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## G = C9×C22.D4order 288 = 25·32

### Direct product of C9 and C22.D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C9×C22.D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C18 — C22×C18 — D4×C18 — C9×C22.D4
 Lower central C1 — C22 — C9×C22.D4
 Upper central C1 — C2×C18 — C9×C22.D4

Generators and relations for C9×C22.D4
G = < a,b,c,d,e | a9=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >

Subgroups: 174 in 117 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], C6, C6 [×2], C6 [×3], C2×C4, C2×C4 [×4], C2×C4 [×2], D4 [×2], C23 [×2], C9, C12 [×5], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, C18, C18 [×2], C18 [×3], C2×C12, C2×C12 [×4], C2×C12 [×2], C3×D4 [×2], C22×C6 [×2], C22.D4, C36 [×5], C2×C18, C2×C18 [×2], C2×C18 [×5], C3×C22⋊C4, C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C22×C12, C6×D4, C2×C36, C2×C36 [×4], C2×C36 [×2], D4×C9 [×2], C22×C18 [×2], C3×C22.D4, C9×C22⋊C4, C9×C22⋊C4 [×2], C9×C4⋊C4 [×2], C22×C36, D4×C18, C9×C22.D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C9, C2×C6 [×7], C2×D4, C4○D4 [×2], C18 [×7], C3×D4 [×2], C22×C6, C22.D4, C2×C18 [×7], C6×D4, C3×C4○D4 [×2], D4×C9 [×2], C22×C18, C3×C22.D4, D4×C18, C9×C4○D4 [×2], C9×C22.D4

Smallest permutation representation of C9×C22.D4
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)
(1 106)(2 107)(3 108)(4 100)(5 101)(6 102)(7 103)(8 104)(9 105)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 64)(17 65)(18 66)(19 59)(20 60)(21 61)(22 62)(23 63)(24 55)(25 56)(26 57)(27 58)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 109)(35 110)(36 111)(37 125)(38 126)(39 118)(40 119)(41 120)(42 121)(43 122)(44 123)(45 124)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)(53 91)(54 92)(73 142)(74 143)(75 144)(76 136)(77 137)(78 138)(79 139)(80 140)(81 141)(82 129)(83 130)(84 131)(85 132)(86 133)(87 134)(88 135)(89 127)(90 128)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 37)(9 38)(10 135)(11 127)(12 128)(13 129)(14 130)(15 131)(16 132)(17 133)(18 134)(19 144)(20 136)(21 137)(22 138)(23 139)(24 140)(25 141)(26 142)(27 143)(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)(91 116)(92 117)(93 109)(94 110)(95 111)(96 112)(97 113)(98 114)(99 115)(100 121)(101 122)(102 123)(103 124)(104 125)(105 126)(106 118)(107 119)(108 120)
(1 57 46 70)(2 58 47 71)(3 59 48 72)(4 60 49 64)(5 61 50 65)(6 62 51 66)(7 63 52 67)(8 55 53 68)(9 56 54 69)(10 124 23 115)(11 125 24 116)(12 126 25 117)(13 118 26 109)(14 119 27 110)(15 120 19 111)(16 121 20 112)(17 122 21 113)(18 123 22 114)(28 85 42 76)(29 86 43 77)(30 87 44 78)(31 88 45 79)(32 89 37 80)(33 90 38 81)(34 82 39 73)(35 83 40 74)(36 84 41 75)(91 127 104 140)(92 128 105 141)(93 129 106 142)(94 130 107 143)(95 131 108 144)(96 132 100 136)(97 133 101 137)(98 134 102 138)(99 135 103 139)
(10 23)(11 24)(12 25)(13 26)(14 27)(15 19)(16 20)(17 21)(18 22)(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)(91 116)(92 117)(93 109)(94 110)(95 111)(96 112)(97 113)(98 114)(99 115)(100 121)(101 122)(102 123)(103 124)(104 125)(105 126)(106 118)(107 119)(108 120)(127 140)(128 141)(129 142)(130 143)(131 144)(132 136)(133 137)(134 138)(135 139)

