metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.5D36, C42⋊7D9, C36.28D4, C12.39D12, (C4×C36)⋊5C2, D18⋊C4⋊1C2, (C4×C12).7S3, C18.4(C2×D4), C2.6(C2×D36), (C2×D36).2C2, (C2×C4).65D18, C6.33(C2×D12), C9⋊1(C4.4D4), (C2×Dic18)⋊1C2, C18.5(C4○D4), (C2×C12).339D6, C3.(C42⋊7S3), C6.75(C4○D12), (C2×C18).16C23, (C2×C36).88C22, C2.7(D36⋊5C2), (C2×Dic9).3C22, (C22×D9).2C22, C22.37(C22×D9), (C2×C6).173(C22×S3), SmallGroup(288,85)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊7D9
G = < a,b,c,d | a4=b4=c9=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 604 in 114 conjugacy classes, 44 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C9, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, D9, C18, C18, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C4.4D4, Dic9, C36, C36, D18, C2×C18, D6⋊C4, C4×C12, C2×Dic6, C2×D12, Dic18, D36, C2×Dic9, C2×C36, C2×C36, C22×D9, C42⋊7S3, D18⋊C4, C4×C36, C2×Dic18, C2×D36, C42⋊7D9
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, D12, C22×S3, C4.4D4, D18, C2×D12, C4○D12, D36, C22×D9, C42⋊7S3, C2×D36, D36⋊5C2, C42⋊7D9
►On 144 points
Generators in S144
(1 95 23 77)(2 96 24 78)(3 97 25 79)(4 98 26 80)(5 99 27 81)(6 91 19 73)(7 92 20 74)(8 93 21 75)(9 94 22 76)(10 100 28 82)(11 101 29 83)(12 102 30 84)(13 103 31 85)(14 104 32 86)(15 105 33 87)(16 106 34 88)(17 107 35 89)(18 108 36 90)(37 127 55 109)(38 128 56 110)(39 129 57 111)(40 130 58 112)(41 131 59 113)(42 132 60 114)(43 133 61 115)(44 134 62 116)(45 135 63 117)(46 136 64 118)(47 137 65 119)(48 138 66 120)(49 139 67 121)(50 140 68 122)(51 141 69 123)(52 142 70 124)(53 143 71 125)(54 144 72 126)
(1 50 14 41)(2 51 15 42)(3 52 16 43)(4 53 17 44)(5 54 18 45)(6 46 10 37)(7 47 11 38)(8 48 12 39)(9 49 13 40)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)(73 118 82 109)(74 119 83 110)(75 120 84 111)(76 121 85 112)(77 122 86 113)(78 123 87 114)(79 124 88 115)(80 125 89 116)(81 126 90 117)(91 136 100 127)(92 137 101 128)(93 138 102 129)(94 139 103 130)(95 140 104 131)(96 141 105 132)(97 142 106 133)(98 143 107 134)(99 144 108 135)
(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 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 35)(29 34)(30 33)(31 32)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 63)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 72)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(81 90)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(99 108)(109 143)(110 142)(111 141)(112 140)(113 139)(114 138)(115 137)(116 136)(117 144)(118 134)(119 133)(120 132)(121 131)(122 130)(123 129)(124 128)(125 127)(126 135)
G:=sub<Sym(144)| (1,95,23,77)(2,96,24,78)(3,97,25,79)(4,98,26,80)(5,99,27,81)(6,91,19,73)(7,92,20,74)(8,93,21,75)(9,94,22,76)(10,100,28,82)(11,101,29,83)(12,102,30,84)(13,103,31,85)(14,104,32,86)(15,105,33,87)(16,106,34,88)(17,107,35,89)(18,108,36,90)(37,127,55,109)(38,128,56,110)(39,129,57,111)(40,130,58,112)(41,131,59,113)(42,132,60,114)(43,133,61,115)(44,134,62,116)(45,135,63,117)(46,136,64,118)(47,137,65,119)(48,138,66,120)(49,139,67,121)(50,140,68,122)(51,141,69,123)(52,142,70,124)(53,143,71,125)(54,144,72,126), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63)(73,118,82,109)(74,119,83,110)(75,120,84,111)(76,121,85,112)(77,122,86,113)(78,123,87,114)(79,124,88,115)(80,125,89,116)(81,126,90,117)(91,136,100,127)(92,137,101,128)(93,138,102,129)(94,139,103,130)(95,140,104,131)(96,141,105,132)(97,142,106,133)(98,143,107,134)(99,144,108,135), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(81,90)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(99,108)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,144)(118,134)(119,133)(120,132)(121,131)(122,130)(123,129)(124,128)(125,127)(126,135)>;
G:=Group( (1,95,23,77)(2,96,24,78)(3,97,25,79)(4,98,26,80)(5,99,27,81)(6,91,19,73)(7,92,20,74)(8,93,21,75)(9,94,22,76)(10,100,28,82)(11,101,29,83)(12,102,30,84)(13,103,31,85)(14,104,32,86)(15,105,33,87)(16,106,34,88)(17,107,35,89)(18,108,36,90)(37,127,55,109)(38,128,56,110)(39,129,57,111)(40,130,58,112)(41,131,59,113)(42,132,60,114)(43,133,61,115)(44,134,62,116)(45,135,63,117)(46,136,64,118)(47,137,65,119)(48,138,66,120)(49,139,67,121)(50,140,68,122)(51,141,69,123)(52,142,70,124)(53,143,71,125)(54,144,72,126), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63)(73,118,82,109)(74,119,83,110)(75,120,84,111)(76,121,85,112)(77,122,86,113)(78,123,87,114)(79,124,88,115)(80,125,89,116)(81,126,90,117)(91,136,100,127)(92,137,101,128)(93,138,102,129)(94,139,103,130)(95,140,104,131)(96,141,105,132)(97,142,106,133)(98,143,107,134)(99,144,108,135), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(81,90)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(99,108)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,144)(118,134)(119,133)(120,132)(121,131)(122,130)(123,129)(124,128)(125,127)(126,135) );
G=PermutationGroup([[(1,95,23,77),(2,96,24,78),(3,97,25,79),(4,98,26,80),(5,99,27,81),(6,91,19,73),(7,92,20,74),(8,93,21,75),(9,94,22,76),(10,100,28,82),(11,101,29,83),(12,102,30,84),(13,103,31,85),(14,104,32,86),(15,105,33,87),(16,106,34,88),(17,107,35,89),(18,108,36,90),(37,127,55,109),(38,128,56,110),(39,129,57,111),(40,130,58,112),(41,131,59,113),(42,132,60,114),(43,133,61,115),(44,134,62,116),(45,135,63,117),(46,136,64,118),(47,137,65,119),(48,138,66,120),(49,139,67,121),(50,140,68,122),(51,141,69,123),(52,142,70,124),(53,143,71,125),(54,144,72,126)], [(1,50,14,41),(2,51,15,42),(3,52,16,43),(4,53,17,44),(5,54,18,45),(6,46,10,37),(7,47,11,38),(8,48,12,39),(9,49,13,40),(19,64,28,55),(20,65,29,56),(21,66,30,57),(22,67,31,58),(23,68,32,59),(24,69,33,60),(25,70,34,61),(26,71,35,62),(27,72,36,63),(73,118,82,109),(74,119,83,110),(75,120,84,111),(76,121,85,112),(77,122,86,113),(78,123,87,114),(79,124,88,115),(80,125,89,116),(81,126,90,117),(91,136,100,127),(92,137,101,128),(93,138,102,129),(94,139,103,130),(95,140,104,131),(96,141,105,132),(97,142,106,133),(98,143,107,134),(99,144,108,135)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,26),(20,25),(21,24),(22,23),(28,35),(29,34),(30,33),(31,32),(37,62),(38,61),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,63),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,72),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82),(81,90),(91,107),(92,106),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100),(99,108),(109,143),(110,142),(111,141),(112,140),(113,139),(114,138),(115,137),(116,136),(117,144),(118,134),(119,133),(120,132),(121,131),(122,130),(123,129),(124,128),(125,127),(126,135)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12L | 18A | ··· | 18I | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 2 | 2 | ··· | 2 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | C4○D4 | D9 | D12 | D18 | C4○D12 | D36 | D36⋊5C2 |
| kernel | C42⋊7D9 | D18⋊C4 | C4×C36 | C2×Dic18 | C2×D36 | C4×C12 | C36 | C2×C12 | C18 | C42 | C12 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 3 | 4 | 9 | 8 | 12 | 24 |
Matrix representation of C42⋊7D9 ►in GL4(𝔽37) generated by
| 7 | 14 | 0 | 0 |
| 23 | 30 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 31 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 30 | 23 |
| 0 | 0 | 14 | 7 |
| 26 | 6 | 0 | 0 |
| 31 | 20 | 0 | 0 |
| 0 | 0 | 17 | 11 |
| 0 | 0 | 26 | 6 |
| 26 | 6 | 0 | 0 |
| 17 | 11 | 0 | 0 |
| 0 | 0 | 17 | 11 |
| 0 | 0 | 31 | 20 |
G:=sub<GL(4,GF(37))| [7,23,0,0,14,30,0,0,0,0,31,0,0,0,0,31],[6,0,0,0,0,6,0,0,0,0,30,14,0,0,23,7],[26,31,0,0,6,20,0,0,0,0,17,26,0,0,11,6],[26,17,0,0,6,11,0,0,0,0,17,31,0,0,11,20] >;
C42⋊7D9 in GAP, Magma, Sage, TeX
C_4^2\rtimes_7D_9
% in TeX
G:=Group("C4^2:7D9"); // GroupNames label
G:=SmallGroup(288,85);
// by ID
G=gap.SmallGroup(288,85);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,254,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^9=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations