direct product, metacyclic, nilpotent (class 2), monomial
Aliases: C2×C8×3- 1+2, C72⋊8C6, C18⋊2C24, C36.6C12, C62.10C12, C12.39C62, (C2×C72)⋊C3, (C6×C24).C3, C9⋊3(C2×C24), C3.2(C6×C24), (C3×C6).6C24, C6.5(C3×C24), C32.(C2×C24), (C3×C24).10C6, C24.19(C3×C6), (C2×C36).11C6, (C6×C12).19C6, C6.15(C6×C12), (C2×C18).5C12, C36.28(C2×C6), (C3×C12).14C12, C18.13(C2×C12), C12.16(C3×C12), (C2×C24).2C32, C4.3(C4×3- 1+2), (C4×3- 1+2).6C4, C22.2(C4×3- 1+2), C4.5(C22×3- 1+2), (C22×3- 1+2).4C4, (C4×3- 1+2).21C22, (C3×C12).67(C2×C6), (C3×C6).33(C2×C12), (C2×C12).31(C3×C6), (C2×C6).16(C3×C12), C2.2(C2×C4×3- 1+2), (C2×C4×3- 1+2).11C2, (C2×C4).5(C2×3- 1+2), (C2×3- 1+2).13(C2×C4), SmallGroup(432,211)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C8×3- 1+2
G = < a,b,c,d | a2=b8=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Subgroups: 110 in 88 conjugacy classes, 77 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C9, C32, C12, C12, C2×C6, C2×C6, C2×C8, C18, C3×C6, C3×C6, C24, C24, C2×C12, C2×C12, 3- 1+2, C36, C2×C18, C3×C12, C62, C2×C24, C2×C24, C2×3- 1+2, C2×3- 1+2, C72, C2×C36, C3×C24, C6×C12, C4×3- 1+2, C22×3- 1+2, C2×C72, C6×C24, C8×3- 1+2, C2×C4×3- 1+2, C2×C8×3- 1+2
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C32, C12, C2×C6, C2×C8, C3×C6, C24, C2×C12, 3- 1+2, C3×C12, C62, C2×C24, C2×3- 1+2, C3×C24, C6×C12, C4×3- 1+2, C22×3- 1+2, C6×C24, C8×3- 1+2, C2×C4×3- 1+2, C2×C8×3- 1+2
►On 144 points
Generators in S144
(1 78)(2 79)(3 80)(4 81)(5 73)(6 74)(7 75)(8 76)(9 77)(10 114)(11 115)(12 116)(13 117)(14 109)(15 110)(16 111)(17 112)(18 113)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 121)(29 122)(30 123)(31 124)(32 125)(33 126)(34 118)(35 119)(36 120)(37 88)(38 89)(39 90)(40 82)(41 83)(42 84)(43 85)(44 86)(45 87)(46 97)(47 98)(48 99)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 64)(62 65)(63 66)(127 139)(128 140)(129 141)(130 142)(131 143)(132 144)(133 136)(134 137)(135 138)
(1 21 63 36 42 138 48 10)(2 22 55 28 43 139 49 11)(3 23 56 29 44 140 50 12)(4 24 57 30 45 141 51 13)(5 25 58 31 37 142 52 14)(6 26 59 32 38 143 53 15)(7 27 60 33 39 144 54 16)(8 19 61 34 40 136 46 17)(9 20 62 35 41 137 47 18)(64 118 82 133 97 112 76 100)(65 119 83 134 98 113 77 101)(66 120 84 135 99 114 78 102)(67 121 85 127 91 115 79 103)(68 122 86 128 92 116 80 104)(69 123 87 129 93 117 81 105)(70 124 88 130 94 109 73 106)(71 125 89 131 95 110 74 107)(72 126 90 132 96 111 75 108)
(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)
(2 8 5)(3 6 9)(11 17 14)(12 15 18)(19 25 22)(20 23 26)(28 34 31)(29 32 35)(37 43 40)(38 41 44)(46 52 49)(47 50 53)(55 61 58)(56 59 62)(64 70 67)(65 68 71)(73 79 76)(74 77 80)(82 88 85)(83 86 89)(91 97 94)(92 95 98)(100 106 103)(101 104 107)(109 115 112)(110 113 116)(118 124 121)(119 122 125)(127 133 130)(128 131 134)(136 142 139)(137 140 143)
