direct product, metabelian, nilpotent (class 2), monomial
Aliases: C32×C8○D4, C8.7C62, D4.(C3×C12), (C2×C24)⋊15C6, (C6×C24)⋊23C2, C4.5(C6×C12), C24.40(C2×C6), (C3×D4).6C12, Q8.2(C3×C12), C12.40(C2×C12), M4(2)⋊5(C3×C6), C62.66(C2×C4), (C2×C4).24C62, C4.12(C2×C62), C22.1(C6×C12), (C3×Q8).10C12, (D4×C32).4C4, (Q8×C32).4C4, (C3×M4(2))⋊11C6, (C3×C24).75C22, C6.42(C22×C12), C12.68(C22×C6), (C3×C12).195C23, (C6×C12).373C22, (C32×M4(2))⋊17C2, (C2×C8)⋊7(C3×C6), C2.7(C2×C6×C12), C4○D4.5(C3×C6), (C2×C6).11(C2×C12), (C3×C4○D4).22C6, (C3×C12).122(C2×C4), (C2×C12).160(C2×C6), (C32×C4○D4).9C2, (C3×C6).134(C22×C4), SmallGroup(288,828)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C8○D4
G = < a,b,c,d,e | a3=b3=c8=e2=1, d2=c4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c4d >
Subgroups: 204 in 186 conjugacy classes, 168 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, D4, Q8, C32, C12, C2×C6, C2×C8, M4(2), C4○D4, C3×C6, C3×C6, C24, C2×C12, C3×D4, C3×Q8, C8○D4, C3×C12, C3×C12, C62, C2×C24, C3×M4(2), C3×C4○D4, C3×C24, C3×C24, C6×C12, D4×C32, Q8×C32, C3×C8○D4, C6×C24, C32×M4(2), C32×C4○D4, C32×C8○D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C32, C12, C2×C6, C22×C4, C3×C6, C2×C12, C22×C6, C8○D4, C3×C12, C62, C22×C12, C6×C12, C2×C62, C3×C8○D4, C2×C6×C12, C32×C8○D4
►On 144 points
Generators in S144
(1 103 95)(2 104 96)(3 97 89)(4 98 90)(5 99 91)(6 100 92)(7 101 93)(8 102 94)(9 106 130)(10 107 131)(11 108 132)(12 109 133)(13 110 134)(14 111 135)(15 112 136)(16 105 129)(17 121 113)(18 122 114)(19 123 115)(20 124 116)(21 125 117)(22 126 118)(23 127 119)(24 128 120)(25 49 41)(26 50 42)(27 51 43)(28 52 44)(29 53 45)(30 54 46)(31 55 47)(32 56 48)(33 137 81)(34 138 82)(35 139 83)(36 140 84)(37 141 85)(38 142 86)(39 143 87)(40 144 88)(57 77 65)(58 78 66)(59 79 67)(60 80 68)(61 73 69)(62 74 70)(63 75 71)(64 76 72)
(1 23 47)(2 24 48)(3 17 41)(4 18 42)(5 19 43)(6 20 44)(7 21 45)(8 22 46)(9 138 78)(10 139 79)(11 140 80)(12 141 73)(13 142 74)(14 143 75)(15 144 76)(16 137 77)(25 97 121)(26 98 122)(27 99 123)(28 100 124)(29 101 125)(30 102 126)(31 103 127)(32 104 128)(33 57 129)(34 58 130)(35 59 131)(36 60 132)(37 61 133)(38 62 134)(39 63 135)(40 64 136)(49 89 113)(50 90 114)(51 91 115)(52 92 116)(53 93 117)(54 94 118)(55 95 119)(56 96 120)(65 105 81)(66 106 82)(67 107 83)(68 108 84)(69 109 85)(70 110 86)(71 111 87)(72 112 88)
(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 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 35 37 39)(34 36 38 40)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 95 93 91)(90 96 94 92)(97 103 101 99)(98 104 102 100)(105 107 109 111)(106 108 110 112)(113 119 117 115)(114 120 118 116)(121 127 125 123)(122 128 126 124)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)
(1 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(9 124)(10 125)(11 126)(12 127)(13 128)(14 121)(15 122)(16 123)(17 135)(18 136)(19 129)(20 130)(21 131)(22 132)(23 133)(24 134)(25 143)(26 144)(27 137)(28 138)(29 139)(30 140)(31 141)(32 142)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)(49 87)(50 88)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 89)(72 90)(73 103)(74 104)(75 97)(76 98)(77 99)(78 100)(79 101)(80 102)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(111 113)(112 114)