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), (1,106)(2,107)(3,108)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,64)(17,65)(18,66)(19,59)(20,60)(21,61)(22,62)(23,63)(24,55)(25,56)(26,57)(27,58)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,109)(35,110)(36,111)(37,125)(38,126)(39,118)(40,119)(41,120)(42,121)(43,122)(44,123)(45,124)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,91)(54,92)(73,142)(74,143)(75,144)(76,136)(77,137)(78,138)(79,139)(80,140)(81,141)(82,129)(83,130)(84,131)(85,132)(86,133)(87,134)(88,135)(89,127)(90,128), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,37)(9,38)(10,135)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,144)(20,136)(21,137)(22,138)(23,139)(24,140)(25,141)(26,142)(27,143)(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)(91,116)(92,117)(93,109)(94,110)(95,111)(96,112)(97,113)(98,114)(99,115)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,118)(107,119)(108,120), (1,57,46,70)(2,58,47,71)(3,59,48,72)(4,60,49,64)(5,61,50,65)(6,62,51,66)(7,63,52,67)(8,55,53,68)(9,56,54,69)(10,124,23,115)(11,125,24,116)(12,126,25,117)(13,118,26,109)(14,119,27,110)(15,120,19,111)(16,121,20,112)(17,122,21,113)(18,123,22,114)(28,85,42,76)(29,86,43,77)(30,87,44,78)(31,88,45,79)(32,89,37,80)(33,90,38,81)(34,82,39,73)(35,83,40,74)(36,84,41,75)(91,127,104,140)(92,128,105,141)(93,129,106,142)(94,130,107,143)(95,131,108,144)(96,132,100,136)(97,133,101,137)(98,134,102,138)(99,135,103,139), (10,23)(11,24)(12,25)(13,26)(14,27)(15,19)(16,20)(17,21)(18,22)(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)(91,116)(92,117)(93,109)(94,110)(95,111)(96,112)(97,113)(98,114)(99,115)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,118)(107,119)(108,120)(127,140)(128,141)(129,142)(130,143)(131,144)(132,136)(133,137)(134,138)(135,139)>;

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), (1,106)(2,107)(3,108)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,64)(17,65)(18,66)(19,59)(20,60)(21,61)(22,62)(23,63)(24,55)(25,56)(26,57)(27,58)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,109)(35,110)(36,111)(37,125)(38,126)(39,118)(40,119)(41,120)(42,121)(43,122)(44,123)(45,124)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,91)(54,92)(73,142)(74,143)(75,144)(76,136)(77,137)(78,138)(79,139)(80,140)(81,141)(82,129)(83,130)(84,131)(85,132)(86,133)(87,134)(88,135)(89,127)(90,128), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,37)(9,38)(10,135)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,144)(20,136)(21,137)(22,138)(23,139)(24,140)(25,141)(26,142)(27,143)(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)(91,116)(92,117)(93,109)(94,110)(95,111)(96,112)(97,113)(98,114)(99,115)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,118)(107,119)(108,120), (1,57,46,70)(2,58,47,71)(3,59,48,72)(4,60,49,64)(5,61,50,65)(6,62,51,66)(7,63,52,67)(8,55,53,68)(9,56,54,69)(10,124,23,115)(11,125,24,116)(12,126,25,117)(13,118,26,109)(14,119,27,110)(15,120,19,111)(16,121,20,112)(17,122,21,113)(18,123,22,114)(28,85,42,76)(29,86,43,77)(30,87,44,78)(31,88,45,79)(32,89,37,80)(33,90,38,81)(34,82,39,73)(35,83,40,74)(36,84,41,75)(91,127,104,140)(92,128,105,141)(93,129,106,142)(94,130,107,143)(95,131,108,144)(96,132,100,136)(97,133,101,137)(98,134,102,138)(99,135,103,139), (10,23)(11,24)(12,25)(13,26)(14,27)(15,19)(16,20)(17,21)(18,22)(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)(91,116)(92,117)(93,109)(94,110)(95,111)(96,112)(97,113)(98,114)(99,115)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,118)(107,119)(108,120)(127,140)(128,141)(129,142)(130,143)(131,144)(132,136)(133,137)(134,138)(135,139) );

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)], [(1,106),(2,107),(3,108),(4,100),(5,101),(6,102),(7,103),(8,104),(9,105),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,64),(17,65),(18,66),(19,59),(20,60),(21,61),(22,62),(23,63),(24,55),(25,56),(26,57),(27,58),(28,112),(29,113),(30,114),(31,115),(32,116),(33,117),(34,109),(35,110),(36,111),(37,125),(38,126),(39,118),(40,119),(41,120),(42,121),(43,122),(44,123),(45,124),(46,93),(47,94),(48,95),(49,96),(50,97),(51,98),(52,99),(53,91),(54,92),(73,142),(74,143),(75,144),(76,136),(77,137),(78,138),(79,139),(80,140),(81,141),(82,129),(83,130),(84,131),(85,132),(86,133),(87,134),(88,135),(89,127),(90,128)], [(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,37),(9,38),(10,135),(11,127),(12,128),(13,129),(14,130),(15,131),(16,132),(17,133),(18,134),(19,144),(20,136),(21,137),(22,138),(23,139),(24,140),(25,141),(26,142),(27,143),(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),(91,116),(92,117),(93,109),(94,110),(95,111),(96,112),(97,113),(98,114),(99,115),(100,121),(101,122),(102,123),(103,124),(104,125),(105,126),(106,118),(107,119),(108,120)], [(1,57,46,70),(2,58,47,71),(3,59,48,72),(4,60,49,64),(5,61,50,65),(6,62,51,66),(7,63,52,67),(8,55,53,68),(9,56,54,69),(10,124,23,115),(11,125,24,116),(12,126,25,117),(13,118,26,109),(14,119,27,110),(15,120,19,111),(16,121,20,112),(17,122,21,113),(18,123,22,114),(28,85,42,76),(29,86,43,77),(30,87,44,78),(31,88,45,79),(32,89,37,80),(33,90,38,81),(34,82,39,73),(35,83,40,74),(36,84,41,75),(91,127,104,140),(92,128,105,141),(93,129,106,142),(94,130,107,143),(95,131,108,144),(96,132,100,136),(97,133,101,137),(98,134,102,138),(99,135,103,139)], [(10,23),(11,24),(12,25),(13,26),(14,27),(15,19),(16,20),(17,21),(18,22),(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),(91,116),(92,117),(93,109),(94,110),(95,111),(96,112),(97,113),(98,114),(99,115),(100,121),(101,122),(102,123),(103,124),(104,125),(105,126),(106,118),(107,119),(108,120),(127,140),(128,141),(129,142),(130,143),(131,144),(132,136),(133,137),(134,138),(135,139)])

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 4A 4B 4C 4D 4E 4F 4G 6A ··· 6F 6G 6H 6I 6J 6K 6L 9A ··· 9F 12A ··· 12H 12I ··· 12N 18A ··· 18R 18S ··· 18AD 18AE ··· 18AJ 36A ··· 36X 36Y ··· 36AP order 1 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 1 1 2 2 4 1 1 2 2 2 2 4 4 4 1 ··· 1 2 2 2 2 4 4 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 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 type + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 C9 C18 C18 C18 C18 D4 C4○D4 C3×D4 C3×C4○D4 D4×C9 C9×C4○D4 kernel C9×C22.D4 C9×C22⋊C4 C9×C4⋊C4 C22×C36 D4×C18 C3×C22.D4 C3×C22⋊C4 C3×C4⋊C4 C22×C12 C6×D4 C22.D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C2×C18 C18 C2×C6 C6 C22 C2 # reps 1 3 2 1 1 2 6 4 2 2 6 18 12 6 6 2 4 4 8 12 24

Matrix representation of C9×C22.D4 in GL4(𝔽37) generated by

 26 0 0 0 0 26 0 0 0 0 9 0 0 0 0 9
,
 36 0 0 0 0 36 0 0 0 0 6 22 0 0 27 31
,
 1 0 0 0 0 1 0 0 0 0 36 0 0 0 0 36
,
 31 0 0 0 12 6 0 0 0 0 36 21 0 0 0 1
,
 1 1 0 0 0 36 0 0 0 0 1 0 0 0 23 36
G:=sub<GL(4,GF(37))| [26,0,0,0,0,26,0,0,0,0,9,0,0,0,0,9],[36,0,0,0,0,36,0,0,0,0,6,27,0,0,22,31],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[31,12,0,0,0,6,0,0,0,0,36,0,0,0,21,1],[1,0,0,0,1,36,0,0,0,0,1,23,0,0,0,36] >;

C9×C22.D4 in GAP, Magma, Sage, TeX

C_9\times C_2^2.D_4
% in TeX

G:=Group("C9xC2^2.D4");
// GroupNames label

G:=SmallGroup(288,173);
// by ID

G=gap.SmallGroup(288,173);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,365,1094,142,360]);
// Polycyclic

G:=Group<a,b,c,d,e|a^9=b^2=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;
// generators/relations

׿
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