direct product, metacyclic, nilpotent (class 3), monomial
Aliases: C32×C8.C4, C24.3C12, C62.11Q8, C4.8(C6×C12), C8.1(C3×C12), (C6×C24).17C2, (C2×C24).20C6, (C3×C24).14C4, C12.91(C3×D4), C12.57(C2×C12), (C3×C12).186D4, (C2×C4).19C62, C22.(Q8×C32), C4.19(D4×C32), M4(2).2(C3×C6), (C6×C12).366C22, (C3×M4(2)).10C6, (C32×M4(2)).6C2, C6.20(C3×C4⋊C4), (C2×C8).5(C3×C6), (C2×C6).5(C3×Q8), C2.5(C32×C4⋊C4), (C3×C6).49(C4⋊C4), (C2×C12).153(C2×C6), (C3×C12).142(C2×C4), SmallGroup(288,326)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C8.C4
G = < a,b,c,d | a3=b3=c8=1, d4=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 108 in 90 conjugacy classes, 72 normal (20 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C8, C2×C4, C32, C12, C2×C6, C2×C8, M4(2), C3×C6, C3×C6, C24, C24, C2×C12, C8.C4, C3×C12, C62, C2×C24, C3×M4(2), C3×C24, C3×C24, C6×C12, C3×C8.C4, C6×C24, C32×M4(2), C32×C8.C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C32, C12, C2×C6, C4⋊C4, C3×C6, C2×C12, C3×D4, C3×Q8, C8.C4, C3×C12, C62, C3×C4⋊C4, C6×C12, D4×C32, Q8×C32, C3×C8.C4, C32×C4⋊C4, C32×C8.C4
►On 144 points
Generators in S144
(1 73 61)(2 74 62)(3 75 63)(4 76 64)(5 77 57)(6 78 58)(7 79 59)(8 80 60)(9 32 52)(10 25 53)(11 26 54)(12 27 55)(13 28 56)(14 29 49)(15 30 50)(16 31 51)(17 69 46)(18 70 47)(19 71 48)(20 72 41)(21 65 42)(22 66 43)(23 67 44)(24 68 45)(33 112 104)(34 105 97)(35 106 98)(36 107 99)(37 108 100)(38 109 101)(39 110 102)(40 111 103)(81 142 89)(82 143 90)(83 144 91)(84 137 92)(85 138 93)(86 139 94)(87 140 95)(88 141 96)(113 130 121)(114 131 122)(115 132 123)(116 133 124)(117 134 125)(118 135 126)(119 136 127)(120 129 128)
(1 31 17)(2 32 18)(3 25 19)(4 26 20)(5 27 21)(6 28 22)(7 29 23)(8 30 24)(9 47 62)(10 48 63)(11 41 64)(12 42 57)(13 43 58)(14 44 59)(15 45 60)(16 46 61)(33 123 140)(34 124 141)(35 125 142)(36 126 143)(37 127 144)(38 128 137)(39 121 138)(40 122 139)(49 67 79)(50 68 80)(51 69 73)(52 70 74)(53 71 75)(54 72 76)(55 65 77)(56 66 78)(81 98 134)(82 99 135)(83 100 136)(84 101 129)(85 102 130)(86 103 131)(87 104 132)(88 97 133)(89 106 117)(90 107 118)(91 108 119)(92 109 120)(93 110 113)(94 111 114)(95 112 115)(96 105 116)
(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)
(1 101 3 99 5 97 7 103)(2 100 4 98 6 104 8 102)(9 119 11 117 13 115 15 113)(10 118 12 116 14 114 16 120)(17 84 19 82 21 88 23 86)(18 83 20 81 22 87 24 85)(25 135 27 133 29 131 31 129)(26 134 28 132 30 130 32 136)(33 80 39 74 37 76 35 78)(34 79 40 73 38 75 36 77)(41 89 43 95 45 93 47 91)(42 96 44 94 46 92 48 90)(49 122 51 128 53 126 55 124)(50 121 52 127 54 125 56 123)(57 105 59 111 61 109 63 107)(58 112 60 110 62 108 64 106)(65 141 67 139 69 137 71 143)(66 140 68 138 70 144 72 142)
