direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C22⋊Q8, C36.62D4, C4⋊C4⋊3C18, (C2×C18)⋊2Q8, (Q8×C18)⋊8C2, (C2×Q8)⋊3C18, C4.13(D4×C9), C2.6(D4×C18), C6.69(C6×D4), C6.20(C6×Q8), C2.3(Q8×C18), C22⋊2(Q8×C9), C18.69(C2×D4), C12.72(C3×D4), (C6×Q8).16C6, C18.20(C2×Q8), C22⋊C4.1C18, (C22×C4).7C18, C18.42(C4○D4), (C22×C36).15C2, C23.13(C2×C18), (C2×C18).77C23, (C22×C12).29C6, (C2×C36).123C22, (C22×C18).28C22, C22.12(C22×C18), (C9×C4⋊C4)⋊12C2, C3.(C3×C22⋊Q8), C2.5(C9×C4○D4), (C3×C4⋊C4).13C6, (C2×C6).4(C3×Q8), (C3×C22⋊Q8).C3, (C2×C4).3(C2×C18), C6.42(C3×C4○D4), (C2×C12).64(C2×C6), (C9×C22⋊C4).4C2, (C3×C22⋊C4).7C6, (C2×C6).82(C22×C6), (C22×C6).47(C2×C6), SmallGroup(288,172)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C22⋊Q8
G = < a,b,c,d,e | a9=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 150 in 111 conjugacy classes, 72 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C9, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C18, C18, C2×C12, C2×C12, C2×C12, C3×Q8, C22×C6, C22⋊Q8, C36, C36, C2×C18, C2×C18, C2×C18, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C6×Q8, C2×C36, C2×C36, C2×C36, Q8×C9, C22×C18, C3×C22⋊Q8, C9×C22⋊C4, C9×C4⋊C4, C9×C4⋊C4, C22×C36, Q8×C18, C9×C22⋊Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C9, C2×C6, C2×D4, C2×Q8, C4○D4, C18, C3×D4, C3×Q8, C22×C6, C22⋊Q8, C2×C18, C6×D4, C6×Q8, C3×C4○D4, D4×C9, Q8×C9, C22×C18, C3×C22⋊Q8, D4×C18, Q8×C18, C9×C4○D4, C9×C22⋊Q8
►On 144 points
Generators in S144
(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 43)(2 44)(3 45)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 23)(11 24)(12 25)(13 26)(14 27)(15 19)(16 20)(17 21)(18 22)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 46)(35 47)(36 48)(55 80)(56 81)(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 82)(71 83)(72 84)(91 104)(92 105)(93 106)(94 107)(95 108)(96 100)(97 101)(98 102)(99 103)(109 118)(110 119)(111 120)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)(127 140)(128 141)(129 142)(130 143)(131 144)(132 136)(133 137)(134 138)(135 139)
(1 34)(2 35)(3 36)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 139)(11 140)(12 141)(13 142)(14 143)(15 144)(16 136)(17 137)(18 138)(19 131)(20 132)(21 133)(22 134)(23 135)(24 127)(25 128)(26 129)(27 130)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 46)(44 47)(45 48)(55 89)(56 90)(57 82)(58 83)(59 84)(60 85)(61 86)(62 87)(63 88)(64 76)(65 77)(66 78)(67 79)(68 80)(69 81)(70 73)(71 74)(72 75)(91 125)(92 126)(93 118)(94 119)(95 120)(96 121)(97 122)(98 123)(99 124)(100 112)(101 113)(102 114)(103 115)(104 116)(105 117)(106 109)(107 110)(108 111)
(1 57 43 73)(2 58 44 74)(3 59 45 75)(4 60 37 76)(5 61 38 77)(6 62 39 78)(7 63 40 79)(8 55 41 80)(9 56 42 81)(10 99 135 115)(11 91 127 116)(12 92 128 117)(13 93 129 109)(14 94 130 110)(15 95 131 111)(16 96 132 112)(17 97 133 113)(18 98 134 114)(19 108 144 120)(20 100 136 121)(21 101 137 122)(22 102 138 123)(23 103 139 124)(24 104 140 125)(25 105 141 126)(26 106 142 118)(27 107 143 119)(28 85 49 64)(29 86 50 65)(30 87 51 66)(31 88 52 67)(32 89 53 68)(33 90 54 69)(34 82 46 70)(35 83 47 71)(36 84 48 72)
