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## G = C9×C8.C22order 288 = 25·32

### Direct product of C9 and C8.C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C9×C8.C22
 Chief series C1 — C2 — C6 — C12 — C36 — D4×C9 — C9×SD16 — C9×C8.C22
 Lower central C1 — C2 — C4 — C9×C8.C22
 Upper central C1 — C18 — C2×C36 — C9×C8.C22

Generators and relations for C9×C8.C22
G = < a,b,c,d | a9=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >

Subgroups: 126 in 90 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×2], Q8, C9, C12 [×2], C12 [×3], C2×C6, C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C18, C18 [×2], C24 [×2], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C3×Q8 [×2], C3×Q8, C8.C22, C36 [×2], C36 [×3], C2×C18, C2×C18, C3×M4(2), C3×SD16 [×2], C3×Q16 [×2], C6×Q8, C3×C4○D4, C72 [×2], C2×C36, C2×C36 [×2], D4×C9, D4×C9, Q8×C9, Q8×C9 [×2], Q8×C9, C3×C8.C22, C9×M4(2), C9×SD16 [×2], C9×Q16 [×2], Q8×C18, C9×C4○D4, C9×C8.C22
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C9, C2×C6 [×7], C2×D4, C18 [×7], C3×D4 [×2], C22×C6, C8.C22, C2×C18 [×7], C6×D4, D4×C9 [×2], C22×C18, C3×C8.C22, D4×C18, C9×C8.C22

Smallest permutation representation of C9×C8.C22
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)
(1 118 59 131 46 112 43 102)(2 119 60 132 47 113 44 103)(3 120 61 133 48 114 45 104)(4 121 62 134 49 115 37 105)(5 122 63 135 50 116 38 106)(6 123 55 127 51 117 39 107)(7 124 56 128 52 109 40 108)(8 125 57 129 53 110 41 100)(9 126 58 130 54 111 42 101)(10 65 36 75 20 94 137 90)(11 66 28 76 21 95 138 82)(12 67 29 77 22 96 139 83)(13 68 30 78 23 97 140 84)(14 69 31 79 24 98 141 85)(15 70 32 80 25 99 142 86)(16 71 33 81 26 91 143 87)(17 72 34 73 27 92 144 88)(18 64 35 74 19 93 136 89)
(10 137)(11 138)(12 139)(13 140)(14 141)(15 142)(16 143)(17 144)(18 136)(19 35)(20 36)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(37 62)(38 63)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(73 88)(74 89)(75 90)(76 82)(77 83)(78 84)(79 85)(80 86)(81 87)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 109)(118 131)(119 132)(120 133)(121 134)(122 135)(123 127)(124 128)(125 129)(126 130)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 73)(8 74)(9 75)(10 126)(11 118)(12 119)(13 120)(14 121)(15 122)(16 123)(17 124)(18 125)(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 69)(38 70)(39 71)(40 72)(41 64)(42 65)(43 66)(44 67)(45 68)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(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), (1,118,59,131,46,112,43,102)(2,119,60,132,47,113,44,103)(3,120,61,133,48,114,45,104)(4,121,62,134,49,115,37,105)(5,122,63,135,50,116,38,106)(6,123,55,127,51,117,39,107)(7,124,56,128,52,109,40,108)(8,125,57,129,53,110,41,100)(9,126,58,130,54,111,42,101)(10,65,36,75,20,94,137,90)(11,66,28,76,21,95,138,82)(12,67,29,77,22,96,139,83)(13,68,30,78,23,97,140,84)(14,69,31,79,24,98,141,85)(15,70,32,80,25,99,142,86)(16,71,33,81,26,91,143,87)(17,72,34,73,27,92,144,88)(18,64,35,74,19,93,136,89), (10,137)(11,138)(12,139)(13,140)(14,141)(15,142)(16,143)(17,144)(18,136)(19,35)(20,36)(21,28)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(37,62)(38,63)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(73,88)(74,89)(75,90)(76,82)(77,83)(78,84)(79,85)(80,86)(81,87)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,109)(118,131)(119,132)(120,133)(121,134)(122,135)(123,127)(124,128)(125,129)(126,130), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,73)(8,74)(9,75)(10,126)(11,118)(12,119)(13,120)(14,121)(15,122)(16,123)(17,124)(18,125)(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)(37,69)(38,70)(39,71)(40,72)(41,64)(42,65)(43,66)(44,67)(45,68)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(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), (1,118,59,131,46,112,43,102)(2,119,60,132,47,113,44,103)(3,120,61,133,48,114,45,104)(4,121,62,134,49,115,37,105)(5,122,63,135,50,116,38,106)(6,123,55,127,51,117,39,107)(7,124,56,128,52,109,40,108)(8,125,57,129,53,110,41,100)(9,126,58,130,54,111,42,101)(10,65,36,75,20,94,137,90)(11,66,28,76,21,95,138,82)(12,67,29,77,22,96,139,83)(13,68,30,78,23,97,140,84)(14,69,31,79,24,98,141,85)(15,70,32,80,25,99,142,86)(16,71,33,81,26,91,143,87)(17,72,34,73,27,92,144,88)(18,64,35,74,19,93,136,89), (10,137)(11,138)(12,139)(13,140)(14,141)(15,142)(16,143)(17,144)(18,136)(19,35)(20,36)(21,28)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(37,62)(38,63)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(73,88)(74,89)(75,90)(76,82)(77,83)(78,84)(79,85)(80,86)(81,87)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,109)(118,131)(119,132)(120,133)(121,134)(122,135)(123,127)(124,128)(125,129)(126,130), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,73)(8,74)(9,75)(10,126)(11,118)(12,119)(13,120)(14,121)(15,122)(16,123)(17,124)(18,125)(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)(37,69)(38,70)(39,71)(40,72)(41,64)(42,65)(43,66)(44,67)(45,68)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(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)], [(1,118,59,131,46,112,43,102),(2,119,60,132,47,113,44,103),(3,120,61,133,48,114,45,104),(4,121,62,134,49,115,37,105),(5,122,63,135,50,116,38,106),(6,123,55,127,51,117,39,107),(7,124,56,128,52,109,40,108),(8,125,57,129,53,110,41,100),(9,126,58,130,54,111,42,101),(10,65,36,75,20,94,137,90),(11,66,28,76,21,95,138,82),(12,67,29,77,22,96,139,83),(13,68,30,78,23,97,140,84),(14,69,31,79,24,98,141,85),(15,70,32,80,25,99,142,86),(16,71,33,81,26,91,143,87),(17,72,34,73,27,92,144,88),(18,64,35,74,19,93,136,89)], [(10,137),(11,138),(12,139),(13,140),(14,141),(15,142),(16,143),(17,144),(18,136),(19,35),(20,36),(21,28),(22,29),(23,30),(24,31),(25,32),(26,33),(27,34),(37,62),(38,63),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,61),(73,88),(74,89),(75,90),(76,82),(77,83),(78,84),(79,85),(80,86),(81,87),(100,110),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,109),(118,131),(119,132),(120,133),(121,134),(122,135),(123,127),(124,128),(125,129),(126,130)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,81),(7,73),(8,74),(9,75),(10,126),(11,118),(12,119),(13,120),(14,121),(15,122),(16,123),(17,124),(18,125),(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),(37,69),(38,70),(39,71),(40,72),(41,64),(42,65),(43,66),(44,67),(45,68),(46,82),(47,83),(48,84),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(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)])

