metabelian, soluble, monomial, A-group
Aliases: C21.A4, C3.(C7⋊A4), C7⋊(C3.A4), C22⋊(C7⋊C9), (C2×C14)⋊2C9, (C2×C42).2C3, (C2×C6).(C7⋊C3), SmallGroup(252,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C14 — C21.A4 |
Generators and relations for C21.A4
G = < a,b,c,d | a21=b2=c2=1, d3=a7, ab=ba, ac=ca, dad-1=a4, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of C21.A4
►On 126 points
Generators in S126
(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)(64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)
(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)(61 72)(62 73)(63 74)(85 125)(86 126)(87 106)(88 107)(89 108)(90 109)(91 110)(92 111)(93 112)(94 113)(95 114)(96 115)(97 116)(98 117)(99 118)(100 119)(101 120)(102 121)(103 122)(104 123)(105 124)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 41)(17 42)(18 22)(19 23)(20 24)(21 25)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)(61 72)(62 73)(63 74)
(1 111 44 8 118 51 15 125 58)(2 106 48 9 113 55 16 120 62)(3 122 52 10 108 59 17 115 45)(4 117 56 11 124 63 18 110 49)(5 112 60 12 119 46 19 126 53)(6 107 43 13 114 50 20 121 57)(7 123 47 14 109 54 21 116 61)(22 91 81 29 98 67 36 105 74)(23 86 64 30 93 71 37 100 78)(24 102 68 31 88 75 38 95 82)(25 97 72 32 104 79 39 90 65)(26 92 76 33 99 83 40 85 69)(27 87 80 34 94 66 41 101 73)(28 103 84 35 89 70 42 96 77)
G:=sub<Sym(126)| (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)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126), (43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74)(85,125)(86,126)(87,106)(88,107)(89,108)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114)(96,115)(97,116)(98,117)(99,118)(100,119)(101,120)(102,121)(103,122)(104,123)(105,124), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,22)(19,23)(20,24)(21,25)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74), (1,111,44,8,118,51,15,125,58)(2,106,48,9,113,55,16,120,62)(3,122,52,10,108,59,17,115,45)(4,117,56,11,124,63,18,110,49)(5,112,60,12,119,46,19,126,53)(6,107,43,13,114,50,20,121,57)(7,123,47,14,109,54,21,116,61)(22,91,81,29,98,67,36,105,74)(23,86,64,30,93,71,37,100,78)(24,102,68,31,88,75,38,95,82)(25,97,72,32,104,79,39,90,65)(26,92,76,33,99,83,40,85,69)(27,87,80,34,94,66,41,101,73)(28,103,84,35,89,70,42,96,77)>;
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)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126), (43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74)(85,125)(86,126)(87,106)(88,107)(89,108)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114)(96,115)(97,116)(98,117)(99,118)(100,119)(101,120)(102,121)(103,122)(104,123)(105,124), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,22)(19,23)(20,24)(21,25)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74), (1,111,44,8,118,51,15,125,58)(2,106,48,9,113,55,16,120,62)(3,122,52,10,108,59,17,115,45)(4,117,56,11,124,63,18,110,49)(5,112,60,12,119,46,19,126,53)(6,107,43,13,114,50,20,121,57)(7,123,47,14,109,54,21,116,61)(22,91,81,29,98,67,36,105,74)(23,86,64,30,93,71,37,100,78)(24,102,68,31,88,75,38,95,82)(25,97,72,32,104,79,39,90,65)(26,92,76,33,99,83,40,85,69)(27,87,80,34,94,66,41,101,73)(28,103,84,35,89,70,42,96,77) );
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),(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)], [(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71),(61,72),(62,73),(63,74),(85,125),(86,126),(87,106),(88,107),(89,108),(90,109),(91,110),(92,111),(93,112),(94,113),(95,114),(96,115),(97,116),(98,117),(99,118),(100,119),(101,120),(102,121),(103,122),(104,123),(105,124)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,41),(17,42),(18,22),(19,23),(20,24),(21,25),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71),(61,72),(62,73),(63,74)], [(1,111,44,8,118,51,15,125,58),(2,106,48,9,113,55,16,120,62),(3,122,52,10,108,59,17,115,45),(4,117,56,11,124,63,18,110,49),(5,112,60,12,119,46,19,126,53),(6,107,43,13,114,50,20,121,57),(7,123,47,14,109,54,21,116,61),(22,91,81,29,98,67,36,105,74),(23,86,64,30,93,71,37,100,78),(24,102,68,31,88,75,38,95,82),(25,97,72,32,104,79,39,90,65),(26,92,76,33,99,83,40,85,69),(27,87,80,34,94,66,41,101,73),(28,103,84,35,89,70,42,96,77)]])
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | 7B | 9A | ··· | 9F | 14A | ··· | 14F | 21A | 21B | 21C | 21D | 42A | ··· | 42L |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | 7 | 9 | ··· | 9 | 14 | ··· | 14 | 21 | 21 | 21 | 21 | 42 | ··· | 42 |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 28 | ··· | 28 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 |
36 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | |||||||
| image | C1 | C3 | C9 | A4 | C7⋊C3 | C3.A4 | C7⋊C9 | C7⋊A4 | C21.A4 |
| kernel | C21.A4 | C2×C42 | C2×C14 | C21 | C2×C6 | C7 | C22 | C3 | C1 |
| # reps | 1 | 2 | 6 | 1 | 2 | 2 | 4 | 6 | 12 |
Matrix representation of C21.A4 ►in GL6(𝔽127)
| 20 | 20 | 79 | 0 | 0 | 0 |
| 79 | 59 | 107 | 0 | 0 | 0 |
| 107 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 79 | 8 | 0 |
| 0 | 0 | 0 | 64 | 0 | 64 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 123 | 126 | 0 |
| 0 | 0 | 0 | 123 | 0 | 126 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 126 | 0 | 0 |
| 0 | 0 | 0 | 0 | 126 | 0 |
| 0 | 0 | 0 | 4 | 0 | 1 |
| 81 | 61 | 107 | 0 | 0 | 0 |
| 22 | 44 | 20 | 0 | 0 | 0 |
| 107 | 2 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 123 | 125 | 0 |
| 0 | 0 | 0 | 10 | 4 | 1 |
| 0 | 0 | 0 | 71 | 4 | 0 |
G:=sub<GL(6,GF(127))| [20,79,107,0,0,0,20,59,0,0,0,0,79,107,0,0,0,0,0,0,0,32,79,64,0,0,0,0,8,0,0,0,0,0,0,64],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,123,123,0,0,0,0,126,0,0,0,0,0,0,126],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,126,0,4,0,0,0,0,126,0,0,0,0,0,0,1],[81,22,107,0,0,0,61,44,2,0,0,0,107,20,2,0,0,0,0,0,0,123,10,71,0,0,0,125,4,4,0,0,0,0,1,0] >;
C21.A4 in GAP, Magma, Sage, TeX
C_{21}.A_4 % in TeX
G:=Group("C21.A4"); // GroupNames label
G:=SmallGroup(252,11);
// by ID
G=gap.SmallGroup(252,11);
# by ID
G:=PCGroup([5,-3,-3,-2,2,-7,15,272,543,1804]);
// Polycyclic
G:=Group<a,b,c,d|a^21=b^2=c^2=1,d^3=a^7,a*b=b*a,a*c=c*a,d*a*d^-1=a^4,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
Export