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G = C23.9D18order 288 = 25·32

4th non-split extension by C23 of D18 acting via D18/C9=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D18.4D4, C23.9D18, C2.8(D4×D9), D18⋊C45C2, C4⋊Dic94C2, C22⋊C43D9, (C2×C12).3D6, C6.80(S3×D4), (C2×C4).26D18, C18.19(C2×D4), Dic9⋊C410C2, C18.8(C4○D4), (C2×C36).3C22, C3.(C23.9D6), (C22×C6).43D6, C6.78(C4○D12), C18.D44C2, C2.8(D42D9), (C2×C18).24C23, C91(C22.D4), C6.76(D42S3), C2.10(D365C2), (C2×Dic9).5C22, C22.42(C22×D9), (C22×C18).13C22, (C22×D9).17C22, (C2×C4×D9)⋊10C2, (C9×C22⋊C4)⋊5C2, (C2×C9⋊D4).3C2, (C3×C22⋊C4).7S3, (C2×C6).181(C22×S3), SmallGroup(288,93)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C23.9D18
C1C3C9C18C2×C18C22×D9C2×C4×D9 — C23.9D18
C9C2×C18 — C23.9D18
C1C22C22⋊C4

Generators and relations for C23.9D18
 G = < a,b,c,d,e | a2=b2=c2=1, d18=e2=b, ab=ba, dad-1=ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d17 >

Subgroups: 524 in 117 conjugacy classes, 40 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], S3 [×2], C6 [×3], C6, C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C9, Dic3 [×3], C12 [×2], D6 [×4], C2×C6, C2×C6 [×3], C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, D9 [×2], C18 [×3], C18, C4×S3 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×C6, C22.D4, Dic9 [×3], C36 [×2], D18 [×2], D18 [×2], C2×C18, C2×C18 [×3], Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, C4×D9 [×2], C2×Dic9 [×3], C9⋊D4 [×2], C2×C36 [×2], C22×D9, C22×C18, C23.9D6, Dic9⋊C4, C4⋊Dic9, D18⋊C4, C18.D4, C9×C22⋊C4, C2×C4×D9, C2×C9⋊D4, C23.9D18
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], D9, C22×S3, C22.D4, D18 [×3], C4○D12, S3×D4, D42S3, C22×D9, C23.9D6, D365C2, D4×D9, D42D9, C23.9D18

Smallest permutation representation of C23.9D18
On 144 points
Generators in S144
(1 79)(2 125)(3 81)(4 127)(5 83)(6 129)(7 85)(8 131)(9 87)(10 133)(11 89)(12 135)(13 91)(14 137)(15 93)(16 139)(17 95)(18 141)(19 97)(20 143)(21 99)(22 109)(23 101)(24 111)(25 103)(26 113)(27 105)(28 115)(29 107)(30 117)(31 73)(32 119)(33 75)(34 121)(35 77)(36 123)(37 78)(38 124)(39 80)(40 126)(41 82)(42 128)(43 84)(44 130)(45 86)(46 132)(47 88)(48 134)(49 90)(50 136)(51 92)(52 138)(53 94)(54 140)(55 96)(56 142)(57 98)(58 144)(59 100)(60 110)(61 102)(62 112)(63 104)(64 114)(65 106)(66 116)(67 108)(68 118)(69 74)(70 120)(71 76)(72 122)
(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 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)(121 139)(122 140)(123 141)(124 142)(125 143)(126 144)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 37)(73 118)(74 119)(75 120)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 127)(83 128)(84 129)(85 130)(86 131)(87 132)(88 133)(89 134)(90 135)(91 136)(92 137)(93 138)(94 139)(95 140)(96 141)(97 142)(98 143)(99 144)(100 109)(101 110)(102 111)(103 112)(104 113)(105 114)(106 115)(107 116)(108 117)
(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 18 19 36)(2 35 20 17)(3 16 21 34)(4 33 22 15)(5 14 23 32)(6 31 24 13)(7 12 25 30)(8 29 26 11)(9 10 27 28)(37 38 55 56)(39 72 57 54)(40 53 58 71)(41 70 59 52)(42 51 60 69)(43 68 61 50)(44 49 62 67)(45 66 63 48)(46 47 64 65)(73 84 91 102)(74 101 92 83)(75 82 93 100)(76 99 94 81)(77 80 95 98)(78 97 96 79)(85 108 103 90)(86 89 104 107)(87 106 105 88)(109 120 127 138)(110 137 128 119)(111 118 129 136)(112 135 130 117)(113 116 131 134)(114 133 132 115)(121 144 139 126)(122 125 140 143)(123 142 141 124)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G6A6B6C6D6E9A9B9C12A12B12C12D18A···18I18J···18O36A···36L
order122222234444444666669991212121218···1818···1836···36
size1111418182224181836362224422244442···24···44···4

54 irreducible representations

dim1111111122222222224444
type++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2S3D4D6D6C4○D4D9D18D18C4○D12D365C2S3×D4D42S3D4×D9D42D9
kernelC23.9D18Dic9⋊C4C4⋊Dic9D18⋊C4C18.D4C9×C22⋊C4C2×C4×D9C2×C9⋊D4C3×C22⋊C4D18C2×C12C22×C6C18C22⋊C4C2×C4C23C6C2C6C6C2C2
# reps11111111122143634121133

Matrix representation of C23.9D18 in GL6(𝔽37)

100000
010000
000100
001000
000001
000010
,
100000
010000
0036000
0003600
0000360
0000036
,
100000
010000
001000
000100
0000360
0000036
,
11200000
17310000
0031000
0003100
000060
0000031
,
17310000
11200000
0031000
000600
000060
000006

G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[11,17,0,0,0,0,20,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,6,0,0,0,0,0,0,31],[17,11,0,0,0,0,31,20,0,0,0,0,0,0,31,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,6] >;

C23.9D18 in GAP, Magma, Sage, TeX

C_2^3._9D_{18}
% in TeX

G:=Group("C2^3.9D18");
// GroupNames label

G:=SmallGroup(288,93);
// by ID

G=gap.SmallGroup(288,93);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,254,219,6725,292,9414]);
// Polycyclic

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

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