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G = C23.23D18order 288 = 25·32

8th non-split extension by C23 of D18 acting via D18/D9=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.23D18, (C2×D4).5D9, (C2×C18).7D4, (C6×D4).19S3, (C2×C4).15D18, C18.48(C2×D4), Dic9⋊C414C2, (D4×C18).10C2, (C2×C12).215D6, (C22×C6).48D6, C18.29(C4○D4), C18.D48C2, (C2×C18).50C23, (C2×C36).60C22, (C22×Dic9)⋊5C2, C22.4(C9⋊D4), C95(C22.D4), C6.86(D42S3), C2.15(D42D9), C3.(C23.23D6), C22.57(C22×D9), (C22×C18).18C22, (C2×Dic9).15C22, C2.11(C2×C9⋊D4), C6.95(C2×C3⋊D4), (C2×C6).4(C3⋊D4), (C2×C6).207(C22×S3), SmallGroup(288,145)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C23.23D18
C1C3C9C18C2×C18C2×Dic9C22×Dic9 — C23.23D18
C9C2×C18 — C23.23D18
C1C22C2×D4

Generators and relations for C23.23D18
 G = < a,b,c,d,e | a2=b2=c2=d18=1, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 404 in 117 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], C6, C6 [×2], C6 [×3], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C9, Dic3 [×4], C12, C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C18, C18 [×2], C18 [×3], C2×Dic3 [×6], C2×C12, C3×D4 [×2], C22×C6 [×2], C22.D4, Dic9 [×4], C36, C2×C18, C2×C18 [×2], C2×C18 [×5], Dic3⋊C4 [×2], C6.D4 [×3], C22×Dic3, C6×D4, C2×Dic9 [×4], C2×Dic9 [×2], C2×C36, D4×C9 [×2], C22×C18 [×2], C23.23D6, Dic9⋊C4 [×2], C18.D4, C18.D4 [×2], C22×Dic9, D4×C18, C23.23D18
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], D9, C3⋊D4 [×2], C22×S3, C22.D4, D18 [×3], D42S3 [×2], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C23.23D6, D42D9 [×2], C2×C9⋊D4, C23.23D18

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

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G6A6B6C6D6E6F6G9A9B9C12A12B18A···18I18J···18U36A···36F
order1222222344444446666666999121218···1818···1836···36
size1111224241818181836362224444222442···24···44···4

54 irreducible representations

dim11111222222222244
type++++++++++++--
imageC1C2C2C2C2S3D4D6D6C4○D4D9C3⋊D4D18D18C9⋊D4D42S3D42D9
kernelC23.23D18Dic9⋊C4C18.D4C22×Dic9D4×C18C6×D4C2×C18C2×C12C22×C6C18C2×D4C2×C6C2×C4C23C22C6C2
# reps123111212434361226

Matrix representation of C23.23D18 in GL6(𝔽37)

3600000
0360000
001000
000100
000013
0000036
,
3600000
0360000
001000
000100
000010
000001
,
100000
010000
001000
000100
0000360
0000036
,
100000
9360000
0026600
00312000
000010
00002436
,
36290000
2810000
00322700
0032500
0000310
000046

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,3,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,9,0,0,0,0,0,36,0,0,0,0,0,0,26,31,0,0,0,0,6,20,0,0,0,0,0,0,1,24,0,0,0,0,0,36],[36,28,0,0,0,0,29,1,0,0,0,0,0,0,32,32,0,0,0,0,27,5,0,0,0,0,0,0,31,4,0,0,0,0,0,6] >;

C23.23D18 in GAP, Magma, Sage, TeX

C_2^3._{23}D_{18}
% in TeX

G:=Group("C2^3.23D18");
// GroupNames label

G:=SmallGroup(288,145);
// by ID

G=gap.SmallGroup(288,145);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,219,6725,292,9414]);
// Polycyclic

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

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