G:=sub<Sym(144)| (1,103,95)(2,104,96)(3,97,89)(4,98,90)(5,99,91)(6,100,92)(7,101,93)(8,102,94)(9,106,130)(10,107,131)(11,108,132)(12,109,133)(13,110,134)(14,111,135)(15,112,136)(16,105,129)(17,121,113)(18,122,114)(19,123,115)(20,124,116)(21,125,117)(22,126,118)(23,127,119)(24,128,120)(25,49,41)(26,50,42)(27,51,43)(28,52,44)(29,53,45)(30,54,46)(31,55,47)(32,56,48)(33,137,81)(34,138,82)(35,139,83)(36,140,84)(37,141,85)(38,142,86)(39,143,87)(40,144,88)(57,77,65)(58,78,66)(59,79,67)(60,80,68)(61,73,69)(62,74,70)(63,75,71)(64,76,72), (1,23,47)(2,24,48)(3,17,41)(4,18,42)(5,19,43)(6,20,44)(7,21,45)(8,22,46)(9,138,78)(10,139,79)(11,140,80)(12,141,73)(13,142,74)(14,143,75)(15,144,76)(16,137,77)(25,97,121)(26,98,122)(27,99,123)(28,100,124)(29,101,125)(30,102,126)(31,103,127)(32,104,128)(33,57,129)(34,58,130)(35,59,131)(36,60,132)(37,61,133)(38,62,134)(39,63,135)(40,64,136)(49,89,113)(50,90,114)(51,91,115)(52,92,116)(53,93,117)(54,94,118)(55,95,119)(56,96,120)(65,105,81)(66,106,82)(67,107,83)(68,108,84)(69,109,85)(70,110,86)(71,111,87)(72,112,88), (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), (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,95,93,91)(90,96,94,92)(97,103,101,99)(98,104,102,100)(105,107,109,111)(106,108,110,112)(113,119,117,115)(114,120,118,116)(121,127,125,123)(122,128,126,124)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144), (1,61)(2,62)(3,63)(4,64)(5,57)(6,58)(7,59)(8,60)(9,124)(10,125)(11,126)(12,127)(13,128)(14,121)(15,122)(16,123)(17,135)(18,136)(19,129)(20,130)(21,131)(22,132)(23,133)(24,134)(25,143)(26,144)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42)(49,87)(50,88)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,89)(72,90)(73,103)(74,104)(75,97)(76,98)(77,99)(78,100)(79,101)(80,102)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(111,113)(112,114)>;
G:=Group( (1,103,95)(2,104,96)(3,97,89)(4,98,90)(5,99,91)(6,100,92)(7,101,93)(8,102,94)(9,106,130)(10,107,131)(11,108,132)(12,109,133)(13,110,134)(14,111,135)(15,112,136)(16,105,129)(17,121,113)(18,122,114)(19,123,115)(20,124,116)(21,125,117)(22,126,118)(23,127,119)(24,128,120)(25,49,41)(26,50,42)(27,51,43)(28,52,44)(29,53,45)(30,54,46)(31,55,47)(32,56,48)(33,137,81)(34,138,82)(35,139,83)(36,140,84)(37,141,85)(38,142,86)(39,143,87)(40,144,88)(57,77,65)(58,78,66)(59,79,67)(60,80,68)(61,73,69)(62,74,70)(63,75,71)(64,76,72), (1,23,47)(2,24,48)(3,17,41)(4,18,42)(5,19,43)(6,20,44)(7,21,45)(8,22,46)(9,138,78)(10,139,79)(11,140,80)(12,141,73)(13,142,74)(14,143,75)(15,144,76)(16,137,77)(25,97,121)(26,98,122)(27,99,123)(28,100,124)(29,101,125)(30,102,126)(31,103,127)(32,104,128)(33,57,129)(34,58,130)(35,59,131)(36,60,132)(37,61,133)(38,62,134)(39,63,135)(40,64,136)(49,89,113)(50,90,114)(51,91,115)(52,92,116)(53,93,117)(54,94,118)(55,95,119)(56,96,120)(65,105,81)(66,106,82)(67,107,83)(68,108,84)(69,109,85)(70,110,86)(71,111,87)(72,112,88), (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), (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,95,93,91)(90,96,94,92)(97,103,101,99)(98,104,102,100)(105,107,109,111)(106,108,110,112)(113,119,117,115)(114,120,118,116)(121,127,125,123)(122,128,126,124)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144), (1,61)(2,62)(3,63)(4,64)(5,57)(6,58)(7,59)(8,60)(9,124)(10,125)(11,126)(12,127)(13,128)(14,121)(15,122)(16,123)(17,135)(18,136)(19,129)(20,130)(21,131)(22,132)(23,133)(24,134)(25,143)(26,144)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42)(49,87)(50,88)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,89)(72,90)(73,103)(74,104)(75,97)(76,98)(77,99)(78,100)(79,101)(80,102)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(111,113)(112,114) );
G=PermutationGroup([[(1,103,95),(2,104,96),(3,97,89),(4,98,90),(5,99,91),(6,100,92),(7,101,93),(8,102,94),(9,106,130),(10,107,131),(11,108,132),(12,109,133),(13,110,134),(14,111,135),(15,112,136),(16,105,129),(17,121,113),(18,122,114),(19,123,115),(20,124,116),(21,125,117),(22,126,118),(23,127,119),(24,128,120),(25,49,41),(26,50,42),(27,51,43),(28,52,44),(29,53,45),(30,54,46),(31,55,47),(32,56,48),(33,137,81),(34,138,82),(35,139,83),(36,140,84),(37,141,85),(38,142,86),(39,143,87),(40,144,88),(57,77,65),(58,78,66),(59,79,67),(60,80,68),(61,73,69),(62,74,70),(63,75,71),(64,76,72)], [(1,23,47),(2,24,48),(3,17,41),(4,18,42),(5,19,43),(6,20,44),(7,21,45),(8,22,46),(9,138,78),(10,139,79),(11,140,80),(12,141,73),(13,142,74),(14,143,75),(15,144,76),(16,137,77),(25,97,121),(26,98,122),(27,99,123),(28,100,124),(29,101,125),(30,102,126),(31,103,127),(32,104,128),(33,57,129),(34,58,130),(35,59,131),(36,60,132),(37,61,133),(38,62,134),(39,63,135),(40,64,136),(49,89,113),(50,90,114),(51,91,115),(52,92,116),(53,93,117),(54,94,118),(55,95,119),(56,96,120),(65,105,81),(66,106,82),(67,107,83),(68,108,84),(69,109,85),(70,110,86),(71,111,87),(72,112,88)], [(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)], [(1,7,5,3),(2,8,6,4),(9,11,13,15),(10,12,14,16),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,35,37,39),(34,36,38,40),(41,47,45,43),(42,48,46,44),(49,55,53,51),(50,56,54,52),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,95,93,91),(90,96,94,92),(97,103,101,99),(98,104,102,100),(105,107,109,111),(106,108,110,112),(113,119,117,115),(114,120,118,116),(121,127,125,123),(122,128,126,124),(129,131,133,135),(130,132,134,136),(137,139,141,143),(138,140,142,144)], [(1,61),(2,62),(3,63),(4,64),(5,57),(6,58),(7,59),(8,60),(9,124),(10,125),(11,126),(12,127),(13,128),(14,121),(15,122),(16,123),(17,135),(18,136),(19,129),(20,130),(21,131),(22,132),(23,133),(24,134),(25,143),(26,144),(27,137),(28,138),(29,139),(30,140),(31,141),(32,142),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,41),(40,42),(49,87),(50,88),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,89),(72,90),(73,103),(74,104),(75,97),(76,98),(77,99),(78,100),(79,101),(80,102),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(111,113),(112,114)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6H | 6I | ··· | 6AF | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 12A | ··· | 12P | 12Q | ··· | 12AN | 24A | ··· | 24AF | 24AG | ··· | 24CB |
| order | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | C8○D4 | C3×C8○D4 |
| kernel | C32×C8○D4 | C6×C24 | C32×M4(2) | C32×C4○D4 | C3×C8○D4 | D4×C32 | Q8×C32 | C2×C24 | C3×M4(2) | C3×C4○D4 | C3×D4 | C3×Q8 | C32 | C3 |
| # reps | 1 | 3 | 3 | 1 | 8 | 6 | 2 | 24 | 24 | 8 | 48 | 16 | 4 | 32 |
Matrix representation of C32×C8○D4 ►in GL3(𝔽73) generated by
| 64 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 64 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 46 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 22 |
| 1 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 61 | 46 |
| 1 | 0 | 0 |
| 0 | 12 | 54 |
| 0 | 69 | 61 |
G:=sub<GL(3,GF(73))| [64,0,0,0,1,0,0,0,1],[64,0,0,0,8,0,0,0,8],[46,0,0,0,22,0,0,0,22],[1,0,0,0,27,61,0,0,46],[1,0,0,0,12,69,0,54,61] >;
C32×C8○D4 in GAP, Magma, Sage, TeX
C_3^2\times C_8\circ D_4
% in TeX
G:=Group("C3^2xC8oD4"); // GroupNames label
G:=SmallGroup(288,828);
// by ID
G=gap.SmallGroup(288,828);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,504,1563,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^8=e^2=1,d^2=c^4,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,c*e=e*c,e*d*e=c^4*d>;
// generators/relations