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G = C9×C41D4order 288 = 25·32

Direct product of C9 and C41D4

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

Aliases: C9×C41D4, C366D4, C429C18, C41(D4×C9), (C4×C36)⋊13C2, (C2×D4)⋊3C18, C6.72(C6×D4), C2.9(D4×C18), (D4×C18)⋊12C2, (C4×C12).20C6, (C6×D4).13C6, C18.72(C2×D4), C12.40(C3×D4), C23.6(C2×C18), (C2×C18).82C23, (C2×C36).124C22, (C22×C18).4C22, C22.17(C22×C18), C3.(C3×C41D4), (C2×C4).22(C2×C18), (C3×C41D4).2C3, (C22×C6).9(C2×C6), (C2×C12).140(C2×C6), (C2×C6).87(C22×C6), SmallGroup(288,177)

Series: Derived Chief Lower central Upper central

C1C22 — C9×C41D4
C1C3C6C2×C6C2×C18C22×C18D4×C18 — C9×C41D4
C1C22 — C9×C41D4
C1C2×C18 — C9×C41D4

Generators and relations for C9×C41D4
 G = < a,b,c,d | a9=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 270 in 162 conjugacy classes, 78 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C22, C22 [×12], C6 [×3], C6 [×4], C2×C4 [×3], D4 [×12], C23 [×4], C9, C12 [×6], C2×C6, C2×C6 [×12], C42, C2×D4 [×6], C18 [×3], C18 [×4], C2×C12 [×3], C3×D4 [×12], C22×C6 [×4], C41D4, C36 [×6], C2×C18, C2×C18 [×12], C4×C12, C6×D4 [×6], C2×C36 [×3], D4×C9 [×12], C22×C18 [×4], C3×C41D4, C4×C36, D4×C18 [×6], C9×C41D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×6], C23, C9, C2×C6 [×7], C2×D4 [×3], C18 [×7], C3×D4 [×6], C22×C6, C41D4, C2×C18 [×7], C6×D4 [×3], D4×C9 [×6], C22×C18, C3×C41D4, D4×C18 [×3], C9×C41D4

Smallest permutation representation of C9×C41D4
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)
(1 129 47 142)(2 130 48 143)(3 131 49 144)(4 132 50 136)(5 133 51 137)(6 134 52 138)(7 135 53 139)(8 127 54 140)(9 128 46 141)(10 30 22 39)(11 31 23 40)(12 32 24 41)(13 33 25 42)(14 34 26 43)(15 35 27 44)(16 36 19 45)(17 28 20 37)(18 29 21 38)(55 104 68 91)(56 105 69 92)(57 106 70 93)(58 107 71 94)(59 108 72 95)(60 100 64 96)(61 101 65 97)(62 102 66 98)(63 103 67 99)(73 118 82 109)(74 119 83 110)(75 120 84 111)(76 121 85 112)(77 122 86 113)(78 123 87 114)(79 124 88 115)(80 125 89 116)(81 126 90 117)
(1 70 34 74)(2 71 35 75)(3 72 36 76)(4 64 28 77)(5 65 29 78)(6 66 30 79)(7 67 31 80)(8 68 32 81)(9 69 33 73)(10 115 138 102)(11 116 139 103)(12 117 140 104)(13 109 141 105)(14 110 142 106)(15 111 143 107)(16 112 144 108)(17 113 136 100)(18 114 137 101)(19 121 131 95)(20 122 132 96)(21 123 133 97)(22 124 134 98)(23 125 135 99)(24 126 127 91)(25 118 128 92)(26 119 129 93)(27 120 130 94)(37 86 50 60)(38 87 51 61)(39 88 52 62)(40 89 53 63)(41 90 54 55)(42 82 46 56)(43 83 47 57)(44 84 48 58)(45 85 49 59)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 46)(26 47)(27 48)(28 136)(29 137)(30 138)(31 139)(32 140)(33 141)(34 142)(35 143)(36 144)(37 132)(38 133)(39 134)(40 135)(41 127)(42 128)(43 129)(44 130)(45 131)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)

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

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

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

126 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A···4F6A···6F6G···6N9A···9F12A···12L18A···18R18S···18AP36A···36AJ
order12222222334···46···66···69···912···1218···1818···1836···36
size11114444112···21···14···41···12···21···14···42···2

126 irreducible representations

dim111111111222
type++++
imageC1C2C2C3C6C6C9C18C18D4C3×D4D4×C9
kernelC9×C41D4C4×C36D4×C18C3×C41D4C4×C12C6×D4C41D4C42C2×D4C36C12C4
# reps1162212663661236

Matrix representation of C9×C41D4 in GL4(𝔽37) generated by

7000
0700
0090
0009
,
03600
1000
00360
00036
,
0100
36000
0001
00360
,
0100
1000
0010
00036
G:=sub<GL(4,GF(37))| [7,0,0,0,0,7,0,0,0,0,9,0,0,0,0,9],[0,1,0,0,36,0,0,0,0,0,36,0,0,0,0,36],[0,36,0,0,1,0,0,0,0,0,0,36,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,36] >;

C9×C41D4 in GAP, Magma, Sage, TeX

C_9\times C_4\rtimes_1D_4
% in TeX

G:=Group("C9xC4:1D4");
// GroupNames label

G:=SmallGroup(288,177);
// by ID

G=gap.SmallGroup(288,177);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,365,176,1094,268,360]);
// 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,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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