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G = C9×D4⋊C4order 288 = 25·32

Direct product of C9 and D4⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C9×D4⋊C4, D41C36, C18.13D8, C36.60D4, C18.9SD16, C4⋊C41C18, (C2×C72)⋊4C2, (C2×C8)⋊2C18, (D4×C9)⋊4C4, C2.1(C9×D8), (C2×C24).2C6, C4.1(C2×C36), (C6×D4).8C6, C4.11(D4×C9), C6.13(C3×D8), C36.30(C2×C4), (C2×D4).3C18, (C3×D4).2C12, (D4×C18).9C2, C12.69(C3×D4), (C2×C18).46D4, C2.1(C9×SD16), C6.9(C3×SD16), C22.8(D4×C9), C12.30(C2×C12), C18.24(C22⋊C4), (C2×C36).113C22, (C9×C4⋊C4)⋊10C2, (C3×C4⋊C4).8C6, C3.(C3×D4⋊C4), (C3×D4⋊C4).C3, (C2×C6).55(C3×D4), C2.6(C9×C22⋊C4), (C2×C4).12(C2×C18), C6.24(C3×C22⋊C4), (C2×C12).133(C2×C6), SmallGroup(288,52)

Series: Derived Chief Lower central Upper central

C1C4 — C9×D4⋊C4
C1C2C6C2×C6C2×C12C2×C36C9×C4⋊C4 — C9×D4⋊C4
C1C2C4 — C9×D4⋊C4
C1C2×C18C2×C36 — C9×D4⋊C4

Generators and relations for C9×D4⋊C4
 G = < a,b,c,d | a9=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Subgroups: 138 in 75 conjugacy classes, 42 normal (36 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C6 [×3], C6 [×2], C8, C2×C4, C2×C4, D4 [×2], D4, C23, C9, C12 [×2], C12, C2×C6, C2×C6 [×4], C4⋊C4, C2×C8, C2×D4, C18 [×3], C18 [×2], C24, C2×C12, C2×C12, C3×D4 [×2], C3×D4, C22×C6, D4⋊C4, C36 [×2], C36, C2×C18, C2×C18 [×4], C3×C4⋊C4, C2×C24, C6×D4, C72, C2×C36, C2×C36, D4×C9 [×2], D4×C9, C22×C18, C3×D4⋊C4, C9×C4⋊C4, C2×C72, D4×C18, C9×D4⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], C9, C12 [×2], C2×C6, C22⋊C4, D8, SD16, C18 [×3], C2×C12, C3×D4 [×2], D4⋊C4, C36 [×2], C2×C18, C3×C22⋊C4, C3×D8, C3×SD16, C2×C36, D4×C9 [×2], C3×D4⋊C4, C9×C22⋊C4, C9×D8, C9×SD16, C9×D4⋊C4

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

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

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

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

126 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D6A···6F6G6H6I6J8A8B8C8D9A···9F12A12B12C12D12E12F12G12H18A···18R18S···18AD24A···24H36A···36L36M···36X72A···72X
order1222223344446···6666688889···9121212121212121218···1818···1824···2436···3636···3672···72
size1111441122441···1444422221···1222244441···14···42···22···24···42···2

126 irreducible representations

dim111111111111111222222222222
type+++++++
imageC1C2C2C2C3C4C6C6C6C9C12C18C18C18C36D4D4D8SD16C3×D4C3×D4C3×D8C3×SD16D4×C9D4×C9C9×D8C9×SD16
kernelC9×D4⋊C4C9×C4⋊C4C2×C72D4×C18C3×D4⋊C4D4×C9C3×C4⋊C4C2×C24C6×D4D4⋊C4C3×D4C4⋊C4C2×C8C2×D4D4C36C2×C18C18C18C12C2×C6C6C6C4C22C2C2
# reps111124222686662411222244661212

Matrix representation of C9×D4⋊C4 in GL4(𝔽73) generated by

1000
0100
00160
00016
,
72000
07200
00171
00172
,
72000
39100
00171
00072
,
604800
361300
00012
0060
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[72,0,0,0,0,72,0,0,0,0,1,1,0,0,71,72],[72,39,0,0,0,1,0,0,0,0,1,0,0,0,71,72],[60,36,0,0,48,13,0,0,0,0,0,6,0,0,12,0] >;

C9×D4⋊C4 in GAP, Magma, Sage, TeX

C_9\times D_4\rtimes C_4
% in TeX

G:=Group("C9xD4:C4");
// GroupNames label

G:=SmallGroup(288,52);
// by ID

G=gap.SmallGroup(288,52);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,268,4371,2194,360]);
// Polycyclic

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

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