direct product, p-group, abelian, monomial
Aliases: C2×C43, SmallGroup(128,997)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C2×C43 |
| C1 — C2×C43 |
| C1 — C2×C43 |
Generators and relations for C2×C43
G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 636, all normal (4 characteristic)
C1, C2, C4, C22, C2×C4, C23, C23, C42, C22×C4, C24, C2×C42, C23×C4, C43, C22×C42, C2×C43
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C24, C2×C42, C23×C4, C43, C22×C42, C2×C43
►Regular action on 128 points
Generators in S128
(1 73)(2 74)(3 75)(4 76)(5 128)(6 125)(7 126)(8 127)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(77 81)(78 82)(79 83)(80 84)(85 89)(86 90)(87 91)(88 92)(93 97)(94 98)(95 99)(96 100)(101 105)(102 106)(103 107)(104 108)(109 113)(110 114)(111 115)(112 116)(117 121)(118 122)(119 123)(120 124)
(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)(125 126 127 128)
(1 25 9 85)(2 26 10 86)(3 27 11 87)(4 28 12 88)(5 116 71 54)(6 113 72 55)(7 114 69 56)(8 115 70 53)(13 89 73 29)(14 90 74 30)(15 91 75 31)(16 92 76 32)(17 93 77 33)(18 94 78 34)(19 95 79 35)(20 96 80 36)(21 97 81 37)(22 98 82 38)(23 99 83 39)(24 100 84 40)(41 119 103 57)(42 120 104 58)(43 117 101 59)(44 118 102 60)(45 123 107 61)(46 124 108 62)(47 121 105 63)(48 122 106 64)(49 127 111 66)(50 128 112 67)(51 125 109 68)(52 126 110 65)
(1 109 101 77)(2 110 102 78)(3 111 103 79)(4 112 104 80)(5 124 100 92)(6 121 97 89)(7 122 98 90)(8 123 99 91)(9 51 43 17)(10 52 44 18)(11 49 41 19)(12 50 42 20)(13 55 47 21)(14 56 48 22)(15 53 45 23)(16 54 46 24)(25 68 59 33)(26 65 60 34)(27 66 57 35)(28 67 58 36)(29 72 63 37)(30 69 64 38)(31 70 61 39)(32 71 62 40)(73 113 105 81)(74 114 106 82)(75 115 107 83)(76 116 108 84)(85 125 117 93)(86 126 118 94)(87 127 119 95)(88 128 120 96)
G:=sub<Sym(128)| (1,73)(2,74)(3,75)(4,76)(5,128)(6,125)(7,126)(8,127)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(77,81)(78,82)(79,83)(80,84)(85,89)(86,90)(87,91)(88,92)(93,97)(94,98)(95,99)(96,100)(101,105)(102,106)(103,107)(104,108)(109,113)(110,114)(111,115)(112,116)(117,121)(118,122)(119,123)(120,124), (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)(125,126,127,128), (1,25,9,85)(2,26,10,86)(3,27,11,87)(4,28,12,88)(5,116,71,54)(6,113,72,55)(7,114,69,56)(8,115,70,53)(13,89,73,29)(14,90,74,30)(15,91,75,31)(16,92,76,32)(17,93,77,33)(18,94,78,34)(19,95,79,35)(20,96,80,36)(21,97,81,37)(22,98,82,38)(23,99,83,39)(24,100,84,40)(41,119,103,57)(42,120,104,58)(43,117,101,59)(44,118,102,60)(45,123,107,61)(46,124,108,62)(47,121,105,63)(48,122,106,64)(49,127,111,66)(50,128,112,67)(51,125,109,68)(52,126,110,65), (1,109,101,77)(2,110,102,78)(3,111,103,79)(4,112,104,80)(5,124,100,92)(6,121,97,89)(7,122,98,90)(8,123,99,91)(9,51,43,17)(10,52,44,18)(11,49,41,19)(12,50,42,20)(13,55,47,21)(14,56,48,22)(15,53,45,23)(16,54,46,24)(25,68,59,33)(26,65,60,34)(27,66,57,35)(28,67,58,36)(29,72,63,37)(30,69,64,38)(31,70,61,39)(32,71,62,40)(73,113,105,81)(74,114,106,82)(75,115,107,83)(76,116,108,84)(85,125,117,93)(86,126,118,94)(87,127,119,95)(88,128,120,96)>;
G:=Group( (1,73)(2,74)(3,75)(4,76)(5,128)(6,125)(7,126)(8,127)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(77,81)(78,82)(79,83)(80,84)(85,89)(86,90)(87,91)(88,92)(93,97)(94,98)(95,99)(96,100)(101,105)(102,106)(103,107)(104,108)(109,113)(110,114)(111,115)(112,116)(117,121)(118,122)(119,123)(120,124), (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)(125,126,127,128), (1,25,9,85)(2,26,10,86)(3,27,11,87)(4,28,12,88)(5,116,71,54)(6,113,72,55)(7,114,69,56)(8,115,70,53)(13,89,73,29)(14,90,74,30)(15,91,75,31)(16,92,76,32)(17,93,77,33)(18,94,78,34)(19,95,79,35)(20,96,80,36)(21,97,81,37)(22,98,82,38)(23,99,83,39)(24,100,84,40)(41,119,103,57)(42,120,104,58)(43,117,101,59)(44,118,102,60)(45,123,107,61)(46,124,108,62)(47,121,105,63)(48,122,106,64)(49,127,111,66)(50,128,112,67)(51,125,109,68)(52,126,110,65), (1,109,101,77)(2,110,102,78)(3,111,103,79)(4,112,104,80)(5,124,100,92)(6,121,97,89)(7,122,98,90)(8,123,99,91)(9,51,43,17)(10,52,44,18)(11,49,41,19)(12,50,42,20)(13,55,47,21)(14,56,48,22)(15,53,45,23)(16,54,46,24)(25,68,59,33)(26,65,60,34)(27,66,57,35)(28,67,58,36)(29,72,63,37)(30,69,64,38)(31,70,61,39)(32,71,62,40)(73,113,105,81)(74,114,106,82)(75,115,107,83)(76,116,108,84)(85,125,117,93)(86,126,118,94)(87,127,119,95)(88,128,120,96) );
G=PermutationGroup([[(1,73),(2,74),(3,75),(4,76),(5,128),(6,125),(7,126),(8,127),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(77,81),(78,82),(79,83),(80,84),(85,89),(86,90),(87,91),(88,92),(93,97),(94,98),(95,99),(96,100),(101,105),(102,106),(103,107),(104,108),(109,113),(110,114),(111,115),(112,116),(117,121),(118,122),(119,123),(120,124)], [(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),(125,126,127,128)], [(1,25,9,85),(2,26,10,86),(3,27,11,87),(4,28,12,88),(5,116,71,54),(6,113,72,55),(7,114,69,56),(8,115,70,53),(13,89,73,29),(14,90,74,30),(15,91,75,31),(16,92,76,32),(17,93,77,33),(18,94,78,34),(19,95,79,35),(20,96,80,36),(21,97,81,37),(22,98,82,38),(23,99,83,39),(24,100,84,40),(41,119,103,57),(42,120,104,58),(43,117,101,59),(44,118,102,60),(45,123,107,61),(46,124,108,62),(47,121,105,63),(48,122,106,64),(49,127,111,66),(50,128,112,67),(51,125,109,68),(52,126,110,65)], [(1,109,101,77),(2,110,102,78),(3,111,103,79),(4,112,104,80),(5,124,100,92),(6,121,97,89),(7,122,98,90),(8,123,99,91),(9,51,43,17),(10,52,44,18),(11,49,41,19),(12,50,42,20),(13,55,47,21),(14,56,48,22),(15,53,45,23),(16,54,46,24),(25,68,59,33),(26,65,60,34),(27,66,57,35),(28,67,58,36),(29,72,63,37),(30,69,64,38),(31,70,61,39),(32,71,62,40),(73,113,105,81),(74,114,106,82),(75,115,107,83),(76,116,108,84),(85,125,117,93),(86,126,118,94),(87,127,119,95),(88,128,120,96)]])
128 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4DH |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | + | |
| image | C1 | C2 | C2 | C4 |
| kernel | C2×C43 | C43 | C22×C42 | C2×C42 |
| # reps | 1 | 8 | 7 | 112 |
Matrix representation of C2×C43 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 2 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[3,0,0,0,0,3,0,0,0,0,2,0,0,0,0,4],[1,0,0,0,0,3,0,0,0,0,2,0,0,0,0,3],[4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,2] >;
C2×C43 in GAP, Magma, Sage, TeX
C_2\times C_4^3
% in TeX
G:=Group("C2xC4^3"); // GroupNames label
G:=SmallGroup(128,997);
// by ID
G=gap.SmallGroup(128,997);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,232,352]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations