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## G = C9×C42⋊2C2order 288 = 25·32

### Direct product of C9 and C42⋊2C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C9×C42⋊2C2
 Chief series C1 — C3 — C6 — C2×C6 — C2×C18 — C22×C18 — C9×C22⋊C4 — C9×C42⋊2C2
 Lower central C1 — C22 — C9×C42⋊2C2
 Upper central C1 — C2×C18 — C9×C42⋊2C2

Generators and relations for C9×C422C2
G = < a,b,c,d | a9=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >

Subgroups: 126 in 90 conjugacy classes, 60 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C23, C9, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C18, C18, C2×C12, C22×C6, C422C2, C36, C2×C18, C2×C18, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C36, C22×C18, C3×C422C2, C4×C36, C9×C22⋊C4, C9×C4⋊C4, C9×C422C2
Quotients: C1, C2, C3, C22, C6, C23, C9, C2×C6, C4○D4, C18, C22×C6, C422C2, C2×C18, C3×C4○D4, C22×C18, C3×C422C2, C9×C4○D4, C9×C422C2

Smallest permutation representation of C9×C422C2
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)
(1 119 47 110)(2 120 48 111)(3 121 49 112)(4 122 50 113)(5 123 51 114)(6 124 52 115)(7 125 53 116)(8 126 54 117)(9 118 46 109)(10 79 22 88)(11 80 23 89)(12 81 24 90)(13 73 25 82)(14 74 26 83)(15 75 27 84)(16 76 19 85)(17 77 20 86)(18 78 21 87)(28 106 44 93)(29 107 45 94)(30 108 37 95)(31 100 38 96)(32 101 39 97)(33 102 40 98)(34 103 41 99)(35 104 42 91)(36 105 43 92)(55 140 68 127)(56 141 69 128)(57 142 70 129)(58 143 71 130)(59 144 72 131)(60 136 64 132)(61 137 65 133)(62 138 66 134)(63 139 67 135)
(1 57 28 74)(2 58 29 75)(3 59 30 76)(4 60 31 77)(5 61 32 78)(6 62 33 79)(7 63 34 80)(8 55 35 81)(9 56 36 73)(10 115 134 98)(11 116 135 99)(12 117 127 91)(13 109 128 92)(14 110 129 93)(15 111 130 94)(16 112 131 95)(17 113 132 96)(18 114 133 97)(19 121 144 108)(20 122 136 100)(21 123 137 101)(22 124 138 102)(23 125 139 103)(24 126 140 104)(25 118 141 105)(26 119 142 106)(27 120 143 107)(37 85 49 72)(38 86 50 64)(39 87 51 65)(40 88 52 66)(41 89 53 67)(42 90 54 68)(43 82 46 69)(44 83 47 70)(45 84 48 71)
(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 19)(17 20)(18 21)(55 90)(56 82)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 89)(64 77)(65 78)(66 79)(67 80)(68 81)(69 73)(70 74)(71 75)(72 76)(91 117)(92 109)(93 110)(94 111)(95 112)(96 113)(97 114)(98 115)(99 116)(100 122)(101 123)(102 124)(103 125)(104 126)(105 118)(106 119)(107 120)(108 121)(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), (1,119,47,110)(2,120,48,111)(3,121,49,112)(4,122,50,113)(5,123,51,114)(6,124,52,115)(7,125,53,116)(8,126,54,117)(9,118,46,109)(10,79,22,88)(11,80,23,89)(12,81,24,90)(13,73,25,82)(14,74,26,83)(15,75,27,84)(16,76,19,85)(17,77,20,86)(18,78,21,87)(28,106,44,93)(29,107,45,94)(30,108,37,95)(31,100,38,96)(32,101,39,97)(33,102,40,98)(34,103,41,99)(35,104,42,91)(36,105,43,92)(55,140,68,127)(56,141,69,128)(57,142,70,129)(58,143,71,130)(59,144,72,131)(60,136,64,132)(61,137,65,133)(62,138,66,134)(63,139,67,135), (1,57,28,74)(2,58,29,75)(3,59,30,76)(4,60,31,77)(5,61,32,78)(6,62,33,79)(7,63,34,80)(8,55,35,81)(9,56,36,73)(10,115,134,98)(11,116,135,99)(12,117,127,91)(13,109,128,92)(14,110,129,93)(15,111,130,94)(16,112,131,95)(17,113,132,96)(18,114,133,97)(19,121,144,108)(20,122,136,100)(21,123,137,101)(22,124,138,102)(23,125,139,103)(24,126,140,104)(25,118,141,105)(26,119,142,106)(27,120,143,107)(37,85,49,72)(38,86,50,64)(39,87,51,65)(40,88,52,66)(41,89,53,67)(42,90,54,68)(43,82,46,69)(44,83,47,70)(45,84,48,71), (10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,19)(17,20)(18,21)(55,90)(56,82)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,89)(64,77)(65,78)(66,79)(67,80)(68,81)(69,73)(70,74)(71,75)(72,76)(91,117)(92,109)(93,110)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,122)(101,123)(102,124)(103,125)(104,126)(105,118)(106,119)(107,120)(108,121)(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), (1,119,47,110)(2,120,48,111)(3,121,49,112)(4,122,50,113)(5,123,51,114)(6,124,52,115)(7,125,53,116)(8,126,54,117)(9,118,46,109)(10,79,22,88)(11,80,23,89)(12,81,24,90)(13,73,25,82)(14,74