G:=sub<Sym(144)| (1,73,61)(2,74,62)(3,75,63)(4,76,64)(5,77,57)(6,78,58)(7,79,59)(8,80,60)(9,32,52)(10,25,53)(11,26,54)(12,27,55)(13,28,56)(14,29,49)(15,30,50)(16,31,51)(17,69,46)(18,70,47)(19,71,48)(20,72,41)(21,65,42)(22,66,43)(23,67,44)(24,68,45)(33,112,104)(34,105,97)(35,106,98)(36,107,99)(37,108,100)(38,109,101)(39,110,102)(40,111,103)(81,142,89)(82,143,90)(83,144,91)(84,137,92)(85,138,93)(86,139,94)(87,140,95)(88,141,96)(113,130,121)(114,131,122)(115,132,123)(116,133,124)(117,134,125)(118,135,126)(119,136,127)(120,129,128), (1,31,17)(2,32,18)(3,25,19)(4,26,20)(5,27,21)(6,28,22)(7,29,23)(8,30,24)(9,47,62)(10,48,63)(11,41,64)(12,42,57)(13,43,58)(14,44,59)(15,45,60)(16,46,61)(33,123,140)(34,124,141)(35,125,142)(36,126,143)(37,127,144)(38,128,137)(39,121,138)(40,122,139)(49,67,79)(50,68,80)(51,69,73)(52,70,74)(53,71,75)(54,72,76)(55,65,77)(56,66,78)(81,98,134)(82,99,135)(83,100,136)(84,101,129)(85,102,130)(86,103,131)(87,104,132)(88,97,133)(89,106,117)(90,107,118)(91,108,119)(92,109,120)(93,110,113)(94,111,114)(95,112,115)(96,105,116), (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), (1,101,3,99,5,97,7,103)(2,100,4,98,6,104,8,102)(9,119,11,117,13,115,15,113)(10,118,12,116,14,114,16,120)(17,84,19,82,21,88,23,86)(18,83,20,81,22,87,24,85)(25,135,27,133,29,131,31,129)(26,134,28,132,30,130,32,136)(33,80,39,74,37,76,35,78)(34,79,40,73,38,75,36,77)(41,89,43,95,45,93,47,91)(42,96,44,94,46,92,48,90)(49,122,51,128,53,126,55,124)(50,121,52,127,54,125,56,123)(57,105,59,111,61,109,63,107)(58,112,60,110,62,108,64,106)(65,141,67,139,69,137,71,143)(66,140,68,138,70,144,72,142)>;
G:=Group( (1,73,61)(2,74,62)(3,75,63)(4,76,64)(5,77,57)(6,78,58)(7,79,59)(8,80,60)(9,32,52)(10,25,53)(11,26,54)(12,27,55)(13,28,56)(14,29,49)(15,30,50)(16,31,51)(17,69,46)(18,70,47)(19,71,48)(20,72,41)(21,65,42)(22,66,43)(23,67,44)(24,68,45)(33,112,104)(34,105,97)(35,106,98)(36,107,99)(37,108,100)(38,109,101)(39,110,102)(40,111,103)(81,142,89)(82,143,90)(83,144,91)(84,137,92)(85,138,93)(86,139,94)(87,140,95)(88,141,96)(113,130,121)(114,131,122)(115,132,123)(116,133,124)(117,134,125)(118,135,126)(119,136,127)(120,129,128), (1,31,17)(2,32,18)(3,25,19)(4,26,20)(5,27,21)(6,28,22)(7,29,23)(8,30,24)(9,47,62)(10,48,63)(11,41,64)(12,42,57)(13,43,58)(14,44,59)(15,45,60)(16,46,61)(33,123,140)(34,124,141)(35,125,142)(36,126,143)(37,127,144)(38,128,137)(39,121,138)(40,122,139)(49,67,79)(50,68,80)(51,69,73)(52,70,74)(53,71,75)(54,72,76)(55,65,77)(56,66,78)(81,98,134)(82,99,135)(83,100,136)(84,101,129)(85,102,130)(86,103,131)(87,104,132)(88,97,133)(89,106,117)(90,107,118)(91,108,119)(92,109,120)(93,110,113)(94,111,114)(95,112,115)(96,105,116), (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), (1,101,3,99,5,97,7,103)(2,100,4,98,6,104,8,102)(9,119,11,117,13,115,15,113)(10,118,12,116,14,114,16,120)(17,84,19,82,21,88,23,86)(18,83,20,81,22,87,24,85)(25,135,27,133,29,131,31,129)(26,134,28,132,30,130,32,136)(33,80,39,74,37,76,35,78)(34,79,40,73,38,75,36,77)(41,89,43,95,45,93,47,91)(42,96,44,94,46,92,48,90)(49,122,51,128,53,126,55,124)(50,121,52,127,54,125,56,123)(57,105,59,111,61,109,63,107)(58,112,60,110,62,108,64,106)(65,141,67,139,69,137,71,143)(66,140,68,138,70,144,72,142) );
