direct product, metabelian, nilpotent (class 2), monomial
Aliases: C42×He3, C122⋊2C3, C3.1C122, (C3×C12)⋊6C12, (C6×C12).16C6, C6.10(C6×C12), C32⋊4(C4×C12), C12.13(C3×C12), (C2×C6).24C62, (C4×C12).6C32, C62.31(C2×C6), C22.2(C22×He3), (C22×He3).36C22, C2.1(C2×C4×He3), (C2×C4×He3).16C2, (C2×C4).4(C2×He3), (C3×C6).26(C2×C12), (C2×C12).28(C3×C6), (C2×He3).35(C2×C4), SmallGroup(432,201)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42×He3
G = < a,b,c,d,e | a4=b4=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 285 in 165 conjugacy classes, 105 normal (9 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C32, C12, C12, C2×C6, C2×C6, C42, C3×C6, C2×C12, C2×C12, He3, C3×C12, C62, C4×C12, C4×C12, C2×He3, C6×C12, C4×He3, C22×He3, C122, C2×C4×He3, C42×He3
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, C42, C3×C6, C2×C12, He3, C3×C12, C62, C4×C12, C2×He3, C6×C12, C4×He3, C22×He3, C122, C2×C4×He3, C42×He3
►On 144 points
Generators in S144
(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)(129 130 131 132)(133 134 135 136)(137 138 139 140)(141 142 143 144)
(1 107 38 101)(2 108 39 102)(3 105 40 103)(4 106 37 104)(5 44 95 137)(6 41 96 138)(7 42 93 139)(8 43 94 140)(9 22 54 51)(10 23 55 52)(11 24 56 49)(12 21 53 50)(13 91 69 136)(14 92 70 133)(15 89 71 134)(16 90 72 135)(17 48 65 141)(18 45 66 142)(19 46 67 143)(20 47 68 144)(25 98 114 129)(26 99 115 130)(27 100 116 131)(28 97 113 132)(29 117 78 86)(30 118 79 87)(31 119 80 88)(32 120 77 85)(33 126 74 61)(34 127 75 62)(35 128 76 63)(36 125 73 64)(57 110 123 81)(58 111 124 82)(59 112 121 83)(60 109 122 84)
(9 25 109)(10 26 110)(11 27 111)(12 28 112)(13 67 125)(14 68 126)(15 65 127)(16 66 128)(17 62 71)(18 63 72)(19 64 69)(20 61 70)(21 97 121)(22 98 122)(23 99 123)(24 100 124)(33 133 47)(34 134 48)(35 135 45)(36 136 46)(49 131 58)(50 132 59)(51 129 60)(52 130 57)(53 113 83)(54 114 84)(55 115 81)(56 116 82)(73 91 143)(74 92 144)(75 89 141)(76 90 142)
(1 78 139)(2 79 140)(3 80 137)(4 77 138)(5 105 88)(6 106 85)(7 107 86)(8 108 87)(9 109 25)(10 110 26)(11 111 27)(12 112 28)(13 67 125)(14 68 126)(15 65 127)(16 66 128)(17 62 71)(18 63 72)(19 64 69)(20 61 70)(21 121 97)(22 122 98)(23 123 99)(24 124 100)(29 42 38)(30 43 39)(31 44 40)(32 41 37)(33 133 47)(34 134 48)(35 135 45)(36 136 46)(49 58 131)(50 59 132)(51 60 129)(52 57 130)(53 83 113)(54 84 114)(55 81 115)(56 82 116)(73 91 143)(74 92 144)(75 89 141)(76 90 142)(93 101 117)(94 102 118)(95 103 119)(96 104 120)
(1 109 136)(2 110 133)(3 111 134)(4 112 135)(5 24 127)(6 21 128)(7 22 125)(8 23 126)(9 36 139)(10 33 140)(11 34 137)(12 35 138)(13 107 122)(14 108 123)(15 105 124)(16 106 121)(17 119 131)(18 120 132)(19 117 129)(20 118 130)(25 46 78)(26 47 79)(27 48 80)(28 45 77)(29 114 143)(30 115 144)(31 116 141)(32 113 142)(37 83 90)(38 84 91)(39 81 92)(40 82 89)(41 53 76)(42 54 73)(43 55 74)(44 56 75)(49 62 95)(50 63 96)(51 64 93)(52 61 94)(57 70 102)(58 71 103)(59 72 104)(60 69 101)(65 88 100)(66 85 97)(67 86 98)(68 87 99)
G:=sub<Sym(144)| (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)(129,130,131,132)(133,134,135,136)(137,138,139,140)(141,142,143,144), (1,107,38,101)(2,108,39,102)(3,105,40,103)(4,106,37,104)(5,44,95,137)(6,41,96,138)(7,42,93,139)(8,43,94,140)(9,22,54,51)(10,23,55,52)(11,24,56,49)(12,21,53,50)(13,91,69,136)(14,92,70,133)(15,89,71,134)(16,90,72,135)(17,48,65,141)(18,45,66,142)(19,46,67,143)(20,47,68,144)(25,98,114,129)(26,99,115,130)(27,100,116,131)(28,97,113,132)(29,117,78,86)(30,118,79,87)(31,119,80,88)(32,120,77,85)(33,126,74,61)(34,127,75,62)(35,128,76,63)(36,125,73,64)(57,110,123,81)(58,111,124,82)(59,112,121,83)(60,109,122,84), (9,25,109)(10,26,110)(11,27,111)(12,28,112)(13,67,125)(14,68,126)(15,65,127)(16,66,128)(17,62,71)(18,63,72)(19,64,69)(20,61,70)(21,97,121)(22,98,122)(23,99,123)(24,100,124)(33,133,47)(34,134,48)(35,135,45)(36,136,46)(49,131,58)(50,132,59)(51,129,60)(52,130,57)(53,113,83)(54,114,84)(55,115,81)(56,116,82)(73,91,143)(74,92,144)(75,89,141)(76,90,142), (1,78,139)(2,79,140)(3,80,137)(4,77,138)(5,105,88)(6,106,85)(7,107,86)(8,108,87)(9,109,25)(10,110,26)(11,111,27)(12,112,28)(13,67,125)(14,68,126)(15,65,127)(16,66,128)(17,62,71)(18,63,72)(19,64,69)(20,61,70)(21,121,97)(22,122,98)(23,123,99)(24,124,100)(29,42,38)(30,43,39)(31,44,40)(32,41,37)(33,133,47)(34,134,48)(35,135,45)(36,136,46)(49,58,131)(50,59,132)(51,60,129)(52,57,130)(53,83,113)(54,84,114)(55,81,115)(56,82,116)(73,91,143)(74,92,144)(75,89,141)(76,90,142)(93,101,117)(94,102,118)(95,103,119)(96,104,120), (1,109,136)(2,110,133)(3,111,134)(4,112,135)(5,24,127)(6,21,128)(7,22,125)(8,23,126)(9,36,139)(10,33,140)(11,34,137)(12,35,138)(13,107,122)(14,108,123)(15,105,124)(16,106,121)(17,119,131)(18,120,132)(19,117,129)(20,118,130)(25,46,78)(26,47,79)(27,48,80)(28,45,77)(29,114,143)(30,115,144)(31,116,141)(32,113,142)(37,83,90)(38,84,91)(39,81,92)(40,82,89)(41,53,76)(42,54,73)(43,55,74)(44,56,75)(49,62,95)(50,63,96)(51,64,93)(52,61,94)(57,70,102)(58,71,103)(59,72,104)(60,69,101)(65,88,100)(66,85,97)(67,86,98)(68,87,99)>;
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)(125,126,127,128)(129,130,131,132)(133,134,135,136)(137,138,139,140)(141,142,143,144), (1,107,38,101)(2,108,39,102)(3,105,40,103)(4,106,37,104)(5,44,95,137)(6,41,96,138)(7,42,93,139)(8,43,94,140)(9,22,54,51)(10,23,55,52)(11,24,56,49)(12,21,53,50)(13,91,69,136)(14,92,70,133)(15,89,71,134)(16,90,72,135)(17,48,65,141)(18,45,66,142)(19,46,67,143)(20,47,68,144)(25,98,114,129)(26,99,115,130)(27,100,116,131)(28,97,113,132)(29,117,78,86)(30,118,79,87)(31,119,80,88)(32,120,77,85)(33,126,74,61)(34,127,75,62)(35,128,76,63)(36,125,73,64)(57,110,123,81)(58,111,124,82)(59,112,121,83)(60,109,122,84), (9,25,109)(10,26,110)(11,27,111)(12,28,112)(13,67,125)(14,68,126)(15,65,127)(16,66,128)(17,62,71)(18,63,72)(19,64,69)(20,61,70)(21,97,121)(22,98,122)(23,99,123)(24,100,124)(33,133,47)(34,134,48)(35,135,45)(36,136,46)(49,131,58)(50,132,59)(51,129,60)(52,130,57)(53,113,83)(54,114,84)(55,115,81)(56,116,82)(73,91,143)(74,92,144)(75,89,141)(76,90,142), (1,78,139)(2,79,140)(3,80,137)(4,77,138)(5,105,88)(6,106,85)(7,107,86)(8,108,87)(9,109,25)(10,110,26)(11,111,27)(12,112,28)(13,67,125)(14,68,126)(15,65,127)(16,66,128)(17,62,71)(18,63,72)(19,64,69)(20,61,70)(21,121,97)(22,122,98)(23,123,99)(24,124,100)(29,42,38)(30,43,39)(31,44,40)(32,41,37)(33,133,47)(34,134,48)(35,135,45)(36,136,46)(49,58,131)(50,59,132)(51,60,129)(52,57,130)(53,83,113)(54,84,114)(55,81,115)(56,82,116)(73,91,143)(74,92,144)(75,89,141)(76,90,142)(93,101,117)(94,102,118)(95,103,119)(96,104,120), (1,109,136)(2,110,133)(3,111,134)(4,112,135)(5,24,127)(6,21,128)(7,22,125)(8,23,126)(9,36,139)(10,33,140)(11,34,137)(12,35,138)(13,107,122)(14,108,123)(15,105,124)(16,106,121)(17,119,131)(18,120,132)(19,117,129)(20,118,130)(25,46,78)(26,47,79)(27,48,80)(28,45,77)(29,114,143)(30,115,144)(31,116,141)(32,113,142)(37,83,90)(38,84,91)(39,81,92)(40,82,89)(41,53,76)(42,54,73)(43,55,74)(44,56,75)(49,62,95)(50,63,96)(51,64,93)(52,61,94)(57,70,102)(58,71,103)(59,72,104)(60,69,101)(65,88,100)(66,85,97)(67,86,98)(68,87,99) );