(1 93 43 109)(2 94 44 110)(3 95 45 111)(4 96 37 112)(5 97 38 113)(6 98 39 114)(7 99 40 115)(8 91 41 116)(9 92 42 117)(10 79 135 63)(11 80 127 55)(12 81 128 56)(13 73 129 57)(14 74 130 58)(15 75 131 59)(16 76 132 60)(17 77 133 61)(18 78 134 62)(19 84 144 72)(20 85 136 64)(21 86 137 65)(22 87 138 66)(23 88 139 67)(24 89 140 68)(25 90 141 69)(26 82 142 70)(27 83 143 71)(28 121 49 100)(29 122 50 101)(30 123 51 102)(31 124 52 103)(32 125 53 104)(33 126 54 105)(34 118 46 106)(35 119 47 107)(36 120 48 108)
G:=sub<Sym(144)| (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), (1,43)(2,44)(3,45)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,23)(11,24)(12,25)(13,26)(14,27)(15,19)(16,20)(17,21)(18,22)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,46)(35,47)(36,48)(55,80)(56,81)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,82)(71,83)(72,84)(91,104)(92,105)(93,106)(94,107)(95,108)(96,100)(97,101)(98,102)(99,103)(109,118)(110,119)(111,120)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(127,140)(128,141)(129,142)(130,143)(131,144)(132,136)(133,137)(134,138)(135,139), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,139)(11,140)(12,141)(13,142)(14,143)(15,144)(16,136)(17,137)(18,138)(19,131)(20,132)(21,133)(22,134)(23,135)(24,127)(25,128)(26,129)(27,130)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,46)(44,47)(45,48)(55,89)(56,90)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,76)(65,77)(66,78)(67,79)(68,80)(69,81)(70,73)(71,74)(72,75)(91,125)(92,126)(93,118)(94,119)(95,120)(96,121)(97,122)(98,123)(99,124)(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,109)(107,110)(108,111), (1,57,43,73)(2,58,44,74)(3,59,45,75)(4,60,37,76)(5,61,38,77)(6,62,39,78)(7,63,40,79)(8,55,41,80)(9,56,42,81)(10,99,135,115)(11,91,127,116)(12,92,128,117)(13,93,129,109)(14,94,130,110)(15,95,131,111)(16,96,132,112)(17,97,133,113)(18,98,134,114)(19,108,144,120)(20,100,136,121)(21,101,137,122)(22,102,138,123)(23,103,139,124)(24,104,140,125)(25,105,141,126)(26,106,142,118)(27,107,143,119)(28,85,49,64)(29,86,50,65)(30,87,51,66)(31,88,52,67)(32,89,53,68)(33,90,54,69)(34,82,46,70)(35,83,47,71)(36,84,48,72), (1,93,43,109)(2,94,44,110)(3,95,45,111)(4,96,37,112)(5,97,38,113)(6,98,39,114)(7,99,40,115)(8,91,41,116)(9,92,42,117)(10,79,135,63)(11,80,127,55)(12,81,128,56)(13,73,129,57)(14,74,130,58)(15,75,131,59)(16,76,132,60)(17,77,133,61)(18,78,134,62)(19,84,144,72)(20,85,136,64)(21,86,137,65)(22,87,138,66)(23,88,139,67)(24,89,140,68)(25,90,141,69)(26,82,142,70)(27,83,143,71)(28,121,49,100)(29,122,50,101)(30,123,51,102)(31,124,52,103)(32,125,53,104)(33,126,54,105)(34,118,46,106)(35,119,47,107)(36,120,48,108)>;
