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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

Aliases: C9×C4○D8, D83C18, Q163C18, C36.69D4, SD163C18, C72.28C22, C36.47C23, (C2×C8)⋊4C18, (C9×D8)⋊7C2, (C2×C72)⋊12C2, C4○D43C18, C8.6(C2×C18), (C9×Q16)⋊7C2, (C3×D8).7C6, C4.20(D4×C9), C6.77(C6×D4), (C2×C24).19C6, C24.26(C2×C6), (C9×SD16)⋊7C2, D4.2(C2×C18), C12.87(C3×D4), (C2×C18).11D4, C18.77(C2×D4), C2.14(D4×C18), Q8.5(C2×C18), (C3×Q16).7C6, C22.1(D4×C9), C4.4(C22×C18), (C3×SD16).4C6, C12.47(C22×C6), (D4×C9).12C22, (Q8×C9).13C22, (C2×C36).130C22, C3.(C3×C4○D8), (C3×C4○D8).C3, (C9×C4○D4)⋊6C2, (C2×C6).14(C3×D4), (C2×C4).29(C2×C18), (C3×C4○D4).13C6, (C3×D4).13(C2×C6), (C3×Q8).26(C2×C6), (C2×C12).147(C2×C6), SmallGroup(288,185)

Series: Derived Chief Lower central Upper central

C1C4 — C9×C4○D8
C1C2C6C12C36D4×C9C9×D8 — C9×C4○D8
C1C2C4 — C9×C4○D8
C1C36C2×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 2A2B2C2D3A3B4A4B4C4D4E6A6B6C6D6E6F6G6H8A8B8C8D9A···9F12A12B12C12D12E12F12G12H12I12J18A···18F18G···18L18M···18X24A···24H36A···36L36M···36R36S···36AD72A···72X
order1222233444446666666688889···91212121212121212121218···1818···1818···1824···2436···3636···3636···3672···72
size1124411112441122444422221···111112244441···12···24···42···21···12···24···42···2

126 irreducible representations

dim111111111111111111222222222
type++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6C9C18C18C18C18C18D4D4C3×D4C3×D4C4○D8D4×C9D4×C9C3×C4○D8C9×C4○D8
kernelC9×C4○D8C2×C72C9×D8C9×SD16C9×Q16C9×C4○D4C3×C4○D8C2×C24C3×D8C3×SD16C3×Q16C3×C4○D4C4○D8C2×C8D8SD16Q16C4○D4C36C2×C18C12C2×C6C9C4C22C3C1
# reps111212222424666126121122466824

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

370
037
,
270
027
,
5716
5757
,
5716
1616
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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