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G = C18.D8order 288 = 25·32

2nd non-split extension by C18 of D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D363C4, C18.7D8, C36.2D4, C4.10D36, C12.2D12, C18.7SD16, C4⋊C41D9, C4.2(C4×D9), C12.4(C4×S3), C36.4(C2×C4), C91(D4⋊C4), C2.2(D4⋊D9), (C2×D36).5C2, (C2×C12).40D6, (C2×C18).32D4, (C2×C4).38D18, C3.(C6.D8), C6.11(D6⋊C4), C2.6(D18⋊C4), C6.14(D4⋊S3), C18.4(C22⋊C4), (C2×C36).18C22, C6.8(Q82S3), C2.2(Q82D9), C22.15(C9⋊D4), (C2×C9⋊C8)⋊1C2, (C9×C4⋊C4)⋊1C2, (C3×C4⋊C4).4S3, (C2×C6).70(C3⋊D4), SmallGroup(288,17)

Series: Derived Chief Lower central Upper central

C1C36 — C18.D8
C1C3C9C18C2×C18C2×C36C2×D36 — C18.D8
C9C18C36 — C18.D8
C1C22C2×C4C4⋊C4

Generators and relations for C18.D8
 G = < a,b,c | a18=b8=c2=1, bab-1=cac=a-1, cbc=a9b-1 >

Subgroups: 448 in 75 conjugacy classes, 32 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], S3 [×2], C6 [×3], C8, C2×C4, C2×C4, D4 [×3], C23, C9, C12 [×2], C12, D6 [×4], C2×C6, C4⋊C4, C2×C8, C2×D4, D9 [×2], C18 [×3], C3⋊C8, D12 [×3], C2×C12, C2×C12, C22×S3, D4⋊C4, C36 [×2], C36, D18 [×4], C2×C18, C2×C3⋊C8, C3×C4⋊C4, C2×D12, C9⋊C8, D36 [×2], D36, C2×C36, C2×C36, C22×D9, C6.D8, C2×C9⋊C8, C9×C4⋊C4, C2×D36, C18.D8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D8, SD16, D9, C4×S3, D12, C3⋊D4, D4⋊C4, D18, D6⋊C4, D4⋊S3, Q82S3, C4×D9, D36, C9⋊D4, C6.D8, D18⋊C4, D4⋊D9, Q82D9, C18.D8

Smallest permutation representation of C18.D8
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 48 31 127 60 82 110 108)(2 47 32 144 61 81 111 107)(3 46 33 143 62 80 112 106)(4 45 34 142 63 79 113 105)(5 44 35 141 64 78 114 104)(6 43 36 140 65 77 115 103)(7 42 19 139 66 76 116 102)(8 41 20 138 67 75 117 101)(9 40 21 137 68 74 118 100)(10 39 22 136 69 73 119 99)(11 38 23 135 70 90 120 98)(12 37 24 134 71 89 121 97)(13 54 25 133 72 88 122 96)(14 53 26 132 55 87 123 95)(15 52 27 131 56 86 124 94)(16 51 28 130 57 85 125 93)(17 50 29 129 58 84 126 92)(18 49 30 128 59 83 109 91)
(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(19 122)(20 121)(21 120)(22 119)(23 118)(24 117)(25 116)(26 115)(27 114)(28 113)(29 112)(30 111)(31 110)(32 109)(33 126)(34 125)(35 124)(36 123)(37 92)(38 91)(39 108)(40 107)(41 106)(42 105)(43 104)(44 103)(45 102)(46 101)(47 100)(48 99)(49 98)(50 97)(51 96)(52 95)(53 94)(54 93)(55 65)(56 64)(57 63)(58 62)(59 61)(66 72)(67 71)(68 70)(73 127)(74 144)(75 143)(76 142)(77 141)(78 140)(79 139)(80 138)(81 137)(82 136)(83 135)(84 134)(85 133)(86 132)(87 131)(88 130)(89 129)(90 128)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D6A6B6C8A8B8C8D9A9B9C12A···12F18A···18I36A···36R
order12222234444666888899912···1218···1836···36
size1111363622244222181818182224···42···24···4

54 irreducible representations

dim11111222222222222224444
type+++++++++++++++++
imageC1C2C2C2C4S3D4D4D6D8SD16D9C4×S3D12C3⋊D4D18C4×D9D36C9⋊D4D4⋊S3Q82S3D4⋊D9Q82D9
kernelC18.D8C2×C9⋊C8C9×C4⋊C4C2×D36D36C3×C4⋊C4C36C2×C18C2×C12C18C18C4⋊C4C12C12C2×C6C2×C4C4C4C22C6C6C2C2
# reps11114111122322236661133

Matrix representation of C18.D8 in GL4(𝔽73) generated by

34500
283100
0010
0001
,
83400
266500
00048
003841
,
704200
45300
0010
004872
G:=sub<GL(4,GF(73))| [3,28,0,0,45,31,0,0,0,0,1,0,0,0,0,1],[8,26,0,0,34,65,0,0,0,0,0,38,0,0,48,41],[70,45,0,0,42,3,0,0,0,0,1,48,0,0,0,72] >;

C18.D8 in GAP, Magma, Sage, TeX

C_{18}.D_8
% in TeX

G:=Group("C18.D8");
// GroupNames label

G:=SmallGroup(288,17);
// by ID

G=gap.SmallGroup(288,17);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,675,346,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^18=b^8=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=a^9*b^-1>;
// generators/relations

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