Copied to
clipboard

G = C2×Q82D9order 288 = 25·32

Direct product of C2 and Q82D9

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Q82D9, Q84D18, C183SD16, C36.19D4, C36.15C23, D36.9C22, C9⋊C89C22, (C2×Q8)⋊3D9, C94(C2×SD16), (Q8×C18)⋊1C2, (C2×D36).8C2, (C2×C12).63D6, (C2×C18).42D4, (C2×C4).55D18, C18.54(C2×D4), C4.9(C9⋊D4), (C6×Q8).12S3, (C3×Q8).54D6, (Q8×C9)⋊3C22, C4.15(C22×D9), C12.16(C3⋊D4), C12.54(C22×S3), (C2×C36).41C22, C6.10(Q82S3), C22.24(C9⋊D4), (C2×C9⋊C8)⋊6C2, C3.(C2×Q82S3), C2.18(C2×C9⋊D4), C6.102(C2×C3⋊D4), (C2×C6).81(C3⋊D4), SmallGroup(288,152)

Series: Derived Chief Lower central Upper central

C1C36 — C2×Q82D9
C1C3C9C18C36D36C2×D36 — C2×Q82D9
C9C18C36 — C2×Q82D9
C1C22C2×C4C2×Q8

Generators and relations for C2×Q82D9
 G = < a,b,c,d,e | a2=b4=d9=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

Subgroups: 512 in 102 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×2], Q8, C23, C9, C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C8, SD16 [×4], C2×D4, C2×Q8, D9 [×2], C18, C18 [×2], C3⋊C8 [×2], D12 [×3], C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, C22×S3, C2×SD16, C36 [×2], C36 [×2], D18 [×4], C2×C18, C2×C3⋊C8, Q82S3 [×4], C2×D12, C6×Q8, C9⋊C8 [×2], D36 [×2], D36, C2×C36, C2×C36, Q8×C9 [×2], Q8×C9, C22×D9, C2×Q82S3, C2×C9⋊C8, Q82D9 [×4], C2×D36, Q8×C18, C2×Q82D9
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], C2×D4, D9, C3⋊D4 [×2], C22×S3, C2×SD16, D18 [×3], Q82S3 [×2], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C2×Q82S3, Q82D9 [×2], C2×C9⋊D4, C2×Q82D9

Smallest permutation representation of C2×Q82D9
On 144 points
Generators in S144
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(1 28 10 19)(2 29 11 20)(3 30 12 21)(4 31 13 22)(5 32 14 23)(6 33 15 24)(7 34 16 25)(8 35 17 26)(9 36 18 27)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)(73 100 82 91)(74 101 83 92)(75 102 84 93)(76 103 85 94)(77 104 86 95)(78 105 87 96)(79 106 88 97)(80 107 89 98)(81 108 90 99)(109 127 118 136)(110 128 119 137)(111 129 120 138)(112 130 121 139)(113 131 122 140)(114 132 123 141)(115 133 124 142)(116 134 125 143)(117 135 126 144)
(1 118 10 109)(2 119 11 110)(3 120 12 111)(4 121 13 112)(5 122 14 113)(6 123 15 114)(7 124 16 115)(8 125 17 116)(9 126 18 117)(19 136 28 127)(20 137 29 128)(21 138 30 129)(22 139 31 130)(23 140 32 131)(24 141 33 132)(25 142 34 133)(26 143 35 134)(27 144 36 135)(37 73 46 82)(38 74 47 83)(39 75 48 84)(40 76 49 85)(41 77 50 86)(42 78 51 87)(43 79 52 88)(44 80 53 89)(45 81 54 90)(55 91 64 100)(56 92 65 101)(57 93 66 102)(58 94 67 103)(59 95 68 104)(60 96 69 105)(61 97 70 106)(62 98 71 107)(63 99 72 108)
(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 81)(2 80)(3 79)(4 78)(5 77)(6 76)(7 75)(8 74)(9 73)(10 90)(11 89)(12 88)(13 87)(14 86)(15 85)(16 84)(17 83)(18 82)(19 108)(20 107)(21 106)(22 105)(23 104)(24 103)(25 102)(26 101)(27 100)(28 99)(29 98)(30 97)(31 96)(32 95)(33 94)(34 93)(35 92)(36 91)(37 135)(38 134)(39 133)(40 132)(41 131)(42 130)(43 129)(44 128)(45 127)(46 144)(47 143)(48 142)(49 141)(50 140)(51 139)(52 138)(53 137)(54 136)(55 117)(56 116)(57 115)(58 114)(59 113)(60 112)(61 111)(62 110)(63 109)(64 126)(65 125)(66 124)(67 123)(68 122)(69 121)(70 120)(71 119)(72 118)

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

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

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

54 conjugacy classes

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

54 irreducible representations

dim11111222222222222244
type+++++++++++++++
imageC1C2C2C2C2S3D4D4D6D6SD16D9C3⋊D4C3⋊D4D18D18C9⋊D4C9⋊D4Q82S3Q82D9
kernelC2×Q82D9C2×C9⋊C8Q82D9C2×D36Q8×C18C6×Q8C36C2×C18C2×C12C3×Q8C18C2×Q8C12C2×C6C2×C4Q8C4C22C6C2
# reps11411111124322366626

Matrix representation of C2×Q82D9 in GL6(𝔽73)

100000
010000
0072000
0007200
000010
000001
,
7200000
0720000
0072000
0007200
000001
0000720
,
30600000
13430000
0007100
0036000
000066
0000667
,
45420000
3130000
001000
000100
000010
000001
,
45420000
70280000
0072000
000100
000010
0000072

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[30,13,0,0,0,0,60,43,0,0,0,0,0,0,0,36,0,0,0,0,71,0,0,0,0,0,0,0,6,6,0,0,0,0,6,67],[45,31,0,0,0,0,42,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[45,70,0,0,0,0,42,28,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;

C2×Q82D9 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_2D_9
% in TeX

G:=Group("C2xQ8:2D9");
// GroupNames label

G:=SmallGroup(288,152);
// by ID

G=gap.SmallGroup(288,152);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,100,675,185,80,6725,292,9414]);
// Polycyclic

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

׿
×
𝔽