Copied to
clipboard

G = D4×C2×C18order 288 = 25·32

Direct product of C2×C18 and D4

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

Aliases: D4×C2×C18, C246C18, C364C23, C18.16C24, C4⋊(C22×C18), C6.80(C6×D4), C234(C2×C18), (C2×C18)⋊2C23, (C23×C18)⋊2C2, (C22×C4)⋊7C18, (C6×D4).27C6, (C2×C36)⋊15C22, (C22×C36)⋊12C2, C2.1(C23×C18), (C23×C6).10C6, C6.16(C23×C6), C12.50(C22×C6), (C22×C12).31C6, C222(C22×C18), (C22×C18)⋊6C22, C3.(D4×C2×C6), (D4×C2×C6).2C3, (C2×C4)⋊4(C2×C18), (C2×C6).69(C3×D4), (C3×D4).19(C2×C6), (C2×C6).7(C22×C6), (C2×C12).154(C2×C6), (C22×C6).49(C2×C6), SmallGroup(288,368)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C2×C18
C1C3C6C18C2×C18D4×C9D4×C18 — D4×C2×C18
C1C2 — D4×C2×C18
C1C22×C18 — D4×C2×C18

Generators and relations for D4×C2×C18
 G = < a,b,c,d | a2=b18=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 474 in 354 conjugacy classes, 234 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×15], C22 [×24], C6, C6 [×6], C6 [×8], C2×C4 [×6], D4 [×16], C23, C23 [×12], C23 [×8], C9, C12 [×4], C2×C6 [×15], C2×C6 [×24], C22×C4, C2×D4 [×12], C24 [×2], C18, C18 [×6], C18 [×8], C2×C12 [×6], C3×D4 [×16], C22×C6, C22×C6 [×12], C22×C6 [×8], C22×D4, C36 [×4], C2×C18 [×15], C2×C18 [×24], C22×C12, C6×D4 [×12], C23×C6 [×2], C2×C36 [×6], D4×C9 [×16], C22×C18, C22×C18 [×12], C22×C18 [×8], D4×C2×C6, C22×C36, D4×C18 [×12], C23×C18 [×2], D4×C2×C18
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], C23 [×15], C9, C2×C6 [×35], C2×D4 [×6], C24, C18 [×15], C3×D4 [×4], C22×C6 [×15], C22×D4, C2×C18 [×35], C6×D4 [×6], C23×C6, D4×C9 [×4], C22×C18 [×15], D4×C2×C6, D4×C18 [×6], C23×C18, D4×C2×C18

Smallest permutation representation of D4×C2×C18
On 144 points
Generators in S144
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 122)(20 123)(21 124)(22 125)(23 126)(24 109)(25 110)(26 111)(27 112)(28 113)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 120)(36 121)(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)(73 102)(74 103)(75 104)(76 105)(77 106)(78 107)(79 108)(80 91)(81 92)(82 93)(83 94)(84 95)(85 96)(86 97)(87 98)(88 99)(89 100)(90 101)
(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)
(1 142 110 106)(2 143 111 107)(3 144 112 108)(4 127 113 91)(5 128 114 92)(6 129 115 93)(7 130 116 94)(8 131 117 95)(9 132 118 96)(10 133 119 97)(11 134 120 98)(12 135 121 99)(13 136 122 100)(14 137 123 101)(15 138 124 102)(16 139 125 103)(17 140 126 104)(18 141 109 105)(19 89 61 54)(20 90 62 37)(21 73 63 38)(22 74 64 39)(23 75 65 40)(24 76 66 41)(25 77 67 42)(26 78 68 43)(27 79 69 44)(28 80 70 45)(29 81 71 46)(30 82 72 47)(31 83 55 48)(32 84 56 49)(33 85 57 50)(34 86 58 51)(35 87 59 52)(36 88 60 53)
(1 34)(2 35)(3 36)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(37 128)(38 129)(39 130)(40 131)(41 132)(42 133)(43 134)(44 135)(45 136)(46 137)(47 138)(48 139)(49 140)(50 141)(51 142)(52 143)(53 144)(54 127)(55 125)(56 126)(57 109)(58 110)(59 111)(60 112)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 121)(70 122)(71 123)(72 124)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)(79 99)(80 100)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 91)(90 92)

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

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

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

180 conjugacy classes

class 1 2A···2G2H···2O3A3B4A4B4C4D6A···6N6O···6AD9A···9F12A···12H18A···18AP18AQ···18CL36A···36X
order12···22···23344446···66···69···912···1218···1818···1836···36
size11···12···21122221···12···21···12···21···12···22···2

180 irreducible representations

dim111111111111222
type+++++
imageC1C2C2C2C3C6C6C6C9C18C18C18D4C3×D4D4×C9
kernelD4×C2×C18C22×C36D4×C18C23×C18D4×C2×C6C22×C12C6×D4C23×C6C22×D4C22×C4C2×D4C24C2×C18C2×C6C22
# reps11122222446672124824

Matrix representation of D4×C2×C18 in GL4(𝔽37) generated by

36000
03600
0010
0001
,
11000
01000
00330
00033
,
36000
03600
00362
00361
,
36000
0100
00360
00361
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[11,0,0,0,0,10,0,0,0,0,33,0,0,0,0,33],[36,0,0,0,0,36,0,0,0,0,36,36,0,0,2,1],[36,0,0,0,0,1,0,0,0,0,36,36,0,0,0,1] >;

D4×C2×C18 in GAP, Magma, Sage, TeX

D_4\times C_2\times C_{18}
% in TeX

G:=Group("D4xC2xC18");
// GroupNames label

G:=SmallGroup(288,368);
// by ID

G=gap.SmallGroup(288,368);
# by ID

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

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

׿
×
𝔽