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## G = C4⋊C4×D9order 288 = 25·32

### Direct product of C4⋊C4 and D9

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4⋊C4×D9
G = < a,b,c,d | a4=b4=c9=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 552 in 138 conjugacy classes, 60 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×6], S3 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, C9, Dic3 [×4], C12 [×2], C12 [×2], D6 [×6], C2×C6, C4⋊C4, C4⋊C4 [×3], C22×C4 [×3], D9 [×4], C18 [×3], C4×S3 [×8], C2×Dic3 [×3], C2×C12, C2×C12 [×2], C22×S3, C2×C4⋊C4, Dic9 [×2], Dic9 [×2], C36 [×2], C36 [×2], D18 [×6], C2×C18, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4, S3×C2×C4 [×3], C4×D9 [×4], C4×D9 [×4], C2×Dic9, C2×Dic9 [×2], C2×C36, C2×C36 [×2], C22×D9, S3×C4⋊C4, Dic9⋊C4 [×2], C4⋊Dic9, C9×C4⋊C4, C2×C4×D9, C2×C4×D9 [×2], C4⋊C4×D9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D9, C4×S3 [×2], C22×S3, C2×C4⋊C4, D18 [×3], S3×C2×C4, S3×D4, S3×Q8, C4×D9 [×2], C22×D9, S3×C4⋊C4, C2×C4×D9, D4×D9, Q8×D9, C4⋊C4×D9

Smallest permutation representation of C4⋊C4×D9
On 144 points
Generators in S144
(1 104 14 95)(2 105 15 96)(3 106 16 97)(4 107 17 98)(5 108 18 99)(6 100 10 91)(7 101 11 92)(8 102 12 93)(9 103 13 94)(19 82 28 73)(20 83 29 74)(21 84 30 75)(22 85 31 76)(23 86 32 77)(24 87 33 78)(25 88 34 79)(26 89 35 80)(27 90 36 81)(37 136 46 127)(38 137 47 128)(39 138 48 129)(40 139 49 130)(41 140 50 131)(42 141 51 132)(43 142 52 133)(44 143 53 134)(45 144 54 135)(55 118 64 109)(56 119 65 110)(57 120 66 111)(58 121 67 112)(59 122 68 113)(60 123 69 114)(61 124 70 115)(62 125 71 116)(63 126 72 117)
(1 59 23 41)(2 60 24 42)(3 61 25 43)(4 62 26 44)(5 63 27 45)(6 55 19 37)(7 56 20 38)(8 57 21 39)(9 58 22 40)(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)(73 136 91 118)(74 137 92 119)(75 138 93 120)(76 139 94 121)(77 140 95 122)(78 141 96 123)(79 142 97 124)(80 143 98 125)(81 144 99 126)(82 127 100 109)(83 128 101 110)(84 129 102 111)(85 130 103 112)(86 131 104 113)(87 132 105 114)(88 133 106 115)(89 134 107 116)(90 135 108 117)
(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 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 35)(29 34)(30 33)(31 32)(37 44)(38 43)(39 42)(40 41)(46 53)(47 52)(48 51)(49 50)(55 62)(56 61)(57 60)(58 59)(64 71)(65 70)(66 69)(67 68)(73 80)(74 79)(75 78)(76 77)(82 89)(83 88)(84 87)(85 86)(91 98)(92 97)(93 96)(94 95)(100 107)(101 106)(102 105)(103 104)(109 116)(110 115)(111 114)(112 113)(118 125)(119 124)(120 123)(121 122)(127 134)(128 133)(129 132)(130 131)(136 143)(137 142)(138 141)(139 140)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + + + - + - image C1 C2 C2 C2 C2 C4 S3 D4 Q8 D6 D9 C4×S3 D18 C4×D9 S3×D4 S3×Q8 D4×D9 Q8×D9 kernel C4⋊C4×D9 Dic9⋊C4 C4⋊Dic9 C9×C4⋊C4 C2×C4×D9 C4×D9 C3×C4⋊C4 D18 D18 C2×C12 C4⋊C4 C12 C2×C4 C4 C6 C6 C2 C2 # reps 1 2 1 1 3 8 1 2 2 3 3 4 9 12 1 1 3 3

Matrix representation of C4⋊C4×D9 in GL5(𝔽37)

 36 0 0 0 0 0 11 36 0 0 0 11 26 0 0 0 0 0 1 0 0 0 0 0 1
,
 6 0 0 0 0 0 1 35 0 0 0 0 36 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 6 11 0 0 0 26 17
,
 36 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 26 17 0 0 0 6 11

G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,11,11,0,0,0,36,26,0,0,0,0,0,1,0,0,0,0,0,1],[6,0,0,0,0,0,1,0,0,0,0,35,36,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,26,0,0,0,11,17],[36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,26,6,0,0,0,17,11] >;

C4⋊C4×D9 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\times D_9
% in TeX

G:=Group("C4:C4xD9");
// GroupNames label

G:=SmallGroup(288,101);
// by ID

G=gap.SmallGroup(288,101);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,58,6725,292,9414]);
// Polycyclic

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

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