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G = C422D9order 288 = 25·32

1st semidirect product of C42 and D9 acting via D9/C9=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C422D9, (C4×C36)⋊9C2, (C4×D9)⋊3C4, C4.22(C4×D9), C12.72(C4×S3), C36.27(C2×C4), (C4×C12).19S3, D18⋊C4.7C2, (C4×Dic9)⋊8C2, D18.3(C2×C4), (C2×C4).64D18, Dic9⋊C417C2, C91(C42⋊C2), C18.3(C4○D4), (C2×C12).338D6, C3.(C422S3), C18.3(C22×C4), Dic9.5(C2×C4), C6.73(C4○D12), (C2×C36).71C22, (C2×C18).13C23, C2.2(D365C2), C22.10(C22×D9), (C2×Dic9).21C22, (C22×D9).14C22, C2.5(C2×C4×D9), C6.42(S3×C2×C4), (C2×C4×D9).7C2, (C2×C6).170(C22×S3), SmallGroup(288,82)

Series: Derived Chief Lower central Upper central

C1C18 — C422D9
C1C3C9C18C2×C18C22×D9C2×C4×D9 — C422D9
C9C18 — C422D9
C1C2×C4C42

Generators and relations for C422D9
 G = < a,b,c,d | a4=b4=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 448 in 114 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×7], C23, C9, Dic3 [×4], C12 [×2], C12 [×2], D6 [×4], C2×C6, C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, D9 [×2], C18, C18 [×2], C4×S3 [×4], C2×Dic3 [×3], C2×C12, C2×C12 [×2], C22×S3, C42⋊C2, Dic9 [×2], Dic9 [×2], C36 [×2], C36 [×2], D18 [×2], D18 [×2], C2×C18, C4×Dic3, Dic3⋊C4 [×2], D6⋊C4 [×2], C4×C12, S3×C2×C4, C4×D9 [×4], C2×Dic9, C2×Dic9 [×2], C2×C36, C2×C36 [×2], C22×D9, C422S3, C4×Dic9, Dic9⋊C4 [×2], D18⋊C4 [×2], C4×C36, C2×C4×D9, C422D9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, C4○D4 [×2], D9, C4×S3 [×2], C22×S3, C42⋊C2, D18 [×3], S3×C2×C4, C4○D12 [×2], C4×D9 [×2], C22×D9, C422S3, C2×C4×D9, D365C2 [×2], C422D9

Smallest permutation representation of C422D9
On 144 points
Generators in S144
(1 86 14 77)(2 87 15 78)(3 88 16 79)(4 89 17 80)(5 90 18 81)(6 82 10 73)(7 83 11 74)(8 84 12 75)(9 85 13 76)(19 100 28 91)(20 101 29 92)(21 102 30 93)(22 103 31 94)(23 104 32 95)(24 105 33 96)(25 106 34 97)(26 107 35 98)(27 108 36 99)(37 118 46 109)(38 119 47 110)(39 120 48 111)(40 121 49 112)(41 122 50 113)(42 123 51 114)(43 124 52 115)(44 125 53 116)(45 126 54 117)(55 136 64 127)(56 137 65 128)(57 138 66 129)(58 139 67 130)(59 140 68 131)(60 141 69 132)(61 142 70 133)(62 143 71 134)(63 144 72 135)
(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)(73 127 91 109)(74 128 92 110)(75 129 93 111)(76 130 94 112)(77 131 95 113)(78 132 96 114)(79 133 97 115)(80 134 98 116)(81 135 99 117)(82 136 100 118)(83 137 101 119)(84 138 102 120)(85 139 103 121)(86 140 104 122)(87 141 105 123)(88 142 106 124)(89 143 107 125)(90 144 108 126)
(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 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(45 54)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 72)(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)(109 125)(110 124)(111 123)(112 122)(113 121)(114 120)(115 119)(116 118)(117 126)(127 143)(128 142)(129 141)(130 140)(131 139)(132 138)(133 137)(134 136)(135 144)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I···4N6A6B6C9A9B9C12A···12L18A···18I36A···36AJ
order1222223444444444···466699912···1218···1836···36
size1111181821111222218···182222222···22···22···2

84 irreducible representations

dim1111111222222222
type++++++++++
imageC1C2C2C2C2C2C4S3D6C4○D4D9C4×S3D18C4○D12C4×D9D365C2
kernelC422D9C4×Dic9Dic9⋊C4D18⋊C4C4×C36C2×C4×D9C4×D9C4×C12C2×C12C18C42C12C2×C4C6C4C2
# reps112211813434981224

Matrix representation of C422D9 in GL4(𝔽37) generated by

1000
0100
0060
0006
,
6000
0600
00510
002732
,
312000
171100
003120
001711
,
312000
26600
003120
00266
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,6,0,0,0,0,6],[6,0,0,0,0,6,0,0,0,0,5,27,0,0,10,32],[31,17,0,0,20,11,0,0,0,0,31,17,0,0,20,11],[31,26,0,0,20,6,0,0,0,0,31,26,0,0,20,6] >;

C422D9 in GAP, Magma, Sage, TeX

C_4^2\rtimes_2D_9
% in TeX

G:=Group("C4^2:2D9");
// GroupNames label

G:=SmallGroup(288,82);
// by ID

G=gap.SmallGroup(288,82);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,58,6725,292,9414]);
// Polycyclic

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

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