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## G = C4×C9⋊D4order 288 = 25·32

### Direct product of C4 and C9⋊D4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4×C9⋊D4
G = < a,b,c,d | a4=b9=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 536 in 141 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, D9, C18, C18, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4×D4, Dic9, Dic9, C36, C36, D18, D18, C2×C18, C2×C18, C2×C18, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C4×D9, C2×Dic9, C9⋊D4, C2×C36, C2×C36, C22×D9, C22×C18, C4×C3⋊D4, C4×Dic9, Dic9⋊C4, D18⋊C4, C18.D4, C2×C4×D9, C2×C9⋊D4, C22×C36, C4×C9⋊D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, D9, C4×S3, C3⋊D4, C22×S3, C4×D4, D18, S3×C2×C4, C4○D12, C2×C3⋊D4, C4×D9, C9⋊D4, C22×D9, C4×C3⋊D4, C2×C4×D9, D365C2, C2×C9⋊D4, C4×C9⋊D4

Smallest permutation representation of C4×C9⋊D4
On 144 points
Generators in S144
(1 58 22 40)(2 59 23 41)(3 60 24 42)(4 61 25 43)(5 62 26 44)(6 63 27 45)(7 55 19 37)(8 56 20 38)(9 57 21 39)(10 64 28 46)(11 65 29 47)(12 66 30 48)(13 67 31 49)(14 68 32 50)(15 69 33 51)(16 70 34 52)(17 71 35 53)(18 72 36 54)(73 127 91 109)(74 128 92 110)(75 129 93 111)(76 130 94 112)(77 131 95 113)(78 132 96 114)(79 133 97 115)(80 134 98 116)(81 135 99 117)(82 136 100 118)(83 137 101 119)(84 138 102 120)(85 139 103 121)(86 140 104 122)(87 141 105 123)(88 142 106 124)(89 143 107 125)(90 144 108 126)
(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 94 13 103)(2 93 14 102)(3 92 15 101)(4 91 16 100)(5 99 17 108)(6 98 18 107)(7 97 10 106)(8 96 11 105)(9 95 12 104)(19 79 28 88)(20 78 29 87)(21 77 30 86)(22 76 31 85)(23 75 32 84)(24 74 33 83)(25 73 34 82)(26 81 35 90)(27 80 36 89)(37 133 46 142)(38 132 47 141)(39 131 48 140)(40 130 49 139)(41 129 50 138)(42 128 51 137)(43 127 52 136)(44 135 53 144)(45 134 54 143)(55 115 64 124)(56 114 65 123)(57 113 66 122)(58 112 67 121)(59 111 68 120)(60 110 69 119)(61 109 70 118)(62 117 71 126)(63 116 72 125)
(2 9)(3 8)(4 7)(5 6)(10 16)(11 15)(12 14)(17 18)(19 25)(20 24)(21 23)(26 27)(28 34)(29 33)(30 32)(35 36)(37 43)(38 42)(39 41)(44 45)(46 52)(47 51)(48 50)(53 54)(55 61)(56 60)(57 59)(62 63)(64 70)(65 69)(66 68)(71 72)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 90)(81 89)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 100)(98 108)(99 107)(109 124)(110 123)(111 122)(112 121)(113 120)(114 119)(115 118)(116 126)(117 125)(127 142)(128 141)(129 140)(130 139)(131 138)(132 137)(133 136)(134 144)(135 143)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 C4○D4 D9 C3⋊D4 C4×S3 D18 D18 C4○D12 C9⋊D4 C4×D9 D36⋊5C2 kernel C4×C9⋊D4 C4×Dic9 Dic9⋊C4 D18⋊C4 C18.D4 C2×C4×D9 C2×C9⋊D4 C22×C36 C9⋊D4 C22×C12 C36 C2×C12 C22×C6 C18 C22×C4 C12 C2×C6 C2×C4 C23 C6 C4 C22 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 2 1 2 3 4 4 6 3 4 12 12 12

Matrix representation of C4×C9⋊D4 in GL4(𝔽37) generated by

 31 0 0 0 0 31 0 0 0 0 36 0 0 0 0 36
,
 36 36 0 0 1 0 0 0 0 0 6 26 0 0 11 17
,
 7 14 0 0 7 30 0 0 0 0 30 23 0 0 30 7
,
 1 0 0 0 36 36 0 0 0 0 1 0 0 0 36 36
G:=sub<GL(4,GF(37))| [31,0,0,0,0,31,0,0,0,0,36,0,0,0,0,36],[36,1,0,0,36,0,0,0,0,0,6,11,0,0,26,17],[7,7,0,0,14,30,0,0,0,0,30,30,0,0,23,7],[1,36,0,0,0,36,0,0,0,0,1,36,0,0,0,36] >;

C4×C9⋊D4 in GAP, Magma, Sage, TeX

C_4\times C_9\rtimes D_4
% in TeX

G:=Group("C4xC9:D4");
// GroupNames label

G:=SmallGroup(288,138);
// by ID

G=gap.SmallGroup(288,138);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,58,6725,292,9414]);
// Polycyclic

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

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