G:=sub<Sym(144)| (1,78)(2,79)(3,80)(4,81)(5,73)(6,74)(7,75)(8,76)(9,77)(10,114)(11,115)(12,116)(13,117)(14,109)(15,110)(16,111)(17,112)(18,113)(19,100)(20,101)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,121)(29,122)(30,123)(31,124)(32,125)(33,126)(34,118)(35,119)(36,120)(37,88)(38,89)(39,90)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(46,97)(47,98)(48,99)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,64)(62,65)(63,66)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)(133,136)(134,137)(135,138), (1,21,63,36,42,138,48,10)(2,22,55,28,43,139,49,11)(3,23,56,29,44,140,50,12)(4,24,57,30,45,141,51,13)(5,25,58,31,37,142,52,14)(6,26,59,32,38,143,53,15)(7,27,60,33,39,144,54,16)(8,19,61,34,40,136,46,17)(9,20,62,35,41,137,47,18)(64,118,82,133,97,112,76,100)(65,119,83,134,98,113,77,101)(66,120,84,135,99,114,78,102)(67,121,85,127,91,115,79,103)(68,122,86,128,92,116,80,104)(69,123,87,129,93,117,81,105)(70,124,88,130,94,109,73,106)(71,125,89,131,95,110,74,107)(72,126,90,132,96,111,75,108), (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), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,25,22)(20,23,26)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(46,52,49)(47,50,53)(55,61,58)(56,59,62)(64,70,67)(65,68,71)(73,79,76)(74,77,80)(82,88,85)(83,86,89)(91,97,94)(92,95,98)(100,106,103)(101,104,107)(109,115,112)(110,113,116)(118,124,121)(119,122,125)(127,133,130)(128,131,134)(136,142,139)(137,140,143)>;
G:=Group( (1,78)(2,79)(3,80)(4,81)(5,73)(6,74)(7,75)(8,76)(9,77)(10,114)(11,115)(12,116)(13,117)(14,109)(15,110)(16,111)(17,112)(18,113)(19,100)(20,101)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,121)(29,122)(30,123)(31,124)(32,125)(33,126)(34,118)(35,119)(36,120)(37,88)(38,89)(39,90)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(46,97)(47,98)(48,99)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,64)(62,65)(63,66)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)(133,136)(134,137)(135,138), (1,21,63,36,42,138,48,10)(2,22,55,28,43,139,49,11)(3,23,56,29,44,140,50,12)(4,24,57,30,45,141,51,13)(5,25,58,31,37,142,52,14)(6,26,59,32,38,143,53,15)(7,27,60,33,39,144,54,16)(8,19,61,34,40,136,46,17)(9,20,62,35,41,137,47,18)(64,118,82,133,97,112,76,100)(65,119,83,134,98,113,77,101)(66,120,84,135,99,114,78,102)(67,121,85,127,91,115,79,103)(68,122,86,128,92,116,80,104)(69,123,87,129,93,117,81,105)(70,124,88,130,94,109,73,106)(71,125,89,131,95,110,74,107)(72,126,90,132,96,111,75,108), (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), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,25,22)(20,23,26)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(46,52,49)(47,50,53)(55,61,58)(56,59,62)(64,70,67)(65,68,71)(73,79,76)(74,77,80)(82,88,85)(83,86,89)(91,97,94)(92,95,98)(100,106,103)(101,104,107)(109,115,112)(110,113,116)(118,124,121)(119,122,125)(127,133,130)(128,131,134)(136,142,139)(137,140,143) );