G:=Group( (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), (1,43)(2,44)(3,45)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,23)(11,24)(12,25)(13,26)(14,27)(15,19)(16,20)(17,21)(18,22)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,46)(35,47)(36,48)(55,80)(56,81)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,82)(71,83)(72,84)(91,104)(92,105)(93,106)(94,107)(95,108)(96,100)(97,101)(98,102)(99,103)(109,118)(110,119)(111,120)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(127,140)(128,141)(129,142)(130,143)(131,144)(132,136)(133,137)(134,138)(135,139), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,139)(11,140)(12,141)(13,142)(14,143)(15,144)(16,136)(17,137)(18,138)(19,131)(20,132)(21,133)(22,134)(23,135)(24,127)(25,128)(26,129)(27,130)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,46)(44,47)(45,48)(55,89)(56,90)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,76)(65,77)(66,78)(67,79)(68,80)(69,81)(70,73)(71,74)(72,75)(91,125)(92,126)(93,118)(94,119)(95,120)(96,121)(97,122)(98,123)(99,124)(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,109)(107,110)(108,111), (1,57,43,73)(2,58,44,74)(3,59,45,75)(4,60,37,76)(5,61,38,77)(6,62,39,78)(7,63,40,79)(8,55,41,80)(9,56,42,81)(10,99,135,115)(11,91,127,116)(12,92,128,117)(13,93,129,109)(14,94,130,110)(15,95,131,111)(16,96,132,112)(17,97,133,113)(18,98,134,114)(19,108,144,120)(20,100,136,121)(21,101,137,122)(22,102,138,123)(23,103,139,124)(24,104,140,125)(25,105,141,126)(26,106,142,118)(27,107,143,119)(28,85,49,64)(29,86,50,65)(30,87,51,66)(31,88,52,67)(32,89,53,68)(33,90,54,69)(34,82,46,70)(35,83,47,71)(36,84,48,72), (1,93,43,109)(2,94,44,110)(3,95,45,111)(4,96,37,112)(5,97,38,113)(6,98,39,114)(7,99,40,115)(8,91,41,116)(9,92,42,117)(10,79,135,63)(11,80,127,55)(12,81,128,56)(13,73,129,57)(14,74,130,58)(15,75,131,59)(16,76,132,60)(17,77,133,61)(18,78,134,62)(19,84,144,72)(20,85,136,64)(21,86,137,65)(22,87,138,66)(23,88,139,67)(24,89,140,68)(25,90,141,69)(26,82,142,70)(27,83,143,71)(28,121,49,100)(29,122,50,101)(30,123,51,102)(31,124,52,103)(32,125,53,104)(33,126,54,105)(34,118,46,106)(35,119,47,107)(36,120,48,108) );
G=PermutationGroup([[(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)], [(1,43),(2,44),(3,45),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,23),(11,24),(12,25),(13,26),(14,27),(15,19),(16,20),(17,21),(18,22),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,46),(35,47),(36,48),(55,80),(56,81),(57,73),(58,74),(59,75),(60,76),(61,77),(62,78),(63,79),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,82),(71,83),(72,84),(91,104),(92,105),(93,106),(94,107),(95,108),(96,100),(97,101),(98,102),(99,103),(109,118),(110,119),(111,120),(112,121),(113,122),(114,123),(115,124),(116,125),(117,126),(127,140),(128,141),(129,142),(130,143),(131,144),(132,136),(133,137),(134,138),(135,139)], [(1,34),(2,35),(3,36),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,139),(11,140),(12,141),(13,142),(14,143),(15,144),(16,136),(17,137),(18,138),(19,131),(20,132),(21,133),(22,134),(23,135),(24,127),(25,128),(26,129),(27,130),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(43,