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G = C4⋊C4⋊D9order 288 = 25·32

6th semidirect product of C4⋊C4 and D9 acting via D9/C9=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C46D9, C4⋊Dic97C2, D18⋊C4.4C2, (C2×C4).11D18, Dic9⋊C413C2, C93(C422C2), (C4×Dic9)⋊14C2, (C2×C12).183D6, C6.84(C4○D12), C18.14(C4○D4), (C2×C36).14C22, (C2×C18).39C23, C2.7(Q83D9), C6.84(D42S3), C2.14(D42D9), C6.42(Q83S3), C2.16(D365C2), (C22×D9).8C22, C22.53(C22×D9), (C2×Dic9).35C22, (C9×C4⋊C4)⋊9C2, C3.(C4⋊C4⋊S3), (C3×C4⋊C4).16S3, (C2×C6).196(C22×S3), SmallGroup(288,108)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C4⋊C4⋊D9
C1C3C9C18C2×C18C22×D9D18⋊C4 — C4⋊C4⋊D9
C9C2×C18 — C4⋊C4⋊D9
C1C22C4⋊C4

Generators and relations for C4⋊C4⋊D9
 G = < a,b,c,d | a4=b4=c9=d2=1, bab-1=a-1, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 396 in 90 conjugacy classes, 38 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], S3, C6 [×3], C2×C4 [×3], C2×C4 [×3], C23, C9, Dic3 [×3], C12 [×3], D6 [×3], C2×C6, C42, C22⋊C4 [×3], C4⋊C4, C4⋊C4 [×2], D9, C18 [×3], C2×Dic3 [×3], C2×C12 [×3], C22×S3, C422C2, Dic9 [×3], C36 [×3], D18 [×3], C2×C18, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4 [×3], C3×C4⋊C4, C2×Dic9 [×3], C2×C36 [×3], C22×D9, C4⋊C4⋊S3, C4×Dic9, Dic9⋊C4, C4⋊Dic9, D18⋊C4 [×3], C9×C4⋊C4, C4⋊C4⋊D9
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], C4○D4 [×3], D9, C22×S3, C422C2, D18 [×3], C4○D12, D42S3, Q83S3, C22×D9, C4⋊C4⋊S3, D365C2, D42D9, Q83D9, C4⋊C4⋊D9

Smallest permutation representation of C4⋊C4⋊D9
On 144 points
Generators in S144
(1 68 14 59)(2 69 15 60)(3 70 16 61)(4 71 17 62)(5 72 18 63)(6 64 10 55)(7 65 11 56)(8 66 12 57)(9 67 13 58)(19 46 28 37)(20 47 29 38)(21 48 30 39)(22 49 31 40)(23 50 32 41)(24 51 33 42)(25 52 34 43)(26 53 35 44)(27 54 36 45)(73 127 82 136)(74 128 83 137)(75 129 84 138)(76 130 85 139)(77 131 86 140)(78 132 87 141)(79 133 88 142)(80 134 89 143)(81 135 90 144)(91 109 100 118)(92 110 101 119)(93 111 102 120)(94 112 103 121)(95 113 104 122)(96 114 105 123)(97 115 106 124)(98 116 107 125)(99 117 108 126)
(1 95 23 77)(2 96 24 78)(3 97 25 79)(4 98 26 80)(5 99 27 81)(6 91 19 73)(7 92 20 74)(8 93 21 75)(9 94 22 76)(10 100 28 82)(11 101 29 83)(12 102 30 84)(13 103 31 85)(14 104 32 86)(15 105 33 87)(16 106 34 88)(17 107 35 89)(18 108 36 90)(37 127 55 109)(38 128 56 110)(39 129 57 111)(40 130 58 112)(41 131 59 113)(42 132 60 114)(43 133 61 115)(44 134 62 116)(45 135 63 117)(46 136 64 118)(47 137 65 119)(48 138 66 120)(49 139 67 121)(50 140 68 122)(51 141 69 123)(52 142 70 124)(53 143 71 125)(54 144 72 126)
(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 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 63)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 72)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(81 90)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(99 108)(109 143)(110 142)(111 141)(112 140)(113 139)(114 138)(115 137)(116 136)(117 144)(118 134)(119 133)(120 132)(121 131)(122 130)(123 129)(124 128)(125 127)(126 135)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I6A6B6C9A9B9C12A···12F18A···18I36A···36R
order12222344444444466699912···1218···1836···36
size1111362224418181818362222224···42···24···4

54 irreducible representations

dim11111122222224444
type++++++++++-+-+
imageC1C2C2C2C2C2S3D6C4○D4D9D18C4○D12D365C2D42S3Q83S3D42D9Q83D9
kernelC4⋊C4⋊D9C4×Dic9Dic9⋊C4C4⋊Dic9D18⋊C4C9×C4⋊C4C3×C4⋊C4C2×C12C18C4⋊C4C2×C4C6C2C6C6C2C2
# reps111131136394121133

Matrix representation of C4⋊C4⋊D9 in GL4(𝔽37) generated by

302300
14700
00310
0006
,
6000
0600
00031
0060
,
312000
171100
0010
0001
,
312000
26600
0010
00036
G:=sub<GL(4,GF(37))| [30,14,0,0,23,7,0,0,0,0,31,0,0,0,0,6],[6,0,0,0,0,6,0,0,0,0,0,6,0,0,31,0],[31,17,0,0,20,11,0,0,0,0,1,0,0,0,0,1],[31,26,0,0,20,6,0,0,0,0,1,0,0,0,0,36] >;

C4⋊C4⋊D9 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes D_9
% in TeX

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

G:=SmallGroup(288,108);
// by ID

G=gap.SmallGroup(288,108);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,254,219,100,6725,292,9414]);
// Polycyclic

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

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