direct product, metabelian, soluble, monomial, A-group
Aliases: C7×C3.A4, C22⋊C63, C21.2A4, C3.(C7×A4), (C2×C14)⋊1C9, (C2×C6).C21, (C2×C42).1C3, SmallGroup(252,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C7×C3.A4 |
Generators and relations for C7×C3.A4
G = < a,b,c,d,e | a7=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >
Smallest permutation representation of C7×C3.A4
►On 126 points
Generators in S126
(1 112 52 93 103 122 20)(2 113 53 94 104 123 21)(3 114 54 95 105 124 22)(4 115 46 96 106 125 23)(5 116 47 97 107 126 24)(6 117 48 98 108 118 25)(7 109 49 99 100 119 26)(8 110 50 91 101 120 27)(9 111 51 92 102 121 19)(10 71 77 86 57 29 42)(11 72 78 87 58 30 43)(12 64 79 88 59 31 44)(13 65 80 89 60 32 45)(14 66 81 90 61 33 37)(15 67 73 82 62 34 38)(16 68 74 83 63 35 39)(17 69 75 84 55 36 40)(18 70 76 85 56 28 41)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 103 106)(101 104 107)(102 105 108)(109 112 115)(110 113 116)(111 114 117)(118 121 124)(119 122 125)(120 123 126)
(1 79)(3 81)(4 73)(6 75)(7 76)(9 78)(11 121)(12 122)(14 124)(15 125)(17 118)(18 119)(19 72)(20 64)(22 66)(23 67)(25 69)(26 70)(28 99)(30 92)(31 93)(33 95)(34 96)(36 98)(37 105)(38 106)(40 108)(41 100)(43 102)(44 103)(46 62)(48 55)(49 56)(51 58)(52 59)(54 61)(82 115)(84 117)(85 109)(87 111)(88 112)(90 114)
(1 79)(2 80)(4 73)(5 74)(7 76)(8 77)(10 120)(12 122)(13 123)(15 125)(16 126)(18 119)(20 64)(21 65)(23 67)(24 68)(26 70)(27 71)(28 99)(29 91)(31 93)(32 94)(34 96)(35 97)(38 106)(39 107)(41 100)(42 101)(44 103)(45 104)(46 62)(47 63)(49 56)(50 57)(52 59)(53 60)(82 115)(83 116)(85 109)(86 110)(88 112)(89 113)
(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)
G:=sub<Sym(126)| (1,112,52,93,103,122,20)(2,113,53,94,104,123,21)(3,114,54,95,105,124,22)(4,115,46,96,106,125,23)(5,116,47,97,107,126,24)(6,117,48,98,108,118,25)(7,109,49,99,100,119,26)(8,110,50,91,101,120,27)(9,111,51,92,102,121,19)(10,71,77,86,57,29,42)(11,72,78,87,58,30,43)(12,64,79,88,59,31,44)(13,65,80,89,60,32,45)(14,66,81,90,61,33,37)(15,67,73,82,62,34,38)(16,68,74,83,63,35,39)(17,69,75,84,55,36,40)(18,70,76,85,56,28,41), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108)(109,112,115)(110,113,116)(111,114,117)(118,121,124)(119,122,125)(120,123,126), (1,79)(3,81)(4,73)(6,75)(7,76)(9,78)(11,121)(12,122)(14,124)(15,125)(17,118)(18,119)(19,72)(20,64)(22,66)(23,67)(25,69)(26,70)(28,99)(30,92)(31,93)(33,95)(34,96)(36,98)(37,105)(38,106)(40,108)(41,100)(43,102)(44,103)(46,62)(48,55)(49,56)(51,58)(52,59)(54,61)(82,115)(84,117)(85,109)(87,111)(88,112)(90,114), (1,79)(2,80)(4,73)(5,74)(7,76)(8,77)(10,120)(12,122)(13,123)(15,125)(16,126)(18,119)(20,64)(21,65)(23,67)(24,68)(26,70)(27,71)(28,99)(29,91)(31,93)(32,94)(34,96)(35,97)(38,106)(39,107)(41,100)(42,101)(44,103)(45,104)(46,62)(47,63)(49,56)(50,57)(52,59)(53,60)(82,115)(83,116)(85,109)(86,110)(88,112)(89,113), (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)>;
G:=Group( (1,112,52,93,103,122,20)(2,113,53,94,104,123,21)(3,114,54,95,105,124,22)(4,115,46,96,106,125,23)(5,116,47,97,107,126,24)(6,117,48,98,108,118,25)(7,109,49,99,100,119,26)(8,110,50,91,101,120,27)(9,111,51,92,102,121,19)(10,71,77,86,57,29,42)(11,72,78,87,58,30,43)(12,64,79,88,59,31,44)(13,65,80,89,60,32,45)(14,66,81,90,61,33,37)(15,67,73,82,62,34,38)(16,68,74,83,63,35,39)(17,69,75,84,55,36,40)(18,70,76,85,56,28,41), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108)(109,112,115)(110,113,116)(111,114,117)(118,121,124)(119,122,125)(120,123,126), (1,79)(3,81)(4,73)(6,75)(7,76)(9,78)(11,121)(12,122)(14,124)(15,125)(17,