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G = C2×D4.D9order 288 = 25·32

Direct product of C2 and D4.D9

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

Aliases: C2×D4.D9, D4.6D18, C182SD16, C36.14D4, C36.11C23, Dic185C22, C9⋊C87C22, C93(C2×SD16), (C6×D4).3S3, (C2×D4).4D9, (C3×D4).29D6, (D4×C18).3C2, (C2×C4).48D18, (C2×C12).55D6, (C2×C18).38D4, C18.45(C2×D4), C4.5(C9⋊D4), (C2×Dic18)⋊9C2, C12.9(C3⋊D4), (D4×C9).6C22, C4.11(C22×D9), C12.50(C22×S3), C6.10(D4.S3), (C2×C36).33C22, C22.21(C9⋊D4), (C2×C9⋊C8)⋊4C2, C3.(C2×D4.S3), C2.8(C2×C9⋊D4), C6.92(C2×C3⋊D4), (C2×C6).77(C3⋊D4), SmallGroup(288,141)

Series: Derived Chief Lower central Upper central

C1C36 — C2×D4.D9
C1C3C9C18C36Dic18C2×Dic18 — C2×D4.D9
C9C18C36 — C2×D4.D9
C1C22C2×C4C2×D4

Generators and relations for C2×D4.D9
 G = < a,b,c,d,e | a2=b4=c2=d9=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 352 in 102 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C9, Dic3, C12, C2×C6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C18, C18, C18, C3⋊C8, Dic6, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, C2×SD16, Dic9, C36, C2×C18, C2×C18, C2×C3⋊C8, D4.S3, C2×Dic6, C6×D4, C9⋊C8, Dic18, Dic18, C2×Dic9, C2×C36, D4×C9, D4×C9, C22×C18, C2×D4.S3, C2×C9⋊C8, D4.D9, C2×Dic18, D4×C18, C2×D4.D9
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, D9, C3⋊D4, C22×S3, C2×SD16, D18, D4.S3, C2×C3⋊D4, C9⋊D4, C22×D9, C2×D4.S3, D4.D9, C2×C9⋊D4, C2×D4.D9

Smallest permutation representation of C2×D4.D9
On 144 points
Generators in S144
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(91 136)(92 137)(93 138)(94 139)(95 140)(96 141)(97 142)(98 143)(99 144)(100 127)(101 128)(102 129)(103 130)(104 131)(105 132)(106 133)(107 134)(108 135)
(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 91 82 100)(74 92 83 101)(75 93 84 102)(76 94 85 103)(77 95 86 104)(78 96 87 105)(79 97 88 106)(80 98 89 107)(81 99 90 108)(109 136 118 127)(110 137 119 128)(111 138 120 129)(112 139 121 130)(113 140 122 131)(114 141 123 132)(115 142 124 133)(116 143 125 134)(117 144 126 135)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(109 118)(110 119)(111 120)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)
(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)
(1 124 10 115)(2 123 11 114)(3 122 12 113)(4 121 13 112)(5 120 14 111)(6 119 15 110)(7 118 16 109)(8 126 17 117)(9 125 18 116)(19 133 28 142)(20 132 29 141)(21 131 30 140)(22 130 31 139)(23 129 32 138)(24 128 33 137)(25 127 34 136)(26 135 35 144)(27 134 36 143)(37 88 46 79)(38 87 47 78)(39 86 48 77)(40 85 49 76)(41 84 50 75)(42 83 51 74)(43 82 52 73)(44 90 53 81)(45 89 54 80)(55 97 64 106)(56 96 65 105)(57 95 66 104)(58 94 67 103)(59 93 68 102)(60 92 69 101)(61 91 70 100)(62 99 71 108)(63 98 72 107)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D6A6B6C6D6E6F6G8A8B8C8D9A9B9C12A12B18A···18I18J···18U36A···36F
order1222223444466666668888999121218···1818···1836···36
size1111442223636222444418181818222442···24···44···4

54 irreducible representations

dim11111222222222222244
type+++++++++++++--
imageC1C2C2C2C2S3D4D4D6D6SD16D9C3⋊D4C3⋊D4D18D18C9⋊D4C9⋊D4D4.S3D4.D9
kernelC2×D4.D9C2×C9⋊C8D4.D9C2×Dic18D4×C18C6×D4C36C2×C18C2×C12C3×D4C18C2×D4C12C2×C6C2×C4D4C4C22C6C2
# reps11411111124322366626

Matrix representation of C2×D4.D9 in GL5(𝔽73)

720000
01000
00100
00010
00001
,
10000
00100
072000
000720
000072
,
720000
00100
01000
000720
000501
,
10000
01000
00100
00020
0007237
,
720000
066700
0676700
000251
0003348

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,72,50,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,72,0,0,0,0,37],[72,0,0,0,0,0,6,67,0,0,0,67,67,0,0,0,0,0,25,33,0,0,0,1,48] >;

C2×D4.D9 in GAP, Magma, Sage, TeX

C_2\times D_4.D_9
% in TeX

G:=Group("C2xD4.D9");
// GroupNames label

G:=SmallGroup(288,141);
// by ID

G=gap.SmallGroup(288,141);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,675,185,80,6725,292,9414]);
// Polycyclic

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

׿
×
𝔽