G=PermutationGroup([[(1,78),(2,79),(3,80),(4,81),(5,73),(6,74),(7,75),(8,76),(9,77),(10,114),(11,115),(12,116),(13,117),(14,109),(15,110),(16,111),(17,112),(18,113),(19,100),(20,101),(21,102),(22,103),(23,104),(24,105),(25,106),(26,107),(27,108),(28,121),(29,122),(30,123),(31,124),(32,125),(33,126),(34,118),(35,119),(36,120),(37,88),(38,89),(39,90),(40,82),(41,83),(42,84),(43,85),(44,86),(45,87),(46,97),(47,98),(48,99),(49,91),(50,92),(51,93),(52,94),(53,95),(54,96),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(61,64),(62,65),(63,66),(127,139),(128,140),(129,141),(130,142),(131,143),(132,144),(133,136),(134,137),(135,138)], [(1,21,63,36,42,138,48,10),(2,22,55,28,43,139,49,11),(3,23,56,29,44,140,50,12),(4,24,57,30,45,141,51,13),(5,25,58,31,37,142,52,14),(6,26,59,32,38,143,53,15),(7,27,60,33,39,144,54,16),(8,19,61,34,40,136,46,17),(9,20,62,35,41,137,47,18),(64,118,82,133,97,112,76,100),(65,119,83,134,98,113,77,101),(66,120,84,135,99,114,78,102),(67,121,85,127,91,115,79,103),(68,122,86,128,92,116,80,104),(69,123,87,129,93,117,81,105),(70,124,88,130,94,109,73,106),(71,125,89,131,95,110,74,107),(72,126,90,132,96,111,75,108)], [(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)], [(2,8,5),(3,6,9),(11,17,14),(12,15,18),(19,25,22),(20,23,26),(28,34,31),(29,32,35),(37,43,40),(38,41,44),(46,52,49),(47,50,53),(55,61,58),(56,59,62),(64,70,67),(65,68,71),(73,79,76),(74,77,80),(82,88,85),(83,86,89),(91,97,94),(92,95,98),(100,106,103),(101,104,107),(109,115,112),(110,113,116),(118,124,121),(119,122,125),(127,133,130),(128,131,134),(136,142,139),(137,140,143)]])
176 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6L | 8A | ··· | 8H | 9A | ··· | 9F | 12A | ··· | 12H | 12I | ··· | 12P | 18A | ··· | 18R | 24A | ··· | 24P | 24Q | ··· | 24AF | 36A | ··· | 36X | 72A | ··· | 72AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
176 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C3 | C4 | C4 | C6 | C6 | C6 | C6 | C8 | C12 | C12 | C12 | C12 | C24 | C24 | 3- 1+2 | C2×3- 1+2 | C2×3- 1+2 | C4×3- 1+2 | C4×3- 1+2 | C8×3- 1+2 |
| kernel | C2×C8×3- 1+2 | C8×3- 1+2 | C2×C4×3- 1+2 | C2×C72 | C6×C24 | C4×3- 1+2 | C22×3- 1+2 | C72 | C2×C36 | C3×C24 | C6×C12 | C2×3- 1+2 | C36 | C2×C18 | C3×C12 | C62 | C18 | C3×C6 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 6 | 2 | 2 | 2 | 12 | 6 | 4 | 2 | 8 | 12 | 12 | 4 | 4 | 48 | 16 | 2 | 4 | 2 | 4 | 4 | 16 |
Matrix representation of C2×C8×3- 1+2 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 64 | 0 | 0 | 0 |
| 0 | 1 | 71 | 0 |
| 0 | 33 | 72 | 8 |
| 0 | 0 | 72 | 0 |
| 64 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 33 | 8 | 0 |
| 0 | 5 | 0 | 64 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[64,0,0,0,0,1,33,0,0,71,72,72,0,0,8,0],[64,0,0,0,0,1,33,5,0,0,8,0,0,0,0,64] >;
C2×C8×3- 1+2 in GAP, Magma, Sage, TeX
C_2\times C_8\times 3_-^{1+2} % in TeX
G:=Group("C2xC8xES-(3,1)"); // GroupNames label
G:=SmallGroup(432,211);
// by ID
G=gap.SmallGroup(432,211);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,252,394,605,242]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations