Copied to
clipboard

G = C4○D4×C18order 288 = 25·32

Direct product of C18 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C4○D4×C18, C18.18C24, C36.51C23, (C2×D4)⋊7C18, D43(C2×C18), Q84(C2×C18), (C2×Q8)⋊8C18, (D4×C18)⋊16C2, (C22×C4)⋊8C18, (Q8×C18)⋊13C2, (C6×D4).28C6, (C6×Q8).29C6, (C22×C36)⋊13C2, (C2×C36)⋊16C22, (D4×C9)⋊12C22, C2.3(C23×C18), C4.8(C22×C18), C6.18(C23×C6), (C2×C18).6C23, (Q8×C9)⋊11C22, (C22×C12).33C6, C23.14(C2×C18), C12.52(C22×C6), C22.9(C22×C18), (C22×C18).30C22, C3.(C6×C4○D4), (C2×C4)⋊5(C2×C18), (C6×C4○D4).2C3, C6.47(C3×C4○D4), (C3×C4○D4).21C6, (C3×D4).20(C2×C6), (C2×C6).8(C22×C6), (C3×Q8).33(C2×C6), (C2×C12).156(C2×C6), (C22×C6).50(C2×C6), SmallGroup(288,370)

Series: Derived Chief Lower central Upper central

C1C2 — C4○D4×C18
C1C3C6C18C2×C18D4×C9C9×C4○D4 — C4○D4×C18
C1C2 — C4○D4×C18
C1C2×C36 — C4○D4×C18

Generators and relations for C4○D4×C18
 G = < a,b,c,d | a18=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 282 in 246 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×8], C22, C22 [×6], C22 [×6], C6, C6 [×2], C6 [×6], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C9, C12 [×8], C2×C6, C2×C6 [×6], C2×C6 [×6], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C18, C18 [×2], C18 [×6], C2×C12, C2×C12 [×15], C3×D4 [×12], C3×Q8 [×4], C22×C6 [×3], C2×C4○D4, C36 [×8], C2×C18, C2×C18 [×6], C2×C18 [×6], C22×C12 [×3], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], C2×C36, C2×C36 [×15], D4×C9 [×12], Q8×C9 [×4], C22×C18 [×3], C6×C4○D4, C22×C36 [×3], D4×C18 [×3], Q8×C18, C9×C4○D4 [×8], C4○D4×C18
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], C9, C2×C6 [×35], C4○D4 [×2], C24, C18 [×15], C22×C6 [×15], C2×C4○D4, C2×C18 [×35], C3×C4○D4 [×2], C23×C6, C22×C18 [×15], C6×C4○D4, C9×C4○D4 [×2], C23×C18, C4○D4×C18

Smallest permutation representation of C4○D4×C18
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)
(1 130 68 113)(2 131 69 114)(3 132 70 115)(4 133 71 116)(5 134 72 117)(6 135 55 118)(7 136 56 119)(8 137 57 120)(9 138 58 121)(10 139 59 122)(11 140 60 123)(12 141 61 124)(13 142 62 125)(14 143 63 126)(15 144 64 109)(16 127 65 110)(17 128 66 111)(18 129 67 112)(19 105 49 79)(20 106 50 80)(21 107 51 81)(22 108 52 82)(23 91 53 83)(24 92 54 84)(25 93 37 85)(26 94 38 86)(27 95 39 87)(28 96 40 88)(29 97 41 89)(30 98 42 90)(31 99 43 73)(32 100 44 74)(33 101 45 75)(34 102 46 76)(35 103 47 77)(36 104 48 78)
(1 139 68 122)(2 140 69 123)(3 141 70 124)(4 142 71 125)(5 143 72 126)(6 144 55 109)(7 127 56 110)(8 128 57 111)(9 129 58 112)(10 130 59 113)(11 131 60 114)(12 132 61 115)(13 133 62 116)(14 134 63 117)(15 135 64 118)(16 136 65 119)(17 137 66 120)(18 138 67 121)(19 88 49 96)(20 89 50 97)(21 90 51 98)(22 73 52 99)(23 74 53 100)(24 75 54 101)(25 76 37 102)(26 77 38 103)(27 78 39 104)(28 79 40 105)(29 80 41 106)(30 81 42 107)(31 82 43 108)(32 83 44 91)(33 84 45 92)(34 85 46 93)(35 86 47 94)(36 87 48 95)
(1 104)(2 105)(3 106)(4 107)(5 108)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 99)(15 100)(16 101)(17 102)(18 103)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 121)(27 122)(28 123)(29 124)(30 125)(31 126)(32 109)(33 110)(34 111)(35 112)(36 113)(37 137)(38 138)(39 139)(40 140)(41 141)(42 142)(43 143)(44 144)(45 127)(46 128)(47 129)(48 130)(49 131)(50 132)(51 133)(52 134)(53 135)(54 136)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(71 81)(72 82)

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

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

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

180 conjugacy classes

class 1 2A2B2C2D···2I3A3B4A4B4C4D4E···4J6A···6F6G···6R9A···9F12A···12H12I···12T18A···18R18S···18BB36A···36X36Y···36BH
order12222···23344444···46···66···69···912···1212···1218···1818···1836···3636···36
size11112···21111112···21···12···21···11···12···21···12···21···12···2

180 irreducible representations

dim111111111111111222
type+++++
imageC1C2C2C2C2C3C6C6C6C6C9C18C18C18C18C4○D4C3×C4○D4C9×C4○D4
kernelC4○D4×C18C22×C36D4×C18Q8×C18C9×C4○D4C6×C4○D4C22×C12C6×D4C6×Q8C3×C4○D4C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C18C6C2
# reps13318266216618186484824

Matrix representation of C4○D4×C18 in GL3(𝔽37) generated by

3600
0210
0021
,
100
0310
0031
,
100
060
01331
,
100
0320
0734
G:=sub<GL(3,GF(37))| [36,0,0,0,21,0,0,0,21],[1,0,0,0,31,0,0,0,31],[1,0,0,0,6,13,0,0,31],[1,0,0,0,3,7,0,20,34] >;

C4○D4×C18 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{18}
% in TeX

G:=Group("C4oD4xC18");
// GroupNames label

G:=SmallGroup(288,370);
// by ID

G=gap.SmallGroup(288,370);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,701,268,242]);
// Polycyclic

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

׿
×
𝔽