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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, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C9, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C18, C18, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C4⋊D4, C36, C36, C2×C18, C2×C18, C2×C18, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×D4, C2×C36, C2×C36, C2×C36, D4×C9, C22×C18, C22×C18, C3×C4⋊D4, C9×C22⋊C4, C9×C4⋊C4, C22×C36, D4×C18, D4×C18, C9×C4⋊D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, C2×D4, C4○D4, C18, C3×D4, C22×C6, C4⋊D4, C2×C18, C6×D4, C3×C4○D4, D4×C9, C22×C18, C3×C4⋊D4, D4×C18, 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 41 117)(2 94 42 109)(3 95 43 110)(4 96 44 111)(5 97 45 112)(6 98 37 113)(7 99 38 114)(8 91 39 115)(9 92 40 116)(10 79 127 55)(11 80 128 56)(12 81 129 57)(13 73 130 58)(14 74 131 59)(15 75 132 60)(16 76 133 61)(17 77 134 62)(18 78 135 63)(19 83 144 72)(20 84 136 64)(21 85 137 65)(22 86 138 66)(23 87 139 67)(24 88 140 68)(25 89 141 69)(26 90 142 70)(27 82 143 71)(28 120 48 100)(29 121 49 101)(30 122 50 102)(31 123 51 103)(32 124 52 104)(33 125 53 105)(34 126 54 106)(35 118 46 107)(36 119 47 108)
(1 81 34 70)(2 73 35 71)(3 74 36 72)(4 75 28 64)(5 76 29 65)(6 77 30 66)(7 78 31 67)(8 79 32 68)(9 80 33 69)(10 124 140 91)(11 125 141 92)(12 126 142 93)(13 118 143 94)(14 119 144 95)(15 120 136 96)(16 121 137 97)(17 122 138 98)(18 123 139 99)(19 110 131 108)(20 111 132 100)(21 112 133 101)(22 113 134 102)(23 114 135 103)(24 115 127 104)(25 116 128 105)(26 117 129 106)(27 109 130 107)(37 62 50 86)(38 63 51 87)(39 55 52 88)(40 56 53 89)(41 57 54 90)(42 58 46 82)(43 59 47 83)(44 60 48 84)(45 61 49 85)
(10 24)(11 25)(12 26)(13 27)(14 19)(15 20)(16 21)(17 22)(18 23)(55 88)(56 89)(57 90)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 73)(72 74)(91 115)(92 116)(93 117)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 118)(108 119)(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), (1,93,41,117)(2,94,42,109)(3,95,43,110)(4,96,44,111)(5,97,45,112)(6,98,37,113)(7,99,38,114)(8,91,39,115)(9,92,40,116)(10,79,127,55)(11,80,128,56)(12,81,129,57)(13,73,130,58)(14,74,131,59)(15,75,132,60)(16,76,133,61)(17,77,134,62)(18,78,135,63)(19,83,144,72)(20,84,136,64)(21,85,137,65)(22,86,138,66)(23,87,139,67)(24,88,140,68)(25,89,141,69)(26,90,142,70)(27,82,143,71)(28,120,48,100)(29,121,49,101)(30,122,50,102)(31,123,51,103)(32,124,52,104)(33,125,53,105)(34,126,54,106)(35,118,46,107)(36,119,47,108), (1,81,34,70)(2,73,35,71)(3,74,36,72)(4,75,28,64)(5,76,29,65)(6,77,30,66)(7,78,31,67)(8,79,32,68)(9,80,33,69)(10,124,140,91)(11,125,141,92)(12,126,142,93)(13,118,143,94)(14,119,144,95)(15,120,136,96)(16,121,137,97)(17,122,138,98)(18,123,139,99)(19,110,131,108)(20,111,132,100)(21,112,133,101)(22,113,134,102)(23,114,135,103)(24,115,127,104)(25,116,128,105)(26,117,129,106)(27,109,130,107)(37,62,50,86)(38,63,51,87)(39,55,52,88)(40,56,53,89)(41,57,54,90)(42,58,46,82)(43,59,47,83)(44,60,48,84)(45,61,49,85), (10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(55,88)(56,89)(57,90)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,73)(72,74)(91,115)(92,116)(93,117)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,118)(108,119)(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), (1,93,41,117)(2,94,42,109)(3,95,43,110)(4,96,44,111)(5,97,45,112)(6,98,37,113)(7,99,38,114)(8,91,39,115)(9,92,40,116)(10,79,127,55)(11,80,128,56)(12,81,129,57)(13,73,130,58)(14,74,131,59)(15,75,132,60)(16,76,133,61)(17,77,134,62)(18,78,135,63)(19,83,144,72)(20,84,136,64)(21,85,137,65)(22,86,138,66)(23,87,139,67)(24,88,140,68)(25,89,141,69)(26,90,142,70)(27,82,143,71)(28,120,48,100)(29,121,49,101)(30,122,50,102)(31,123,51,103)(32,124,52,104)(33,125,53,105)(34,126,54,106)(35,118,46,107)(36,119,47,108), (1,81,34,70)(2,73,35,71)(3,74,36,72)(4,75,28,64)(5,76,29,65)(6,77,30,66)(7,78,31,67)(8,79,32,68)(9,80,33,69)(10,124,140,91)(11,125,141,92)(12,126,142,93)(13,118,143,94)(14,119,144,95)(15,120,136,96)(16,121,137,97)(17,122,138,98)(18,123,139,99)(19,110,131,108)(20,111,132,100)(21,112,133,101)(22,113,134,102)(23,114,135,103)(24,115,127,104)(25,116,128,105)(26,117,129,106)(27,109,130,107)(37,62,50,86)(38,63,51,87)(39,55,52,88)(40,56,53,89)(41,57,54,90)(42,58,46,82)(43,59,47,83)(44,60,48,84)(45,61,49,85), (10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(55,88)(56,89)(57,90)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,73)(72,74)(91,115)(92,116)(93,117)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,118)(108,119)(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)], [(1,93,41,117),(2,94,42,109),(3,95,43,110),(4,96,44,111),(5,97,45,112),(6,98,37,113),(7,99,38,114),(8,91,39,115),(9,92,40,116),(10,79,127,55),(11,80,128,56),(12,81,129,57),(13,73,130,58),(14,74,131,59),(15,75,132,60),(16,76,133,61),(17,77,134,62),(18,78,135,63),(19,83,144,72),(20,84,136,64),(21,85,137,65),(22,86,138,66),(23,87,139,67),(24,88,140,68),(25,89,141,69),(26,90,142,70),(27,82,143,71),(28,120,48,100),(29,121,49,101),(30,122,50,102),(31,123,51,103),(32,124,52,104),(33,125,53,105),(34,126,54,106),(35,118,46,107),(36,119,47,108)], [(1,81,34,70),(2,73,35,71),(3,74,36,72),(4,75,28,64),(5,76,29,65),(6,77,30,66),(7,78,31,67),(8,79,32,68),(9,80,33,69),(10,124,140,91),(11,125,141,92),(12,126,142,93),(13,118,143,94),(14,119,144,95),(15,120,136,96),(16,121,137,97),(17,122,138,98),(18,123,139,99),(19,110,131,108),(20,111,132,100),(21,112,133,101),(22,113,134,102),(23,114,135,103),(24,115,127,104),(25,116,128,105),(26,117,129,106),(27,109,130,107),(37,62,50,86),(38,63,51,87),(39,55,52,88),(40,56,53,89),(41,57,54,90),(42,58,46,82),(43,59,47,83),(44,60,48,84),(45,61,49,85)], [(10,24),(11,25),(12,26),(13,27),(14,19),(15,20),(16,21),(17,22),(18,23),(55,88),(56,89),(57,90),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(71,73),(72,74),(91,115),(92,116),(93,117),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,120),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,118),(108,119),(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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