direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4×D9, D18.Q8, D18.11D4, C4⋊3(C4×D9), C36⋊1(C2×C4), (C4×D9)⋊1C4, C2.3(D4×D9), C2.2(Q8×D9), C12.8(C4×S3), C6.84(S3×D4), C6.35(S3×Q8), C4⋊Dic9⋊11C2, Dic9⋊3(C2×C4), D18.8(C2×C4), C18.23(C2×D4), (C2×C4).45D18, C18.12(C2×Q8), Dic9⋊C4⋊11C2, (C2×C12).180D6, C18.9(C22×C4), (C2×C36).28C22, (C2×C18).32C23, C22.16(C22×D9), (C2×Dic9).31C22, (C22×D9).34C22, C9⋊1(C2×C4⋊C4), C3.(S3×C4⋊C4), (C9×C4⋊C4)⋊2C2, C6.48(S3×C2×C4), (C2×C4×D9).8C2, C2.11(C2×C4×D9), (C3×C4⋊C4).9S3, (C2×C6).189(C22×S3), SmallGroup(288,101)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4×D9
G = < a,b,c,d | a4=b4=c9=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 552 in 138 conjugacy classes, 60 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, C23, C9, Dic3, C12, C12, D6, C2×C6, C4⋊C4, C4⋊C4, C22×C4, D9, C18, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C2×C4⋊C4, Dic9, Dic9, C36, C36, D18, C2×C18, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, C4×D9, C4×D9, C2×Dic9, C2×Dic9, C2×C36, C2×C36, C22×D9, S3×C4⋊C4, Dic9⋊C4, C4⋊Dic9, C9×C4⋊C4, C2×C4×D9, C2×C4×D9, C4⋊C4×D9
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, D9, C4×S3, C22×S3, C2×C4⋊C4, D18, S3×C2×C4, S3×D4, S3×Q8, C4×D9, C22×D9, S3×C4⋊C4, C2×C4×D9, D4×D9, Q8×D9, C4⋊C4×D9
►On 144 points
Generators in S144
(1 104 14 95)(2 105 15 96)(3 106 16 97)(4 107 17 98)(5 108 18 99)(6 100 10 91)(7 101 11 92)(8 102 12 93)(9 103 13 94)(19 82 28 73)(20 83 29 74)(21 84 30 75)(22 85 31 76)(23 86 32 77)(24 87 33 78)(25 88 34 79)(26 89 35 80)(27 90 36 81)(37 136 46 127)(38 137 47 128)(39 138 48 129)(40 139 49 130)(41 140 50 131)(42 141 51 132)(43 142 52 133)(44 143 53 134)(45 144 54 135)(55 118 64 109)(56 119 65 110)(57 120 66 111)(58 121 67 112)(59 122 68 113)(60 123 69 114)(61 124 70 115)(62 125 71 116)(63 126 72 117)
(1 59 23 41)(2 60 24 42)(3 61 25 43)(4 62 26 44)(5 63 27 45)(6 55 19 37)(7 56 20 38)(8 57 21 39)(9 58 22 40)(10 64 28 46)(11 65 29 47)(12 66 30 48)(13 67 31 49)(14 68 32 50)(15 69 33 51)(16 70 34 52)(17 71 35 53)(18 72 36 54)(73 136 91 118)(74 137 92 119)(75 138 93 120)(76 139 94 121)(77 140 95 122)(78 141 96 123)(79 142 97 124)(80 143 98 125)(81 144 99 126)(82 127 100 109)(83 128 101 110)(84 129 102 111)(85 130 103 112)(86 131 104 113)(87 132 105 114)(88 133 106 115)(89 134 107 116)(90 135 108 117)
(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 44)(38 43)(39 42)(40 41)(46 53)(47 52)(48 51)(49 50)(55 62)(56 61)(57 60)(58 59)(64 71)(65 70)(66 69)(67 68)(73 80)(74 79)(75 78)(76 77)(82 89)(83 88)(84 87)(85 86)(91 98)(92 97)(93 96)(94 95)(100 107)(101 106)(102 105)(103 104)(109 116)(110 115)(111 114)(112 113)(118 125)(119 124)(120 123)(121 122)(127 134)(128 133)(129 132)(130 131)(136 143)(137 142)(138 141)(139 140)