118)(18,119)(19,72)(20,64)(22,66)(23,67)(25,69)(26,70)(28,99)(30,92)(31,93)(33,95)(34,96)(36,98)(37,105)(38,106)(40,108)(41,100)(43,102)(44,103)(46,62)(48,55)(49,56)(51,58)(52,59)(54,61)(82,115)(84,117)(85,109)(87,111)(88,112)(90,114), (1,79)(2,80)(4,73)(5,74)(7,76)(8,77)(10,120)(12,122)(13,123)(15,125)(16,126)(18,119)(20,64)(21,65)(23,67)(24,68)(26,70)(27,71)(28,99)(29,91)(31,93)(32,94)(34,96)(35,97)(38,106)(39,107)(41,100)(42,101)(44,103)(45,104)(46,62)(47,63)(49,56)(50,57)(52,59)(53,60)(82,115)(83,116)(85,109)(86,110)(88,112)(89,113), (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) );
G=PermutationGroup([[(1,112,52,93,103,122,20),(2,113,53,94,104,123,21),(3,114,54,95,105,124,22),(4,115,46,96,106,125,23),(5,116,47,97,107,126,24),(6,117,48,98,108,118,25),(7,109,49,99,100,119,26),(8,110,50,91,101,120,27),(9,111,51,92,102,121,19),(10,71,77,86,57,29,42),(11,72,78,87,58,30,43),(12,64,79,88,59,31,44),(13,65,80,89,60,32,45),(14,66,81,90,61,33,37),(15,67,73,82,62,34,38),(16,68,74,83,63,35,39),(17,69,75,84,55,36,40),(18,70,76,85,56,28,41)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81),(82,85,88),(83,86,89),(84,87,90),(91,94,97),(92,95,98),(93,96,99),(100,103,106),(101,104,107),(102,105,108),(109,112,115),(110,113,116),(111,114,117),(118,121,124),(119,122,125),(120,123,126)], [(1,79),(3,81),(4,73),(6,75),(7,76),(9,78),(11,121),(12,122),(14,124),(15,125),(17,118),(18,119),(19,72),(20,64),(22,66),(23,67),(25,69),(26,70),(28,99),(30,92),(31,93),(33,95),(34,96),(36,98),(37,105),(38,106),(40,108),(41,100),(43,102),(44,103),(46,62),(48,55),(49,56),(51,58),(52,59),(54,61),(82,115),(84,117),(85,109),(87,111),(88,112),(90,114)], [(1,79),(2,80),(4,73),(5,74),(7,76),(8,77),(10,120),(12,122),(13,123),(15,125),(16,126),(18,119),(20,64),(21,65),(23,67),(24,68),(26,70),(27,71),(28,99),(29,91),(31,93),(32,94),(34,96),(35,97),(38,106),(39,107),(41,100),(42,101),(44,103),(45,104),(46,62),(47,63),(49,56),(50,57),(52,59),(53,60),(82,115),(83,116),(85,109),(86,110),(88,112),(89,113)], [(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)]])
84 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | ··· | 7F | 9A | ··· | 9F | 14A | ··· | 14F | 21A | ··· | 21L | 42A | ··· | 42L | 63A | ··· | 63AJ |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 9 | ··· | 9 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 | 63 | ··· | 63 |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 4 | ··· | 4 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C3 | C7 | C9 | C21 | C63 | A4 | C3.A4 | C7×A4 | C7×C3.A4 |
| kernel | C7×C3.A4 | C2×C42 | C3.A4 | C2×C14 | C2×C6 | C22 | C21 | C7 | C3 | C1 |
| # reps | 1 | 2 | 6 | 6 | 12 | 36 | 1 | 2 | 6 | 12 |
Matrix representation of C7×C3.A4 ►in GL3(𝔽127) generated by
| 64 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 107 | 0 | 0 |
| 0 | 107 | 0 |
| 0 | 0 | 107 |
| 126 | 0 | 0 |
| 0 | 126 | 0 |
| 74 | 99 | 1 |
| 126 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 28 | 126 |
| 0 | 1 | 0 |
| 53 | 28 | 125 |
| 117 | 101 | 99 |
G:=sub<GL(3,GF(127))| [64,0,0,0,64,0,0,0,64],[107,0,0,0,107,0,0,0,107],[126,0,74,0,126,99,0,0,1],[126,0,0,0,1,28,0,0,126],[0,53,117,1,28,101,0,125,99] >;
C7×C3.A4 in GAP, Magma, Sage, TeX
C_7\times C_3.A_4
% in TeX
G:=Group("C7xC3.A4"); // GroupNames label
G:=SmallGroup(252,10);
// by ID
G=gap.SmallGroup(252,10);
# by ID
G:=PCGroup([5,-3,-7,-3,-2,2,105,2523,4729]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^3=c^2=d^2=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
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