99 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 8A 8B 9A ··· 9F 12A 12B 12C 12D 12E ··· 12J 18A ··· 18F 18G ··· 18L 18M ··· 18R 24A 24B 24C 24D 36A ··· 36L 36M ··· 36AD 72A ··· 72L order 1 2 2 2 3 3 4 4 4 4 4 6 6 6 6 6 6 8 8 9 ··· 9 12 12 12 12 12 ··· 12 18 ··· 18 18 ··· 18 18 ··· 18 24 24 24 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 2 4 1 1 2 2 4 4 4 1 1 2 2 4 4 4 4 1 ··· 1 2 2 2 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C9 C18 C18 C18 C18 C18 D4 D4 C3×D4 C3×D4 D4×C9 D4×C9 C8.C22 C3×C8.C22 C9×C8.C22 kernel C9×C8.C22 C9×M4(2) C9×SD16 C9×Q16 Q8×C18 C9×C4○D4 C3×C8.C22 C3×M4(2) C3×SD16 C3×Q16 C6×Q8 C3×C4○D4 C8.C22 M4(2) SD16 Q16 C2×Q8 C4○D4 C36 C2×C18 C12 C2×C6 C4 C22 C9 C3 C1 # reps 1 1 2 2 1 1 2 2 4 4 2 2 6 6 12 12 6 6 1 1 2 2 6 6 1 2 6

Matrix representation of C9×C8.C22 in GL4(𝔽73) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 18 11 62 18 14 43 17 62 19 19 30 62 54 19 59 55
,
 1 0 10 0 0 72 0 63 0 0 72 0 0 0 0 1
,
 10 64 17 56 64 10 56 17 71 0 63 9 0 71 9 63
G:=sub<GL(4,GF(73))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[18,14,19,54,11,43,19,19,62,17,30,59,18,62,62,55],[1,0,0,0,0,72,0,0,10,0,72,0,0,63,0,1],[10,64,71,0,64,10,0,71,17,56,63,9,56,17,9,63] >;

C9×C8.C22 in GAP, Magma, Sage, TeX

C_9\times C_8.C_2^2
% in TeX

G:=Group("C9xC8.C2^2");
// GroupNames label

G:=SmallGroup(288,187);
// by ID

G=gap.SmallGroup(288,187);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,365,1016,3110,192,5884,2951,242]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations

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