,26,83)(15,75,27,84)(16,76,19,85)(17,77,20,86)(18,78,21,87)(28,106,44,93)(29,107,45,94)(30,108,37,95)(31,100,38,96)(32,101,39,97)(33,102,40,98)(34,103,41,99)(35,104,42,91)(36,105,43,92)(55,140,68,127)(56,141,69,128)(57,142,70,129)(58,143,71,130)(59,144,72,131)(60,136,64,132)(61,137,65,133)(62,138,66,134)(63,139,67,135), (1,57,28,74)(2,58,29,75)(3,59,30,76)(4,60,31,77)(5,61,32,78)(6,62,33,79)(7,63,34,80)(8,55,35,81)(9,56,36,73)(10,115,134,98)(11,116,135,99)(12,117,127,91)(13,109,128,92)(14,110,129,93)(15,111,130,94)(16,112,131,95)(17,113,132,96)(18,114,133,97)(19,121,144,108)(20,122,136,100)(21,123,137,101)(22,124,138,102)(23,125,139,103)(24,126,140,104)(25,118,141,105)(26,119,142,106)(27,120,143,107)(37,85,49,72)(38,86,50,64)(39,87,51,65)(40,88,52,66)(41,89,53,67)(42,90,54,68)(43,82,46,69)(44,83,47,70)(45,84,48,71), (10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,19)(17,20)(18,21)(55,90)(56,82)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,89)(64,77)(65,78)(66,79)(67,80)(68,81)(69,73)(70,74)(71,75)(72,76)(91,117)(92,109)(93,110)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,122)(101,123)(102,124)(103,125)(104,126)(105,118)(106,119)(107,120)(108,121)(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)], [(1,119,47,110),(2,120,48,111),(3,121,49,112),(4,122,50,113),(5,123,51,114),(6,124,52,115),(7,125,53,116),(8,126,54,117),(9,118,46,109),(10,79,22,88),(11,80,23,89),(12,81,24,90),(13,73,25,82),(14,74,26,83),(15,75,27,84),(16,76,19,85),(17,77,20,86),(18,78,21,87),(28,106,44,93),(29,107,45,94),(30,108,37,95),(31,100,38,96),(32,101,39,97),(33,102,40,98),(34,103,41,99),(35,104,42,91),(36,105,43,92),(55,140,68,127),(56,141,69,128),(57,142,70,129),(58,143,71,130),(59,144,72,131),(60,136,64,132),(61,137,65,133),(62,138,66,134),(63,139,67,135)], [(1,57,28,74),(2,58,29,75),(3,59,30,76),(4,60,31,77),(5,61,32,78),(6,62,33,79),(7,63,34,80),(8,55,35,81),(9,56,36,73),(10,115,134,98),(11,116,135,99),(12,117,127,91),(13,109,128,92),(14,110,129,93),(15,111,130,94),(16,112,131,95),(17,113,132,96),(18,114,133,97),(19,121,144,108),(20,122,136,100),(21,123,137,101),(22,124,138,102),(23,125,139,103),(24,126,140,104),(25,118,141,105),(26,119,142,106),(27,120,143,107),(37,85,49,72),(38,86,50,64),(39,87,51,65),(40,88,52,66),(41,89,53,67),(42,90,54,68),(43,82,46,69),(44,83,47,70),(45,84,48,71)], [(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,19),(17,20),(18,21),(55,90),(56,82),(57,83),(58,84),(59,85),(60,86),(61,87),(62,88),(63,89),(64,77),(65,78),(66,79),(67,80),(68,81),(69,73),(70,74),(71,75),(72,76),(91,117),(92,109),(93,110),(94,111),(95,112),(96,113),(97,114),(98,115),(99,116),(100,122),(101,123),(102,124),(103,125),(104,126),(105,118),(106,119),(107,120),(108,121),(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 3A 3B 4A ··· 4F 4G 4H 4I 6A ··· 6F 6G 6H 9A ··· 9F 12A ··· 12L 12M ··· 12R 18A ··· 18R 18S ··· 18X 36A ··· 36AJ 36AK ··· 36BB order 1 2 2 2 2 3 3 4 ··· 4 4 4 4 6 ··· 6 6 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 1 1 4 1 1 2 ··· 2 4 4 4 1 ··· 1 4 4 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C9 C18 C18 C18 C4○D4 C3×C4○D4 C9×C4○D4 kernel C9×C42⋊2C2 C4×C36 C9×C22⋊C4 C9×C4⋊C4 C3×C42⋊2C2 C4×C12 C3×C22⋊C4 C3×C4⋊C4 C42⋊2C2 C42 C22⋊C4 C4⋊C4 C18 C6 C2 # reps 1 1 3 3 2 2 6 6 6 6 18 18 6 12 36

Matrix representation of C9×C422C2 in GL4(𝔽37) generated by

 1 0 0 0 0 1 0 0 0 0 33 0 0 0 0 33
,
 6 0 0 0 0 6 0 0 0 0 0 1 0 0 36 0
,
 21 35 0 0 35 16 0 0 0 0 31 0 0 0 0 31
,
 1 0 0 0 21 36 0 0 0 0 1 0 0 0 0 36
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,33,0,0,0,0,33],[6,0,0,0,0,6,0,0,0,0,0,36,0,0,1,0],[21,35,0,0,35,16,0,0,0,0,31,0,0,0,0,31],[1,21,0,0,0,36,0,0,0,0,1,0,0,0,0,36] >;

C9×C422C2 in GAP, Magma, Sage, TeX

C_9\times C_4^2\rtimes_2C_2
% in TeX

G:=Group("C9xC4^2:2C2");
// GroupNames label

G:=SmallGroup(288,176);
// by ID

G=gap.SmallGroup(288,176);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,365,512,1094,142,360]);
// Polycyclic

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

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