direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C7×He3, C32⋊C21, C21.7C32, (C3×C21)⋊1C3, C3.1(C3×C21), SmallGroup(189,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×He3
G = < a,b,c,d | a7=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >
Smallest permutation representation of C7×He3
►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)
(8 16 63)(9 17 57)(10 18 58)(11 19 59)(12 20 60)(13 21 61)(14 15 62)(36 52 48)(37 53 49)(38 54 43)(39 55 44)(40 56 45)(41 50 46)(42 51 47)
(1 32 28)(2 33 22)(3 34 23)(4 35 24)(5 29 25)(6 30 26)(7 31 27)(8 63 16)(9 57 17)(10 58 18)(11 59 19)(12 60 20)(13 61 21)(14 62 15)(36 52 48)(37 53 49)(38 54 43)(39 55 44)(40 56 45)(41 50 46)(42 51 47)
(1 58 37)(2 59 38)(3 60 39)(4 61 40)(5 62 41)(6 63 42)(7 57 36)(8 47 26)(9 48 27)(10 49 28)(11 43 22)(12 44 23)(13 45 24)(14 46 25)(15 50 29)(16 51 30)(17 52 31)(18 53 32)(19 54 33)(20 55 34)(21 56 35)
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), (8,16,63)(9,17,57)(10,18,58)(11,19,59)(12,20,60)(13,21,61)(14,15,62)(36,52,48)(37,53,49)(38,54,43)(39,55,44)(40,56,45)(41,50,46)(42,51,47), (1,32,28)(2,33,22)(3,34,23)(4,35,24)(5,29,25)(6,30,26)(7,31,27)(8,63,16)(9,57,17)(10,58,18)(11,59,19)(12,60,20)(13,61,21)(14,62,15)(36,52,48)(37,53,49)(38,54,43)(39,55,44)(40,56,45)(41,50,46)(42,51,47), (1,58,37)(2,59,38)(3,60,39)(4,61,40)(5,62,41)(6,63,42)(7,57,36)(8,47,26)(9,48,27)(10,49,28)(11,43,22)(12,44,23)(13,45,24)(14,46,25)(15,50,29)(16,51,30)(17,52,31)(18,53,32)(19,54,33)(20,55,34)(21,56,35)>;
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), (8,16,63)(9,17,57)(10,18,58)(11,19,59)(12,20,60)(13,21,61)(14,15,62)(36,52,48)(37,53,49)(38,54,43)(39,55,44)(40,56,45)(41,50,46)(42,51,47), (1,32,28)(2,33,22)(3,34,23)(4,35,24)(5,29,25)(6,30,26)(7,31,27)(8,63,16)(9,57,17)(10,58,18)(11,59,19)(12,60,20)(13,61,21)(14,62,15)(36,52,48)(37,53,49)(38,54,43)(39,55,44)(40,56,45)(41,50,46)(42,51,47), (1,58,37)(2,59,38)(3,60,39)(4,61,40)(5,62,41)(6,63,42)(7,57,36)(8,47,26)(9,48,27)(10,49,28)(11,43,22)(12,44,23)(13,45,24)(14,46,25)(15,50,29)(16,51,30)(17,52,31)(18,53,32)(19,54,33)(20,55,34)(21,56,35) );
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)], [(8,16,63),(9,17,57),(10,18,58),(11,19,59),(12,20,60),(13,21,61),(14,15,62),(36,52,48),(37,53,49),(38,54,43),(39,55,44),(40,56,45),(41,50,46),(42,51,47)], [(1,32,28),(2,33,22),(3,34,23),(4,35,24),(5,29,25),(6,30,26),(7,31,27),(8,63,16),(9,57,17),(10,58,18),(11,59,19),(12,60,20),(13,61,21),(14,62,15),(36,52,48),(37,53,49),(38,54,43),(39,55,44),(40,56,45),(41,50,46),(42,51,47)], [(1,58,37),(2,59,38),(3,60,39),(4,61,40),(5,62,41),(6,63,42),(7,57,36),(8,47,26),(9,48,27),(10,49,28),(11,43,22),(12,44,23),(13,45,24),(14,46,25),(15,50,29),(16,51,30),(17,52,31),(18,53,32),(19,54,33),(20,55,34),(21,56,35)]])
C7×He3 is a maximal subgroup of
He3⋊D7 C32⋊D21
77 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3J | 7A | ··· | 7F | 21A | ··· | 21L | 21M | ··· | 21BH |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 7 | ··· | 7 | 21 | ··· | 21 | 21 | ··· | 21 |
| size | 1 | 1 | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C7 | C21 | He3 | C7×He3 |
| kernel | C7×He3 | C3×C21 | He3 | C32 | C7 | C1 |
| # reps | 1 | 8 | 6 | 48 | 2 | 12 |
Matrix representation of C7×He3 ►in GL3(𝔽43) generated by
| 21 | 0 | 0 |
| 0 | 21 | 0 |
| 0 | 0 | 21 |
| 1 | 0 | 0 |
| 1 | 36 | 0 |
| 37 | 0 | 6 |
| 36 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 1 | 35 | 0 |
| 0 | 42 | 1 |
| 0 | 42 | 0 |
G:=sub<GL(3,GF(43))| [21,0,0,0,21,0,0,0,21],[1,1,37,0,36,0,0,0,6],[36,0,0,0,36,0,0,0,36],[1,0,0,35,42,42,0,1,0] >;
C7×He3 in GAP, Magma, Sage, TeX
C_7\times {\rm He}_3 % in TeX
G:=Group("C7xHe3"); // GroupNames label
G:=SmallGroup(189,10);
// by ID
G=gap.SmallGroup(189,10);
# by ID
G:=PCGroup([4,-3,-3,-7,-3,529]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations
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