G=PermutationGroup([[(1,73,61),(2,74,62),(3,75,63),(4,76,64),(5,77,57),(6,78,58),(7,79,59),(8,80,60),(9,32,52),(10,25,53),(11,26,54),(12,27,55),(13,28,56),(14,29,49),(15,30,50),(16,31,51),(17,69,46),(18,70,47),(19,71,48),(20,72,41),(21,65,42),(22,66,43),(23,67,44),(24,68,45),(33,112,104),(34,105,97),(35,106,98),(36,107,99),(37,108,100),(38,109,101),(39,110,102),(40,111,103),(81,142,89),(82,143,90),(83,144,91),(84,137,92),(85,138,93),(86,139,94),(87,140,95),(88,141,96),(113,130,121),(114,131,122),(115,132,123),(116,133,124),(117,134,125),(118,135,126),(119,136,127),(120,129,128)], [(1,31,17),(2,32,18),(3,25,19),(4,26,20),(5,27,21),(6,28,22),(7,29,23),(8,30,24),(9,47,62),(10,48,63),(11,41,64),(12,42,57),(13,43,58),(14,44,59),(15,45,60),(16,46,61),(33,123,140),(34,124,141),(35,125,142),(36,126,143),(37,127,144),(38,128,137),(39,121,138),(40,122,139),(49,67,79),(50,68,80),(51,69,73),(52,70,74),(53,71,75),(54,72,76),(55,65,77),(56,66,78),(81,98,134),(82,99,135),(83,100,136),(84,101,129),(85,102,130),(86,103,131),(87,104,132),(88,97,133),(89,106,117),(90,107,118),(91,108,119),(92,109,120),(93,110,113),(94,111,114),(95,112,115),(96,105,116)], [(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)], [(1,101,3,99,5,97,7,103),(2,100,4,98,6,104,8,102),(9,119,11,117,13,115,15,113),(10,118,12,116,14,114,16,120),(17,84,19,82,21,88,23,86),(18,83,20,81,22,87,24,85),(25,135,27,133,29,131,31,129),(26,134,28,132,30,130,32,136),(33,80,39,74,37,76,35,78),(34,79,40,73,38,75,36,77),(41,89,43,95,45,93,47,91),(42,96,44,94,46,92,48,90),(49,122,51,128,53,126,55,124),(50,121,52,127,54,125,56,123),(57,105,59,111,61,109,63,107),(58,112,60,110,62,108,64,106),(65,141,67,139,69,137,71,143),(66,140,68,138,70,144,72,142)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 3A | ··· | 3H | 4A | 4B | 4C | 6A | ··· | 6H | 6I | ··· | 6P | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | ··· | 12P | 12Q | ··· | 12X | 24A | ··· | 24AF | 24AG | ··· | 24BL |
| order | 1 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 1 | ··· | 1 | 1 | 1 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | Q8 | C3×D4 | C3×Q8 | C8.C4 | C3×C8.C4 |
| kernel | C32×C8.C4 | C6×C24 | C32×M4(2) | C3×C8.C4 | C3×C24 | C2×C24 | C3×M4(2) | C24 | C3×C12 | C62 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 1 | 2 | 8 | 4 | 8 | 16 | 32 | 1 | 1 | 8 | 8 | 4 | 32 |
Matrix representation of C32×C8.C4 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 22 | 0 |
| 0 | 0 | 20 | 10 |
| 27 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 52 | 71 |
| 0 | 0 | 15 | 21 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,22,20,0,0,0,10],[27,0,0,0,0,46,0,0,0,0,52,15,0,0,71,21] >;
C32×C8.C4 in GAP, Magma, Sage, TeX
C_3^2\times C_8.C_4
% in TeX
G:=Group("C3^2xC8.C4"); // GroupNames label
G:=SmallGroup(288,326);
// by ID
G=gap.SmallGroup(288,326);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,260,6304,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=1,d^4=c^4,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^-1>;
// generators/relations