G:=sub<Sym(144)| (1,104,14,95)(2,105,15,96)(3,106,16,97)(4,107,17,98)(5,108,18,99)(6,100,10,91)(7,101,11,92)(8,102,12,93)(9,103,13,94)(19,82,28,73)(20,83,29,74)(21,84,30,75)(22,85,31,76)(23,86,32,77)(24,87,33,78)(25,88,34,79)(26,89,35,80)(27,90,36,81)(37,136,46,127)(38,137,47,128)(39,138,48,129)(40,139,49,130)(41,140,50,131)(42,141,51,132)(43,142,52,133)(44,143,53,134)(45,144,54,135)(55,118,64,109)(56,119,65,110)(57,120,66,111)(58,121,67,112)(59,122,68,113)(60,123,69,114)(61,124,70,115)(62,125,71,116)(63,126,72,117), (1,59,23,41)(2,60,24,42)(3,61,25,43)(4,62,26,44)(5,63,27,45)(6,55,19,37)(7,56,20,38)(8,57,21,39)(9,58,22,40)(10,64,28,46)(11,65,29,47)(12,66,30,48)(13,67,31,49)(14,68,32,50)(15,69,33,51)(16,70,34,52)(17,71,35,53)(18,72,36,54)(73,136,91,118)(74,137,92,119)(75,138,93,120)(76,139,94,121)(77,140,95,122)(78,141,96,123)(79,142,97,124)(80,143,98,125)(81,144,99,126)(82,127,100,109)(83,128,101,110)(84,129,102,111)(85,130,103,112)(86,131,104,113)(87,132,105,114)(88,133,106,115)(89,134,107,116)(90,135,108,117), (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)(37,44)(38,43)(39,42)(40,41)(46,53)(47,52)(48,51)(49,50)(55,62)(56,61)(57,60)(58,59)(64,71)(65,70)(66,69)(67,68)(73,80)(74,79)(75,78)(76,77)(82,89)(83,88)(84,87)(85,86)(91,98)(92,97)(93,96)(94,95)(100,107)(101,106)(102,105)(103,104)(109,116)(110,115)(111,114)(112,113)(118,125)(119,124)(120,123)(121,122)(127,134)(128,133)(129,132)(130,131)(136,143)(137,142)(138,141)(139,140)>;
G:=Group( (1,104,14,95)(2,105,15,96)(3,106,16,97)(4,107,17,98)(5,108,18,99)(6,100,10,91)(7,101,11,92)(8,102,12,93)(9,103,13,94)(19,82,28,73)(20,83,29,74)(21,84,30,75)(22,85,31,76)(23,86,32,77)(24,87,33,78)(25,88,34,79)(26,89,35,80)(27,90,36,81)(37,136,46,127)(38,137,47,128)(39,138,48,129)(40,139,49,130)(41,140,50,131)(42,141,51,132)(43,142,52,133)(44,143,53,134)(45,144,54,135)(55,118,64,109)(56,119,65,110)(57,120,66,111)(58,121,67,112)(59,122,68,113)(60,123,69,114)(61,124,70,115)(62,125,71,116)(63,126,72,117), (1,59,23,41)(2,60,24,42)(3,61,25,43)(4,62,26,44)(5,63,27,45)(6,55,19,37)(7,56,20,38)(8,57,21,39)(9,58,22,40)(10,64,28,46)(11,65,29,47)(12,66,30,48)(13,67,31,49)(14,68,32,50)(15,69,33,51)(16,70,34,52)(17,71,35,53)(18,72,36,54)(73,136,91,118)(74,137,92,119)(75,138,93,120)(76,139,94,121)(77,140,95,122)(78,141,96,123)(79,142,97,124)(80,143,98,125)(81,144,99,126)(82,127,100,109)(83,128,101,110)(84,129,102,111)(85,130,103,112)(86,131,104,113)(87,132,105,114)(88,133,106,115)(89,134,107,116)(90,135,108,117), (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)(37,44)(38,43)(39,42)(40,41)(46,53)(47,52)(48,51)(49,50)(55,62)(56,61)(57,60)(58,59)(64,71)(65,70)(66,69)(67,68)(73,80)(74,79)(75,78)(76,77)(82,89)(83,88)(84,87)(85,86)(91,98)(92,97)(93,96)(94,95)(100,107)(101,106)(102,105)(103,104)(109,116)(110,115)(111,114)(112,113)(118,125)(119,124)(120,123)(121,122)(127,134)(128,133)(129,132)(130,131)(136,143)(137,142)(138,141)(139,140) );
G=PermutationGroup([[(1,104,14,95),(2,105,15,96),(3,106,16,97),(4,107,17,98),(5,108,18,99),(6,100,10,91),(7,101,11,92),(8,102,12,93),(9,103,13,94),(19,82,28,73),(20,83,29,74),(21,84,30,75),(22,85,31,76),(23,86,32,77),(24,87,33,78),(25,88,34,79),(26,89,35,80),(27,90,36,81),(37,136,46,127),(38,137,47,128),(39,138,48,129),(40,139,49,130),(41,140,50,131),(42,141,51,132),(43,142,52,133),(44,143,53,134),(45,144,54,135),(55,118,64,109),(56,119,65,110),(57,120,66,111),(58,121,67,112),(59,122,68,113),(60,123,69,114),(61,124,70,115),(62,125,71,116),(63,126,72,117)], [(1,59,23,41),(2,60,24,42),(3,61,25,43),(4,62,26,44),(5,63,27,45),(6,55,19,37),(7,56,20,38),(8,57,21,39),(9,58,22,40),(10,64,28,46),(11,65,29,47),(12,66,30,48),(13,67,31,49),(14,68,32,50),(15,69,33,51),(16,70,34,52),(17,71,35,53),(18,72,36,54),(73,136,91,118),(74,137,92,119),(75,138,93,120),(76,139,94,121),(77,140,95,122),(78,141,96,123),(79,142,97,124),(80,143,98,125),(81,144,99,126),(82,127,100,109),(83,128,101,110),(84,129,102,111),(85,130,103,112),(86,131,104,113),(87,132,105,114),(88,133,106,115),(89,134,107,116),(90,135,108,117)], [(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),(37,44),(38,43),(39,42),(40,41),(46,53),(47,52),(48,51),(49,50),(55,62),(56,61),(57,60),(58,59),(64,71),(65,70),(66,69),(67,68),(73,80),(74,79),(75,78),(76,77),(82,89),(83,88),(84,87),(85,86),(91,98),(92,97),(93,96),(94,95),(100,107),(101,106),(102,105),(103,104),(109,116),(110,115),(111,114),(112,113),(118,125),(119,124),(120,123),(121,122),(127,134),(128,133),(129,132),(130,131),(136,143),(137,142),(138,141),(139,140)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | ··· | 4L | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 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 | 9 | 9 | 9 | 9 | 2 | 2 | ··· | 2 | 18 | ··· | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D6 | D9 | C4×S3 | D18 | C4×D9 | S3×D4 | S3×Q8 | D4×D9 | Q8×D9 |
| kernel | C4⋊C4×D9 | Dic9⋊C4 | C4⋊Dic9 | C9×C4⋊C4 | C2×C4×D9 | C4×D9 | C3×C4⋊C4 | D18 | D18 | C2×C12 | C4⋊C4 | C12 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 8 | 1 | 2 | 2 | 3 | 3 | 4 | 9 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of C4⋊C4×D9 ►in GL5(𝔽37)
| 36 | 0 | 0 | 0 | 0 |
| 0 | 11 | 36 | 0 | 0 |
| 0 | 11 | 26 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 6 | 0 | 0 | 0 | 0 |
| 0 | 1 | 35 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 6 | 11 |
| 0 | 0 | 0 | 26 | 17 |
| 36 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 26 | 17 |
| 0 | 0 | 0 | 6 | 11 |
G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,11,11,0,0,0,36,26,0,0,0,0,0,1,0,0,0,0,0,1],[6,0,0,0,0,0,1,0,0,0,0,35,36,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,26,0,0,0,11,17],[36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,26,6,0,0,0,17,11] >;
C4⋊C4×D9 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\times D_9
% in TeX
G:=Group("C4:C4xD9"); // GroupNames label
G:=SmallGroup(288,101);
// by ID
G=gap.SmallGroup(288,101);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^9=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations