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## G = C9×C42⋊C2order 288 = 25·32

### Direct product of C9 and C42⋊C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C9×C42⋊C2
 Chief series C1 — C3 — C6 — C2×C6 — C2×C18 — C2×C36 — C9×C22⋊C4 — C9×C42⋊C2
 Lower central C1 — C2 — C9×C42⋊C2
 Upper central C1 — C2×C36 — C9×C42⋊C2

Generators and relations for C9×C42⋊C2
G = < a,b,c,d | a9=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >

Subgroups: 138 in 114 conjugacy classes, 90 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C23, C9, C12 [×4], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C18, C18 [×2], C18 [×2], C2×C12 [×2], C2×C12 [×8], C22×C6, C42⋊C2, C36 [×4], C36 [×4], C2×C18, C2×C18 [×2], C2×C18 [×2], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C22×C12, C2×C36 [×2], C2×C36 [×8], C22×C18, C3×C42⋊C2, C4×C36 [×2], C9×C22⋊C4 [×2], C9×C4⋊C4 [×2], C22×C36, C9×C42⋊C2
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, C9, C12 [×4], C2×C6 [×7], C22×C4, C4○D4 [×2], C18 [×7], C2×C12 [×6], C22×C6, C42⋊C2, C36 [×4], C2×C18 [×7], C22×C12, C3×C4○D4 [×2], C2×C36 [×6], C22×C18, C3×C42⋊C2, C22×C36, C9×C4○D4 [×2], C9×C42⋊C2

Smallest permutation representation of C9×C42⋊C2
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 34 117)(2 107 35 109)(3 108 36 110)(4 100 28 111)(5 101 29 112)(6 102 30 113)(7 103 31 114)(8 104 32 115)(9 105 33 116)(10 79 140 68)(11 80 141 69)(12 81 142 70)(13 73 143 71)(14 74 144 72)(15 75 136 64)(16 76 137 65)(17 77 138 66)(18 78 139 67)(19 83 131 59)(20 84 132 60)(21 85 133 61)(22 86 134 62)(23 87 135 63)(24 88 127 55)(25 89 128 56)(26 90 129 57)(27 82 130 58)(37 122 50 98)(38 123 51 99)(39 124 52 91)(40 125 53 92)(41 126 54 93)(42 118 46 94)(43 119 47 95)(44 120 48 96)(45 121 49 97)
(1 57 41 81)(2 58 42 73)(3 59 43 74)(4 60 44 75)(5 61 45 76)(6 62 37 77)(7 63 38 78)(8 55 39 79)(9 56 40 80)(10 115 127 91)(11 116 128 92)(12 117 129 93)(13 109 130 94)(14 110 131 95)(15 111 132 96)(16 112 133 97)(17 113 134 98)(18 114 135 99)(19 119 144 108)(20 120 136 100)(21 121 137 101)(22 122 138 102)(23 123 139 103)(24 124 140 104)(25 125 141 105)(26 126 142 106)(27 118 143 107)(28 84 48 64)(29 85 49 65)(30 86 50 66)(31 87 51 67)(32 88 52 68)(33 89 53 69)(34 90 54 70)(35 82 46 71)(36 83 47 72)
(10 127)(11 128)(12 129)(13 130)(14 131)(15 132)(16 133)(17 134)(18 135)(19 144)(20 136)(21 137)(22 138)(23 139)(24 140)(25 141)(26 142)(27 143)(91 115)(92 116)(93 117)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 118)(108 119)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E ··· 4N 6A ··· 6F 6G 6H 6I 6J 9A ··· 9F 12A ··· 12H 12I ··· 12AB 18A ··· 18R 18S ··· 18AD 36A ··· 36X 36Y ··· 36CF order 1 2 2 2 2 2 3 3 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 1 1 1 1 2 ··· 2 1 ··· 1 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C9 C12 C18 C18 C18 C18 C36 C4○D4 C3×C4○D4 C9×C4○D4 kernel C9×C42⋊C2 C4×C36 C9×C22⋊C4 C9×C4⋊C4 C22×C36 C3×C42⋊C2 C2×C36 C4×C12 C3×C22⋊C4 C3×C4⋊C4 C22×C12 C42⋊C2 C2×C12 C42 C22⋊C4 C4⋊C4 C22×C4 C2×C4 C18 C6 C2 # reps 1 2 2 2 1 2 8 4 4 4 2 6 16 12 12 12 6 48 4 8 24

Matrix representation of C9×C42⋊C2 in GL4(𝔽37) generated by

 33 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 36 0 0 0 0 6 0 0 0 0 2 1 0 0 32 35
,
 1 0 0 0 0 1 0 0 0 0 31 0 0 0 0 31
,
 36 0 0 0 0 36 0 0 0 0 1 0 0 0 33 36
G:=sub<GL(4,GF(37))| [33,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,6,0,0,0,0,2,32,0,0,1,35],[1,0,0,0,0,1,0,0,0,0,31,0,0,0,0,31],[36,0,0,0,0,36,0,0,0,0,1,33,0,0,0,36] >;

C9×C42⋊C2 in GAP, Magma, Sage, TeX

C_9\times C_4^2\rtimes C_2
% in TeX

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

G:=SmallGroup(288,167);
// by ID

G=gap.SmallGroup(288,167);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,365,142,360]);
// 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,c*d=d*c>;
// generators/relations

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