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G = C9×C4⋊D4order 288 = 25·32

Direct product of C9 and C4⋊D4

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

Aliases: C9×C4⋊D4, C369D4, C42(D4×C9), C4⋊C42C18, (C2×C18)⋊4D4, (C2×D4)⋊2C18, C2.5(D4×C18), C6.68(C6×D4), C222(D4×C9), (D4×C18)⋊11C2, C22⋊C43C18, (C22×C4)⋊6C18, (C6×D4).10C6, C12.71(C3×D4), C18.68(C2×D4), (C22×C36)⋊11C2, C23.2(C2×C18), C18.41(C4○D4), (C2×C18).76C23, (C22×C12).28C6, (C2×C36).122C22, (C22×C18).27C22, C22.11(C22×C18), (C9×C4⋊C4)⋊11C2, C3.(C3×C4⋊D4), (C3×C4⋊D4).C3, C2.4(C9×C4○D4), (C3×C4⋊C4).12C6, (C2×C4).9(C2×C18), C6.41(C3×C4○D4), (C2×C6).13(C3×D4), (C9×C22⋊C4)⋊11C2, (C2×C12).63(C2×C6), (C3×C22⋊C4).6C6, (C22×C6).46(C2×C6), (C2×C6).81(C22×C6), SmallGroup(288,171)

Series: Derived Chief Lower central Upper central

C1C22 — C9×C4⋊D4
C1C3C6C2×C6C2×C18C22×C18D4×C18 — C9×C4⋊D4
C1C22 — C9×C4⋊D4
C1C2×C18 — C9×C4⋊D4

Generators and relations for C9×C4⋊D4
 G = < a,b,c,d | a9=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 222 in 141 conjugacy classes, 72 normal (36 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×2], D4 [×6], C23, C23 [×2], C9, C12 [×2], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×8], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], C18 [×3], C18 [×4], C2×C12 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×6], C22×C6, C22×C6 [×2], C4⋊D4, C36 [×2], C36 [×3], C2×C18, C2×C18 [×2], C2×C18 [×8], C3×C22⋊C4 [×2], C3×C4⋊C4, C22×C12, C6×D4, C6×D4 [×2], C2×C36 [×2], C2×C36 [×2], C2×C36 [×2], D4×C9 [×6], C22×C18, C22×C18 [×2], C3×C4⋊D4, C9×C22⋊C4 [×2], C9×C4⋊C4, C22×C36, D4×C18, D4×C18 [×2], C9×C4⋊D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×4], C23, C9, C2×C6 [×7], C2×D4 [×2], C4○D4, C18 [×7], C3×D4 [×4], C22×C6, C4⋊D4, C2×C18 [×7], C6×D4 [×2], C3×C4○D4, D4×C9 [×4], C22×C18, C3×C4⋊D4, D4×C18 [×2], C9×C4○D4, C9×C4⋊D4

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

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

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

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

126 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M6N9A···9F12A···12H12I12J12K12L18A···18R18S···18AD18AE···18AP36A···36X36Y···36AJ
order12222222334444446···6666666669···912···121212121218···1818···1818···1836···3636···36
size11112244112222441···1222244441···12···244441···12···24···42···24···4

126 irreducible representations

dim111111111111111222222222
type+++++++
imageC1C2C2C2C2C3C6C6C6C6C9C18C18C18C18D4D4C4○D4C3×D4C3×D4C3×C4○D4D4×C9D4×C9C9×C4○D4
kernelC9×C4⋊D4C9×C22⋊C4C9×C4⋊C4C22×C36D4×C18C3×C4⋊D4C3×C22⋊C4C3×C4⋊C4C22×C12C6×D4C4⋊D4C22⋊C4C4⋊C4C22×C4C2×D4C36C2×C18C18C12C2×C6C6C4C22C2
# reps12113242266126618222444121212

Matrix representation of C9×C4⋊D4 in GL5(𝔽37)

120000
010000
001000
00010
00001
,
360000
0313100
00600
000360
000036
,
360000
01000
0353600
000036
00010
,
360000
01000
0353600
00010
000036

G:=sub<GL(5,GF(37))| [12,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1],[36,0,0,0,0,0,31,0,0,0,0,31,6,0,0,0,0,0,36,0,0,0,0,0,36],[36,0,0,0,0,0,1,35,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,36,0],[36,0,0,0,0,0,1,35,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,36] >;

C9×C4⋊D4 in GAP, Magma, Sage, TeX

C_9\times C_4\rtimes D_4
% in TeX

G:=Group("C9xC4:D4");
// GroupNames label

G:=SmallGroup(288,171);
// by ID

G=gap.SmallGroup(288,171);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,365,176,1094,360]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^4=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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