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G = C2×C4.Dic9order 288 = 25·32

Direct product of C2 and C4.Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C2×C4.Dic9
 Chief series C1 — C3 — C9 — C18 — C36 — C9⋊C8 — C2×C9⋊C8 — C2×C4.Dic9
 Lower central C9 — C18 — C2×C4.Dic9
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×C4.Dic9
G = < a,b,c,d | a2=b4=1, c18=b2, d2=c9, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c17 >

Subgroups: 224 in 102 conjugacy classes, 68 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C9, C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C18, C18 [×2], C18 [×2], C3⋊C8 [×4], C2×C12 [×2], C2×C12 [×4], C22×C6, C2×M4(2), C36 [×2], C36 [×2], C2×C18, C2×C18 [×2], C2×C18 [×2], C2×C3⋊C8 [×2], C4.Dic3 [×4], C22×C12, C9⋊C8 [×4], C2×C36 [×2], C2×C36 [×4], C22×C18, C2×C4.Dic3, C2×C9⋊C8 [×2], C4.Dic9 [×4], C22×C36, C2×C4.Dic9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], M4(2) [×2], C22×C4, D9, C2×Dic3 [×6], C22×S3, C2×M4(2), Dic9 [×4], D18 [×3], C4.Dic3 [×2], C22×Dic3, C2×Dic9 [×6], C22×D9, C2×C4.Dic3, C4.Dic9 [×2], C22×Dic9, C2×C4.Dic9

Smallest permutation representation of C2×C4.Dic9
On 144 points
Generators in S144
(1 60)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 72)(14 37)(15 38)(16 39)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(73 136)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 144)(82 109)(83 110)(84 111)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 127)(101 128)(102 129)(103 130)(104 131)(105 132)(106 133)(107 134)(108 135)
(1 10 19 28)(2 11 20 29)(3 12 21 30)(4 13 22 31)(5 14 23 32)(6 15 24 33)(7 16 25 34)(8 17 26 35)(9 18 27 36)(37 46 55 64)(38 47 56 65)(39 48 57 66)(40 49 58 67)(41 50 59 68)(42 51 60 69)(43 52 61 70)(44 53 62 71)(45 54 63 72)(73 100 91 82)(74 101 92 83)(75 102 93 84)(76 103 94 85)(77 104 95 86)(78 105 96 87)(79 106 97 88)(80 107 98 89)(81 108 99 90)(109 136 127 118)(110 137 128 119)(111 138 129 120)(112 139 130 121)(113 140 131 122)(114 141 132 123)(115 142 133 124)(116 143 134 125)(117 144 135 126)
(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)
(1 132 10 141 19 114 28 123)(2 113 11 122 20 131 29 140)(3 130 12 139 21 112 30 121)(4 111 13 120 22 129 31 138)(5 128 14 137 23 110 32 119)(6 109 15 118 24 127 33 136)(7 126 16 135 25 144 34 117)(8 143 17 116 26 125 35 134)(9 124 18 133 27 142 36 115)(37 74 46 83 55 92 64 101)(38 91 47 100 56 73 65 82)(39 108 48 81 57 90 66 99)(40 89 49 98 58 107 67 80)(41 106 50 79 59 88 68 97)(42 87 51 96 60 105 69 78)(43 104 52 77 61 86 70 95)(44 85 53 94 62 103 71 76)(45 102 54 75 63 84 72 93)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 6A ··· 6G 8A ··· 8H 9A 9B 9C 12A ··· 12H 18A ··· 18U 36A ··· 36X order 1 2 2 2 2 2 3 4 4 4 4 4 4 6 ··· 6 8 ··· 8 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 1 1 1 1 2 2 2 ··· 2 18 ··· 18 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - + - + - image C1 C2 C2 C2 C4 C4 S3 Dic3 D6 Dic3 M4(2) D9 Dic9 D18 Dic9 C4.Dic3 C4.Dic9 kernel C2×C4.Dic9 C2×C9⋊C8 C4.Dic9 C22×C36 C2×C36 C22×C18 C22×C12 C2×C12 C2×C12 C22×C6 C18 C22×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 2 4 1 6 2 1 3 3 1 4 3 9 9 3 8 24

Matrix representation of C2×C4.Dic9 in GL3(𝔽73) generated by

 72 0 0 0 72 0 0 0 72
,
 72 0 0 0 27 9 0 0 46
,
 1 0 0 0 61 72 0 0 67
,
 72 0 0 0 48 33 0 4 25
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[72,0,0,0,27,0,0,9,46],[1,0,0,0,61,0,0,72,67],[72,0,0,0,48,4,0,33,25] >;

C2×C4.Dic9 in GAP, Magma, Sage, TeX

C_2\times C_4.{\rm Dic}_9
% in TeX

G:=Group("C2xC4.Dic9");
// GroupNames label

G:=SmallGroup(288,131);
// by ID

G=gap.SmallGroup(288,131);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,422,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=1,c^18=b^2,d^2=c^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^17>;
// generators/relations

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