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G = C36.17D4order 288 = 25·32

17th non-split extension by C36 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C36.17D4, C23.12D18, (C6×D4).7S3, (C2×D4).6D9, (C4×Dic9)⋊5C2, (D4×C18).5C2, C18.49(C2×D4), (C2×C12).59D6, (C2×C4).51D18, C4.7(C9⋊D4), C93(C4.4D4), (C22×C6).49D6, (C2×Dic18)⋊10C2, C18.30(C4○D4), C18.D49C2, C12.12(C3⋊D4), (C2×C18).51C23, (C2×C36).37C22, C3.(C23.12D6), C6.87(D42S3), C2.16(D42D9), C22.58(C22×D9), (C22×C18).19C22, (C2×Dic9).16C22, C6.96(C2×C3⋊D4), C2.12(C2×C9⋊D4), (C2×C6).208(C22×S3), SmallGroup(288,146)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C36.17D4
C1C3C9C18C2×C18C2×Dic9C4×Dic9 — C36.17D4
C9C2×C18 — C36.17D4
C1C22C2×D4

Generators and relations for C36.17D4
 G = < a,b,c | a36=b4=1, c2=a18, bab-1=a17, cac-1=a-1, cbc-1=a18b-1 >

Subgroups: 404 in 114 conjugacy classes, 44 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C9, Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×6], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C18, C18 [×2], C18 [×2], Dic6 [×2], C2×Dic3 [×4], C2×C12, C3×D4 [×2], C22×C6 [×2], C4.4D4, Dic9 [×4], C36 [×2], C2×C18, C2×C18 [×6], C4×Dic3, C6.D4 [×4], C2×Dic6, C6×D4, Dic18 [×2], C2×Dic9 [×4], C2×C36, D4×C9 [×2], C22×C18 [×2], C23.12D6, C4×Dic9, C18.D4 [×4], C2×Dic18, D4×C18, C36.17D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], D9, C3⋊D4 [×2], C22×S3, C4.4D4, D18 [×3], D42S3 [×2], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C23.12D6, D42D9 [×2], C2×C9⋊D4, C36.17D4

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

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H6A6B6C6D6E6F6G9A9B9C12A12B18A···18I18J···18U36A···36F
order1222223444444446666666999121218···1818···1836···36
size1111442221818181836362224444222442···24···44···4

54 irreducible representations

dim11111222222222244
type++++++++++++--
imageC1C2C2C2C2S3D4D6D6C4○D4D9C3⋊D4D18D18C9⋊D4D42S3D42D9
kernelC36.17D4C4×Dic9C18.D4C2×Dic18D4×C18C6×D4C36C2×C12C22×C6C18C2×D4C12C2×C4C23C4C6C2
# reps114111212434361226

Matrix representation of C36.17D4 in GL4(𝔽37) generated by

03600
1000
002526
0003
,
0600
31000
0060
001231
,
31000
0600
00631
001231
G:=sub<GL(4,GF(37))| [0,1,0,0,36,0,0,0,0,0,25,0,0,0,26,3],[0,31,0,0,6,0,0,0,0,0,6,12,0,0,0,31],[31,0,0,0,0,6,0,0,0,0,6,12,0,0,31,31] >;

C36.17D4 in GAP, Magma, Sage, TeX

C_{36}._{17}D_4
% in TeX

G:=Group("C36.17D4");
// GroupNames label

G:=SmallGroup(288,146);
// by ID

G=gap.SmallGroup(288,146);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,135,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^36=b^4=1,c^2=a^18,b*a*b^-1=a^17,c*a*c^-1=a^-1,c*b*c^-1=a^18*b^-1>;
// generators/relations

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