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## G = C9×C4○D8order 288 = 25·32

### Direct product of C9 and C4○D8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C9×C4○D8
 Chief series C1 — C2 — C6 — C12 — C36 — D4×C9 — C9×D8 — C9×C4○D8
 Lower central C1 — C2 — C4 — C9×C4○D8
 Upper central C1 — C36 — C2×C36 — C9×C4○D8

Generators and relations for C9×C4○D8
G = < a,b,c,d | a9=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >

Subgroups: 138 in 93 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C6, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], D4 [×2], Q8 [×2], C9, C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C18, C18 [×3], C24 [×2], C2×C12, C2×C12 [×2], C3×D4 [×2], C3×D4 [×2], C3×Q8 [×2], C4○D8, C36 [×2], C36 [×2], C2×C18, C2×C18 [×2], C2×C24, C3×D8, C3×SD16 [×2], C3×Q16, C3×C4○D4 [×2], C72 [×2], C2×C36, C2×C36 [×2], D4×C9 [×2], D4×C9 [×2], Q8×C9 [×2], C3×C4○D8, C2×C72, C9×D8, C9×SD16 [×2], C9×Q16, C9×C4○D4 [×2], C9×C4○D8
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, C4○D8, C2×C18 [×7], C6×D4, D4×C9 [×2], C22×C18, C3×C4○D8, D4×C18, C9×C4○D8

Smallest permutation representation of C9×C4○D8
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 78 42 89)(2 79 43 90)(3 80 44 82)(4 81 45 83)(5 73 37 84)(6 74 38 85)(7 75 39 86)(8 76 40 87)(9 77 41 88)(10 109 27 120)(11 110 19 121)(12 111 20 122)(13 112 21 123)(14 113 22 124)(15 114 23 125)(16 115 24 126)(17 116 25 118)(18 117 26 119)(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)(46 98 62 69)(47 99 63 70)(48 91 55 71)(49 92 56 72)(50 93 57 64)(51 94 58 65)(52 95 59 66)(53 96 60 67)(54 97 61 68)
(1 131 52 114 42 102 59 125)(2 132 53 115 43 103 60 126)(3 133 54 116 44 104 61 118)(4 134 46 117 45 105 62 119)(5 135 47 109 37 106 63 120)(6 127 48 110 38 107 55 121)(7 128 49 111 39 108 56 122)(8 129 50 112 40 100 57 123)(9 130 51 113 41 101 58 124)(10 73 142 99 27 84 32 70)(11 74 143 91 19 85 33 71)(12 75 144 92 20 86 34 72)(13 76 136 93 21 87 35 64)(14 77 137 94 22 88 36 65)(15 78 138 95 23 89 28 66)(16 79 139 96 24 90 29 67)(17 80 140 97 25 82 30 68)(18 81 141 98 26 83 31 69)
(1 125)(2 126)(3 118)(4 119)(5 120)(6 121)(7 122)(8 123)(9 124)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 82)(26 83)(27 84)(28 95)(29 96)(30 97)(31 98)(32 99)(33 91)(34 92)(35 93)(36 94)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 105)(47 106)(48 107)(49 108)(50 100)(51 101)(52 102)(53 103)(54 104)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)

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

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

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

126 conjugacy classes

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

126 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 2 2 2 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 C4○D8 D4×C9 D4×C9 C3×C4○D8 C9×C4○D8 kernel C9×C4○D8 C2×C72 C9×D8 C9×SD16 C9×Q16 C9×C4○D4 C3×C4○D8 C2×C24 C3×D8 C3×SD16 C3×Q16 C3×C4○D4 C4○D8 C2×C8 D8 SD16 Q16 C4○D4 C36 C2×C18 C12 C2×C6 C9 C4 C22 C3 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 6 6 6 12 6 12 1 1 2 2 4 6 6 8 24

Matrix representation of C9×C4○D8 in GL2(𝔽73) generated by

 37 0 0 37
,
 27 0 0 27
,
 57 16 57 57
,
 57 16 16 16
G:=sub<GL(2,GF(73))| [37,0,0,37],[27,0,0,27],[57,57,16,57],[57,16,16,16] >;

C9×C4○D8 in GAP, Magma, Sage, TeX

C_9\times C_4\circ D_8
% in TeX

G:=Group("C9xC4oD8");
// GroupNames label

G:=SmallGroup(288,185);
// by ID

G=gap.SmallGroup(288,185);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,365,772,192,5884,2951,242]);
// Polycyclic

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

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