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## G = C2×D4.D9order 288 = 25·32

### Direct product of C2 and D4.D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C2×D4.D9
 Chief series C1 — C3 — C9 — C18 — C36 — Dic18 — C2×Dic18 — C2×D4.D9
 Lower central C9 — C18 — C36 — C2×D4.D9
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for C2×D4.D9
G = < a,b,c,d,e | a2=b4=c2=d9=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 352 in 102 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×2], D4, Q8 [×3], C23, C9, Dic3 [×2], C12 [×2], C2×C6, C2×C6 [×4], C2×C8, SD16 [×4], C2×D4, C2×Q8, C18, C18 [×2], C18 [×2], C3⋊C8 [×2], Dic6 [×3], C2×Dic3, C2×C12, C3×D4 [×2], C3×D4, C22×C6, C2×SD16, Dic9 [×2], C36 [×2], C2×C18, C2×C18 [×4], C2×C3⋊C8, D4.S3 [×4], C2×Dic6, C6×D4, C9⋊C8 [×2], Dic18 [×2], Dic18, C2×Dic9, C2×C36, D4×C9 [×2], D4×C9, C22×C18, C2×D4.S3, C2×C9⋊C8, D4.D9 [×4], C2×Dic18, D4×C18, C2×D4.D9
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], C2×D4, D9, C3⋊D4 [×2], C22×S3, C2×SD16, D18 [×3], D4.S3 [×2], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C2×D4.S3, D4.D9 [×2], C2×C9⋊D4, C2×D4.D9

Smallest permutation representation of C2×D4.D9
On 144 points
Generators in S144
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(91 136)(92 137)(93 138)(94 139)(95 140)(96 141)(97 142)(98 143)(99 144)(100 127)(101 128)(102 129)(103 130)(104 131)(105 132)(106 133)(107 134)(108 135)
(1 28 10 19)(2 29 11 20)(3 30 12 21)(4 31 13 22)(5 32 14 23)(6 33 15 24)(7 34 16 25)(8 35 17 26)(9 36 18 27)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)(73 91 82 100)(74 92 83 101)(75 93 84 102)(76 94 85 103)(77 95 86 104)(78 96 87 105)(79 97 88 106)(80 98 89 107)(81 99 90 108)(109 136 118 127)(110 137 119 128)(111 138 120 129)(112 139 121 130)(113 140 122 131)(114 141 123 132)(115 142 124 133)(116 143 125 134)(117 144 126 135)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(109 118)(110 119)(111 120)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)
(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)
(1 124 10 115)(2 123 11 114)(3 122 12 113)(4 121 13 112)(5 120 14 111)(6 119 15 110)(7 118 16 109)(8 126 17 117)(9 125 18 116)(19 133 28 142)(20 132 29 141)(21 131 30 140)(22 130 31 139)(23 129 32 138)(24 128 33 137)(25 127 34 136)(26 135 35 144)(27 134 36 143)(37 88 46 79)(38 87 47 78)(39 86 48 77)(40 85 49 76)(41 84 50 75)(42 83 51 74)(43 82 52 73)(44 90 53 81)(45 89 54 80)(55 97 64 106)(56 96 65 105)(57 95 66 104)(58 94 67 103)(59 93 68 102)(60 92 69 101)(61 91 70 100)(62 99 71 108)(63 98 72 107)

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 SD16 D9 C3⋊D4 C3⋊D4 D18 D18 C9⋊D4 C9⋊D4 D4.S3 D4.D9 kernel C2×D4.D9 C2×C9⋊C8 D4.D9 C2×Dic18 D4×C18 C6×D4 C36 C2×C18 C2×C12 C3×D4 C18 C2×D4 C12 C2×C6 C2×C4 D4 C4 C22 C6 C2 # reps 1 1 4 1 1 1 1 1 1 2 4 3 2 2 3 6 6 6 2 6

Matrix representation of C2×D4.D9 in GL5(𝔽73)

 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 72
,
 72 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 50 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 72 37
,
 72 0 0 0 0 0 6 67 0 0 0 67 67 0 0 0 0 0 25 1 0 0 0 33 48

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,72,50,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,72,0,0,0,0,37],[72,0,0,0,0,0,6,67,0,0,0,67,67,0,0,0,0,0,25,33,0,0,0,1,48] >;

C2×D4.D9 in GAP, Magma, Sage, TeX

C_2\times D_4.D_9
% in TeX

G:=Group("C2xD4.D9");
// GroupNames label

G:=SmallGroup(288,141);
// by ID

G=gap.SmallGroup(288,141);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,675,185,80,6725,292,9414]);
// Polycyclic

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

׿
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