46),(44,47),(45,48),(55,89),(56,90),(57,82),(58,83),(59,84),(60,85),(61,86),(62,87),(63,88),(64,76),(65,77),(66,78),(67,79),(68,80),(69,81),(70,73),(71,74),(72,75),(91,125),(92,126),(93,118),(94,119),(95,120),(96,121),(97,122),(98,123),(99,124),(100,112),(101,113),(102,114),(103,115),(104,116),(105,117),(106,109),(107,110),(108,111)], [(1,57,43,73),(2,58,44,74),(3,59,45,75),(4,60,37,76),(5,61,38,77),(6,62,39,78),(7,63,40,79),(8,55,41,80),(9,56,42,81),(10,99,135,115),(11,91,127,116),(12,92,128,117),(13,93,129,109),(14,94,130,110),(15,95,131,111),(16,96,132,112),(17,97,133,113),(18,98,134,114),(19,108,144,120),(20,100,136,121),(21,101,137,122),(22,102,138,123),(23,103,139,124),(24,104,140,125),(25,105,141,126),(26,106,142,118),(27,107,143,119),(28,85,49,64),(29,86,50,65),(30,87,51,66),(31,88,52,67),(32,89,53,68),(33,90,54,69),(34,82,46,70),(35,83,47,71),(36,84,48,72)], [(1,93,43,109),(2,94,44,110),(3,95,45,111),(4,96,37,112),(5,97,38,113),(6,98,39,114),(7,99,40,115),(8,91,41,116),(9,92,42,117),(10,79,135,63),(11,80,127,55),(12,81,128,56),(13,73,129,57),(14,74,130,58),(15,75,131,59),(16,76,132,60),(17,77,133,61),(18,78,134,62),(19,84,144,72),(20,85,136,64),(21,86,137,65),(22,87,138,66),(23,88,139,67),(24,89,140,68),(25,90,141,69),(26,82,142,70),(27,83,143,71),(28,121,49,100),(29,122,50,101),(30,123,51,102),(31,124,52,103),(32,125,53,104),(33,126,54,105),(34,118,46,106),(35,119,47,107),(36,120,48,108)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 9A | ··· | 9F | 12A | ··· | 12H | 12I | ··· | 12P | 18A | ··· | 18R | 18S | ··· | 18AD | 36A | ··· | 36X | 36Y | ··· | 36AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | C18 | D4 | Q8 | C4○D4 | C3×D4 | C3×Q8 | C3×C4○D4 | D4×C9 | Q8×C9 | C9×C4○D4 |
| kernel | C9×C22⋊Q8 | C9×C22⋊C4 | C9×C4⋊C4 | C22×C36 | Q8×C18 | C3×C22⋊Q8 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C6×Q8 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C36 | C2×C18 | C18 | C12 | C2×C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 3 | 1 | 1 | 2 | 4 | 6 | 2 | 2 | 6 | 12 | 18 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 12 |
Matrix representation of C9×C22⋊Q8 ►in GL4(𝔽37) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 |
| 12 | 36 | 0 | 0 |
| 0 | 0 | 36 | 24 |
| 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 4 |
| 0 | 0 | 0 | 31 |
| 36 | 31 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 13 |
| 0 | 0 | 34 | 36 |
G:=sub<GL(4,GF(37))| [16,0,0,0,0,16,0,0,0,0,10,0,0,0,0,10],[1,12,0,0,0,36,0,0,0,0,36,0,0,0,24,1],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,6,0,0,0,4,31],[36,0,0,0,31,1,0,0,0,0,1,34,0,0,13,36] >;
C9×C22⋊Q8 in GAP, Magma, Sage, TeX
C_9\times C_2^2\rtimes Q_8
% in TeX
G:=Group("C9xC2^2:Q8"); // GroupNames label
G:=SmallGroup(288,172);
// by ID
G=gap.SmallGroup(288,172);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,365,176,1094,360]);
// Polycyclic
G:=Group<a,b,c,d,e|a^9=b^2=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations