)(9,44,37)(10,45,38)(11,46,39)(12,47,40)(13,48,41)(14,49,42)(15,50,22)(16,51,23)(17,52,24)(18,53,25)(19,54,26)(20,55,27)(21,56,28), (1,20,13)(2,21,14)(3,15,8)(4,16,9)(5,17,10)(6,18,11)(7,19,12)(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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)(43,45)(46,49)(47,48)(50,52)(53,56)(54,55)(57,59)(60,63)(61,62)>;
G:=Group( (1,62,34)(2,63,35)(3,57,29)(4,58,30)(5,59,31)(6,60,32)(7,61,33)(8,43,36)(9,44,37)(10,45,38)(11,46,39)(12,47,40)(13,48,41)(14,49,42)(15,50,22)(16,51,23)(17,52,24)(18,53,25)(19,54,26)(20,55,27)(21,56,28), (1,20,13)(2,21,14)(3,15,8)(4,16,9)(5,17,10)(6,18,11)(7,19,12)(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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)(43,45)(46,49)(47,48)(50,52)(53,56)(54,55)(57,59)(60,63)(61,62) );
G=PermutationGroup([[(1,62,34),(2,63,35),(3,57,29),(4,58,30),(5,59,31),(6,60,32),(7,61,33),(8,43,36),(9,44,37),(10,45,38),(11,46,39),(12,47,40),(13,48,41),(14,49,42),(15,50,22),(16,51,23),(17,52,24),(18,53,25),(19,54,26),(20,55,27),(21,56,28)], [(1,20,13),(2,21,14),(3,15,8),(4,16,9),(5,17,10),(6,18,11),(7,19,12),(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)], [(1,7),(2,6),(3,5),(8,10),(11,14),(12,13),(15,17),(18,21),(19,20),(22,24),(25,28),(26,27),(29,31),(32,35),(33,34),(36,38),(39,42),(40,41),(43,45),(46,49),(47,48),(50,52),(53,56),(54,55),(57,59),(60,63),(61,62)]])
C32×D7 is a maximal subgroup of
C32.F7 D7⋊He3
45 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 7A | 7B | 7C | 21A | ··· | 21X |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 7 | 7 | 7 | 21 | ··· | 21 |
| size | 1 | 7 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 2 | ··· | 2 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D7 | C3×D7 |
| kernel | C32×D7 | C3×C21 | C3×D7 | C21 | C32 | C3 |
| # reps | 1 | 1 | 8 | 8 | 3 | 24 |
Matrix representation of C32×D7 ►in GL3(𝔽43) generated by
| 36 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 6 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 42 | 1 |
| 0 | 33 | 9 |
| 42 | 0 | 0 |
| 0 | 42 | 0 |
| 0 | 33 | 1 |
G:=sub<GL(3,GF(43))| [36,0,0,0,6,0,0,0,6],[6,0,0,0,1,0,0,0,1],[1,0,0,0,42,33,0,1,9],[42,0,0,0,42,33,0,0,1] >;
C32×D7 in GAP, Magma, Sage, TeX
C_3^2\times D_7
% in TeX
G:=Group("C3^2xD7"); // GroupNames label
G:=SmallGroup(126,11);
// by ID
G=gap.SmallGroup(126,11);
# by ID
G:=PCGroup([4,-2,-3,-3,-7,1731]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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