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G = C2×C8⋊D9order 288 = 25·32

Direct product of C2 and C8⋊D9

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C8⋊D9, C89D18, C24.89D6, C7211C22, C181M4(2), C36.36C23, (C2×C8)⋊6D9, (C2×C72)⋊9C2, C9⋊C810C22, (C4×D9).3C4, C4.24(C4×D9), C91(C2×M4(2)), C12.74(C4×S3), C36.29(C2×C4), (C2×C24).28S3, D18.5(C2×C4), (C2×C4).99D18, C6.7(C8⋊S3), (C2×C12).411D6, (C2×Dic9).5C4, Dic9.7(C2×C4), (C22×D9).3C4, C4.36(C22×D9), C22.14(C4×D9), C18.13(C22×C4), (C4×D9).14C22, C12.197(C22×S3), (C2×C36).109C22, (C2×C9⋊C8)⋊11C2, C3.(C2×C8⋊S3), C6.52(S3×C2×C4), C2.14(C2×C4×D9), (C2×C4×D9).10C2, (C2×C6).40(C4×S3), (C2×C18).14(C2×C4), SmallGroup(288,111)

Series: Derived Chief Lower central Upper central

C1C18 — C2×C8⋊D9
C1C3C9C18C36C4×D9C2×C4×D9 — C2×C8⋊D9
C9C18 — C2×C8⋊D9
C1C2×C4C2×C8

Generators and relations for C2×C8⋊D9
 G = < a,b,c,d | a2=b8=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 384 in 102 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C9, Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C2×C8, C2×C8, M4(2) [×4], C22×C4, D9 [×2], C18, C18 [×2], C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C2×M4(2), Dic9 [×2], C36 [×2], D18 [×2], D18 [×2], C2×C18, C8⋊S3 [×4], C2×C3⋊C8, C2×C24, S3×C2×C4, C9⋊C8 [×2], C72 [×2], C4×D9 [×4], C2×Dic9, C2×C36, C22×D9, C2×C8⋊S3, C8⋊D9 [×4], C2×C9⋊C8, C2×C72, C2×C4×D9, C2×C8⋊D9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], M4(2) [×2], C22×C4, D9, C4×S3 [×2], C22×S3, C2×M4(2), D18 [×3], C8⋊S3 [×2], S3×C2×C4, C4×D9 [×2], C22×D9, C2×C8⋊S3, C8⋊D9 [×2], C2×C4×D9, C2×C8⋊D9

Smallest permutation representation of C2×C8⋊D9
On 144 points
Generators in S144
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(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)
(1 64 28 46 10 55 19 37)(2 65 29 47 11 56 20 38)(3 66 30 48 12 57 21 39)(4 67 31 49 13 58 22 40)(5 68 32 50 14 59 23 41)(6 69 33 51 15 60 24 42)(7 70 34 52 16 61 25 43)(8 71 35 53 17 62 26 44)(9 72 36 54 18 63 27 45)(73 136 100 118 82 127 91 109)(74 137 101 119 83 128 92 110)(75 138 102 120 84 129 93 111)(76 139 103 121 85 130 94 112)(77 140 104 122 86 131 95 113)(78 141 105 123 87 132 96 114)(79 142 106 124 88 133 97 115)(80 143 107 125 89 134 98 116)(81 144 108 126 90 135 99 117)
(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 9)(2 8)(3 7)(4 6)(10 18)(11 17)(12 16)(13 15)(19 27)(20 26)(21 25)(22 24)(28 36)(29 35)(30 34)(31 33)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(73 81)(74 80)(75 79)(76 78)(82 90)(83 89)(84 88)(85 87)(91 99)(92 98)(93 97)(94 96)(100 108)(101 107)(102 106)(103 105)(109 126)(110 125)(111 124)(112 123)(113 122)(114 121)(115 120)(116 119)(117 118)(127 144)(128 143)(129 142)(130 141)(131 140)(132 139)(133 138)(134 137)(135 136)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F6A6B6C8A8B8C8D8E8F8G8H9A9B9C12A12B12C12D18A···18I24A···24H36A···36L72A···72X
order1222223444444666888888889991212121218···1824···2436···3672···72
size1111181821111181822222221818181822222222···22···22···22···2

84 irreducible representations

dim111111112222222222222
type+++++++++++
imageC1C2C2C2C2C4C4C4S3D6D6M4(2)D9C4×S3C4×S3D18D18C8⋊S3C4×D9C4×D9C8⋊D9
kernelC2×C8⋊D9C8⋊D9C2×C9⋊C8C2×C72C2×C4×D9C4×D9C2×Dic9C22×D9C2×C24C24C2×C12C18C2×C8C12C2×C6C8C2×C4C6C4C22C2
# reps1411142212143226386624

Matrix representation of C2×C8⋊D9 in GL3(𝔽73) generated by

7200
0720
0072
,
7200
0816
05765
,
100
04542
0313
,
7200
04542
07028
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[72,0,0,0,8,57,0,16,65],[1,0,0,0,45,31,0,42,3],[72,0,0,0,45,70,0,42,28] >;

C2×C8⋊D9 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes D_9
% in TeX

G:=Group("C2xC8:D9");
// GroupNames label

G:=SmallGroup(288,111);
// by ID

G=gap.SmallGroup(288,111);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,58,80,6725,292,9414]);
// Polycyclic

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

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