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## G = C9×C8○D4order 288 = 25·32

### Direct product of C9 and C8○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C9×C8○D4
 Chief series C1 — C2 — C6 — C12 — C36 — C72 — C2×C72 — C9×C8○D4
 Lower central C1 — C2 — C9×C8○D4
 Upper central C1 — C72 — C9×C8○D4

Generators and relations for C9×C8○D4
G = < a,b,c,d | a9=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 102 in 93 conjugacy classes, 84 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, D4, Q8, C9, C12, C12, C2×C6, C2×C8, M4(2), C4○D4, C18, C18, C24, C24, C2×C12, C3×D4, C3×Q8, C8○D4, C36, C36, C2×C18, C2×C24, C3×M4(2), C3×C4○D4, C72, C72, C2×C36, D4×C9, Q8×C9, C3×C8○D4, C2×C72, C9×M4(2), C9×C4○D4, C9×C8○D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C9, C12, C2×C6, C22×C4, C18, C2×C12, C22×C6, C8○D4, C36, C2×C18, C22×C12, C2×C36, C22×C18, C3×C8○D4, C22×C36, C9×C8○D4

Smallest permutation representation of C9×C8○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)
(1 131 59 114 51 102 37 123)(2 132 60 115 52 103 38 124)(3 133 61 116 53 104 39 125)(4 134 62 117 54 105 40 126)(5 135 63 109 46 106 41 118)(6 127 55 110 47 107 42 119)(7 128 56 111 48 108 43 120)(8 129 57 112 49 100 44 121)(9 130 58 113 50 101 45 122)(10 86 36 94 22 77 137 65)(11 87 28 95 23 78 138 66)(12 88 29 96 24 79 139 67)(13 89 30 97 25 80 140 68)(14 90 31 98 26 81 141 69)(15 82 32 99 27 73 142 70)(16 83 33 91 19 74 143 71)(17 84 34 92 20 75 144 72)(18 85 35 93 21 76 136 64)
(1 78 51 87)(2 79 52 88)(3 80 53 89)(4 81 54 90)(5 73 46 82)(6 74 47 83)(7 75 48 84)(8 76 49 85)(9 77 50 86)(10 122 22 113)(11 123 23 114)(12 124 24 115)(13 125 25 116)(14 126 26 117)(15 118 27 109)(16 119 19 110)(17 120 20 111)(18 121 21 112)(28 131 138 102)(29 132 139 103)(30 133 140 104)(31 134 141 105)(32 135 142 106)(33 127 143 107)(34 128 144 108)(35 129 136 100)(36 130 137 101)(37 95 59 66)(38 96 60 67)(39 97 61 68)(40 98 62 69)(41 99 63 70)(42 91 55 71)(43 92 56 72)(44 93 57 64)(45 94 58 65)
(1 87)(2 88)(3 89)(4 90)(5 82)(6 83)(7 84)(8 85)(9 86)(10 122)(11 123)(12 124)(13 125)(14 126)(15 118)(16 119)(17 120)(18 121)(19 110)(20 111)(21 112)(22 113)(23 114)(24 115)(25 116)(26 117)(27 109)(28 131)(29 132)(30 133)(31 134)(32 135)(33 127)(34 128)(35 129)(36 130)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 64)(45 65)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C ··· 6H 8A 8B 8C 8D 8E ··· 8J 9A ··· 9F 12A 12B 12C 12D 12E ··· 12J 18A ··· 18F 18G ··· 18X 24A ··· 24H 24I ··· 24T 36A ··· 36L 36M ··· 36AD 72A ··· 72X 72Y ··· 72BH order 1 2 2 2 2 3 3 4 4 4 4 4 6 6 6 ··· 6 8 8 8 8 8 ··· 8 9 ··· 9 12 12 12 12 12 ··· 12 18 ··· 18 18 ··· 18 24 ··· 24 24 ··· 24 36 ··· 36 36 ··· 36 72 ··· 72 72 ··· 72 size 1 1 2 2 2 1 1 1 1 2 2 2 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 ··· 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 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 C3 C4 C4 C6 C6 C6 C9 C12 C12 C18 C18 C18 C36 C36 C8○D4 C3×C8○D4 C9×C8○D4 kernel C9×C8○D4 C2×C72 C9×M4(2) C9×C4○D4 C3×C8○D4 D4×C9 Q8×C9 C2×C24 C3×M4(2) C3×C4○D4 C8○D4 C3×D4 C3×Q8 C2×C8 M4(2) C4○D4 D4 Q8 C9 C3 C1 # reps 1 3 3 1 2 6 2 6 6 2 6 12 4 18 18 6 36 12 4 8 24

Matrix representation of C9×C8○D4 in GL2(𝔽73) generated by

 4 0 0 4
,
 10 0 0 10
,
 72 71 1 1
,
 72 71 0 1
G:=sub<GL(2,GF(73))| [4,0,0,4],[10,0,0,10],[72,1,71,1],[72,0,71,1] >;

C9×C8○D4 in GAP, Magma, Sage, TeX

C_9\times C_8\circ D_4
% in TeX

G:=Group("C9xC8oD4");
// GroupNames label

G:=SmallGroup(288,181);
// by ID

G=gap.SmallGroup(288,181);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,168,1563,192,242]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations

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