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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

Aliases: C9×C8○D4, D4.C36, Q8.2C36, M4(2)⋊5C18, C72.30C22, C36.55C23, (C2×C8)⋊7C18, (C2×C72)⋊15C2, C8.7(C2×C18), C4.5(C2×C36), (D4×C9).2C4, (Q8×C9).2C4, C36.34(C2×C4), C24.39(C2×C6), (C2×C24).31C6, C4○D4.5C18, (C3×D4).5C12, (C3×Q8).9C12, C12.35(C2×C12), C2.7(C22×C36), C22.1(C2×C36), (C9×M4(2))⋊11C2, C4.12(C22×C18), C12.66(C22×C6), C18.35(C22×C4), C6.35(C22×C12), (C2×C36).126C22, (C3×M4(2)).12C6, C3.(C3×C8○D4), (C2×C18).8(C2×C4), (C9×C4○D4).6C2, (C3×C8○D4).2C3, (C2×C4).25(C2×C18), (C2×C6).10(C2×C12), (C3×C4○D4).20C6, (C2×C12).143(C2×C6), SmallGroup(288,181)

Series: Derived Chief Lower central Upper central

C1C2 — C9×C8○D4
C1C2C6C12C36C72C2×C72 — C9×C8○D4
C1C2 — C9×C8○D4
C1C72 — 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 [×3], C3, C4, C4 [×3], C22 [×3], C6, C6 [×3], C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, C9, C12, C12 [×3], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C18, C18 [×3], C24, C24 [×3], C2×C12 [×3], C3×D4 [×3], C3×Q8, C8○D4, C36, C36 [×3], C2×C18 [×3], C2×C24 [×3], C3×M4(2) [×3], C3×C4○D4, C72, C72 [×3], C2×C36 [×3], D4×C9 [×3], Q8×C9, C3×C8○D4, C2×C72 [×3], C9×M4(2) [×3], C9×C4○D4, C9×C8○D4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, C9, C12 [×4], C2×C6 [×7], C22×C4, C18 [×7], C2×C12 [×6], C22×C6, C8○D4, C36 [×4], C2×C18 [×7], C22×C12, C2×C36 [×6], 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 42 102 53 125)(2 132 60 115 43 103 54 126)(3 133 61 116 44 104 46 118)(4 134 62 117 45 105 47 119)(5 135 63 109 37 106 48 120)(6 127 55 110 38 107 49 121)(7 128 56 111 39 108 50 122)(8 129 57 112 40 100 51 123)(9 130 58 113 41 101 52 124)(10 86 34 92 22 75 144 72)(11 87 35 93 23 76 136 64)(12 88 36 94 24 77 137 65)(13 89 28 95 25 78 138 66)(14 90 29 96 26 79 139 67)(15 82 30 97 27 80 140 68)(16 83 31 98 19 81 141 69)(17 84 32 99 20 73 142 70)(18 85 33 91 21 74 143 71)
(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)(10 122 22 111)(11 123 23 112)(12 124 24 113)(13 125 25 114)(14 126 26 115)(15 118 27 116)(16 119 19 117)(17 120 20 109)(18 121 21 110)(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 97 61 68)(47 98 62 69)(48 99 63 70)(49 91 55 71)(50 92 56 72)(51 93 57 64)(52 94 58 65)(53 95 59 66)(54 96 60 67)
(1 89)(2 90)(3 82)(4 83)(5 84)(6 85)(7 86)(8 87)(9 88)(10 122)(11 123)(12 124)(13 125)(14 126)(15 118)(16 119)(17 120)(18 121)(19 117)(20 109)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(28 131)(29 132)(30 133)(31 134)(32 135)(33 127)(34 128)(35 129)(36 130)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 68)(47 69)(48 70)(49 71)(50 72)(51 64)(52 65)(53 66)(54 67)(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), (1,131,59,114,42,102,53,125)(2,132,60,115,43,103,54,126)(3,133,61,116,44,104,46,118)(4,134,62,117,45,105,47,119)(5,135,63,109,37,106,48,120)(6,127,55,110,38,107,49,121)(7,128,56,111,39,108,50,122)(8,129,57,112,40,100,51,123)(9,130,58,113,41,101,52,124)(10,86,34,92,22,75,144,72)(11,87,35,93,23,76,136,64)(12,88,36,94,24,77,137,65)(13,89,28,95,25,78,138,66)(14,90,29,96,26,79,139,67)(15,82,30,97,27,80,140,68)(16,83,31,98,19,81,141,69)(17,84,32,99,20,73,142,70)(18,85,33,91,21,74,143,71), (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)(10,122,22,111)(11,123,23,112)(12,124,24,113)(13,125,25,114)(14,126,26,115)(15,118,27,116)(16,119,19,117)(17,120,20,109)(18,121,21,110)(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)(46,97,61,68)(47,98,62,69)(48,99,63,70)(49,91,55,71)(50,92,56,72)(51,93,57,64)(52,94,58,65)(53,95,59,66)(54,96,60,67), (1,89)(2,90)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,122)(11,123)(12,124)(13,125)(14,126)(15,118)(16,119)(17,120)(18,121)(19,117)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,131)(29,132)(30,133)(31,134)(32,135)(33,127)(34,128)(35,129)(36,130)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,68)(47,69)(48,70)(49,71)(50,72)(51,64)(52,65)(53,66)(54,67)(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), (1,131,59,114,42,102,53,125)(2,132,60,115,43,103,54,126)(3,133,61,116,44,104,46,118)(4,134,62,117,45,105,47,119)(5,135,63,109,37,106,48,120)(6,127,55,110,38,107,49,121)(7,128,56,111,39,108,50,122)(8,129,57,112,40,100,51,123)(9,130,58,113,41,101,52,124)(10,86,34,92,22,75,144,72)(11,87,35,93,23,76,136,64)(12,88,36,94,24,77,137,65)(13,89,28,95,25,78,138,66)(14,90,29,96,26,79,139,67)(15,82,30,97,27,80,140,68)(16,83,31,98,19,81,141,69)(17,84,32,99,20,73,142,70)(18,85,33,91,21,74,143,71), (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)(10,122,22,111)(11,123,23,112)(12,124,24,113)(13,125,25,114)(14,126,26,115)(15,118,27,116)(16,119,19,117)(17,120,20,109)(18,121,21,110)(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)(46,97,61,68)(47,98,62,69)(48,99,63,70)(49,91,55,71)(50,92,56,72)(51,93,57,64)(52,94,58,65)(53,95,59,66)(54,96,60,67), (1,89)(2,90)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,122)(11,123)(12,124)(13,125)(14,126)(15,118)(16,119)(17,120)(18,121)(19,117)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,131)(29,132)(30,133)(31,134)(32,135)(33,127)(34,128)(35,129)(36,130)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,68)(47,69)(48,70)(49,71)(50,72)(51,64)(52,65)(53,66)(54,67)(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)], [(1,131,59,114,42,102,53,125),(2,132,60,115,43,103,54,126),(3,133,61,116,44,104,46,118),(4,134,62,117,45,105,47,119),(5,135,63,109,37,106,48,120),(6,127,55,110,38,107,49,121),(7,128,56,111,39,108,50,122),(8,129,57,112,40,100,51,123),(9,130,58,113,41,101,52,124),(10,86,34,92,22,75,144,72),(11,87,35,93,23,76,136,64),(12,88,36,94,24,77,137,65),(13,89,28,95,25,78,138,66),(14,90,29,96,26,79,139,67),(15,82,30,97,27,80,140,68),(16,83,31,98,19,81,141,69),(17,84,32,99,20,73,142,70),(18,85,33,91,21,74,143,71)], [(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),(10,122,22,111),(11,123,23,112),(12,124,24,113),(13,125,25,114),(14,126,26,115),(15,118,27,116),(16,119,19,117),(17,120,20,109),(18,121,21,110),(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),(46,97,61,68),(47,98,62,69),(48,99,63,70),(49,91,55,71),(50,92,56,72),(51,93,57,64),(52,94,58,65),(53,95,59,66),(54,96,60,67)], [(1,89),(2,90),(3,82),(4,83),(5,84),(6,85),(7,86),(8,87),(9,88),(10,122),(11,123),(12,124),(13,125),(14,126),(15,118),(16,119),(17,120),(18,121),(19,117),(20,109),(21,110),(22,111),(23,112),(24,113),(25,114),(26,115),(27,116),(28,131),(29,132),(30,133),(31,134),(32,135),(33,127),(34,128),(35,129),(36,130),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,79),(44,80),(45,81),(46,68),(47,69),(48,70),(49,71),(50,72),(51,64),(52,65),(53,66),(54,67),(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 2A2B2C2D3A3B4A4B4C4D4E6A6B6C···6H8A8B8C8D8E···8J9A···9F12A12B12C12D12E···12J18A···18F18G···18X24A···24H24I···24T36A···36L36M···36AD72A···72X72Y···72BH
order122223344444666···688888···89···91212121212···1218···1818···1824···2424···2436···3636···3672···7272···72
size112221111222112···211112···21···111112···21···12···21···12···21···12···21···12···2

180 irreducible representations

dim111111111111111111222
type++++
imageC1C2C2C2C3C4C4C6C6C6C9C12C12C18C18C18C36C36C8○D4C3×C8○D4C9×C8○D4
kernelC9×C8○D4C2×C72C9×M4(2)C9×C4○D4C3×C8○D4D4×C9Q8×C9C2×C24C3×M4(2)C3×C4○D4C8○D4C3×D4C3×Q8C2×C8M4(2)C4○D4D4Q8C9C3C1
# reps133126266261241818636124824

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

40
04
,
100
010
,
7271
11
,
7271
01
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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