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),(125,126,127,128),(129,130,131,132),(133,134,135,136),(137,138,139,140),(141,142,143,144)], [(1,107,38,101),(2,108,39,102),(3,105,40,103),(4,106,37,104),(5,44,95,137),(6,41,96,138),(7,42,93,139),(8,43,94,140),(9,22,54,51),(10,23,55,52),(11,24,56,49),(12,21,53,50),(13,91,69,136),(14,92,70,133),(15,89,71,134),(16,90,72,135),(17,48,65,141),(18,45,66,142),(19,46,67,143),(20,47,68,144),(25,98,114,129),(26,99,115,130),(27,100,116,131),(28,97,113,132),(29,117,78,86),(30,118,79,87),(31,119,80,88),(32,120,77,85),(33,126,74,61),(34,127,75,62),(35,128,76,63),(36,125,73,64),(57,110,123,81),(58,111,124,82),(59,112,121,83),(60,109,122,84)], [(9,25,109),(10,26,110),(11,27,111),(12,28,112),(13,67,125),(14,68,126),(15,65,127),(16,66,128),(17,62,71),(18,63,72),(19,64,69),(20,61,70),(21,97,121),(22,98,122),(23,99,123),(24,100,124),(33,133,47),(34,134,48),(35,135,45),(36,136,46),(49,131,58),(50,132,59),(51,129,60),(52,130,57),(53,113,83),(54,114,84),(55,115,81),(56,116,82),(73,91,143),(74,92,144),(75,89,141),(76,90,142)], [(1,78,139),(2,79,140),(3,80,137),(4,77,138),(5,105,88),(6,106,85),(7,107,86),(8,108,87),(9,109,25),(10,110,26),(11,111,27),(12,112,28),(13,67,125),(14,68,126),(15,65,127),(16,66,128),(17,62,71),(18,63,72),(19,64,69),(20,61,70),(21,121,97),(22,122,98),(23,123,99),(24,124,100),(29,42,38),(30,43,39),(31,44,40),(32,41,37),(33,133,47),(34,134,48),(35,135,45),(36,136,46),(49,58,131),(50,59,132),(51,60,129),(52,57,130),(53,83,113),(54,84,114),(55,81,115),(56,82,116),(73,91,143),(74,92,144),(75,89,141),(76,90,142),(93,101,117),(94,102,118),(95,103,119),(96,104,120)], [(1,109,136),(2,110,133),(3,111,134),(4,112,135),(5,24,127),(6,21,128),(7,22,125),(8,23,126),(9,36,139),(10,33,140),(11,34,137),(12,35,138),(13,107,122),(14,108,123),(15,105,124),(16,106,121),(17,119,131),(18,120,132),(19,117,129),(20,118,130),(25,46,78),(26,47,79),(27,48,80),(28,45,77),(29,114,143),(30,115,144),(31,116,141),(32,113,142),(37,83,90),(38,84,91),(39,81,92),(40,82,89),(41,53,76),(42,54,73),(43,55,74),(44,56,75),(49,62,95),(50,63,96),(51,64,93),(52,61,94),(57,70,102),(58,71,103),(59,72,104),(60,69,101),(65,88,100),(66,85,97),(67,86,98),(68,87,99)]])
176 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3J | 4A | ··· | 4L | 6A | ··· | 6F | 6G | ··· | 6AD | 12A | ··· | 12X | 12Y | ··· | 12DP |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 |
176 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | + | |||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | He3 | C2×He3 | C4×He3 |
| kernel | C42×He3 | C2×C4×He3 | C122 | C4×He3 | C6×C12 | C3×C12 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 8 | 12 | 24 | 96 | 2 | 6 | 24 |
Matrix representation of C42×He3 ►in GL5(𝔽13)
| 8 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 3 | 0 |
| 0 | 0 | 12 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 | 0 |
| 0 | 0 | 0 | 4 | 1 |
| 0 | 0 | 0 | 10 | 0 |
G:=sub<GL(5,GF(13))| [8,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[5,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[9,0,0,0,0,0,3,0,0,0,0,0,1,9,12,0,0,0,3,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[3,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,2,4,10,0,0,0,1,0] >;
C42×He3 in GAP, Magma, Sage, TeX
C_4^2\times {\rm He}_3 % in TeX
G:=Group("C4^2xHe3"); // GroupNames label
G:=SmallGroup(432,201);
// by ID
G=gap.SmallGroup(432,201);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,252,512,1109]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^3=d^3=e^3=1,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,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations