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G = C23.8D18order 288 = 25·32

3rd non-split extension by C23 of D18 acting via D18/C9=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.8D18, C4⋊Dic93C2, (C2×C4).6D18, Dic9⋊C48C2, C22⋊C4.2D9, C92(C422C2), (C4×Dic9)⋊10C2, C18.7(C4○D4), (C2×C12).175D6, (C2×C36).2C22, (C22×C6).39D6, C6.77(C4○D12), C2.7(D42D9), (C2×C18).20C23, C3.(C23.8D6), C6.74(D42S3), C2.9(D365C2), C18.D4.3C2, (C22×C18).9C22, C22.40(C22×D9), (C2×Dic9).25C22, (C9×C22⋊C4).2C2, (C3×C22⋊C4).6S3, (C2×C6).177(C22×S3), SmallGroup(288,89)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C23.8D18
C1C3C9C18C2×C18C2×Dic9C4×Dic9 — C23.8D18
C9C2×C18 — C23.8D18
C1C22C22⋊C4

Generators and relations for C23.8D18
 G = < a,b,c,d,e | a2=b2=c2=1, d18=b, e2=cb=bc, eae-1=ab=ba, dad-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d17 >

Subgroups: 316 in 90 conjugacy classes, 38 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], C23, C9, Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×3], C42, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×3], C18 [×3], C18, C2×Dic3 [×4], C2×C12 [×2], C22×C6, C422C2, Dic9 [×4], C36 [×2], C2×C18, C2×C18 [×3], C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C3×C22⋊C4, C2×Dic9 [×4], C2×C36 [×2], C22×C18, C23.8D6, C4×Dic9, Dic9⋊C4 [×2], C4⋊Dic9, C18.D4 [×2], C9×C22⋊C4, C23.8D18
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], C4○D4 [×3], D9, C22×S3, C422C2, D18 [×3], C4○D12, D42S3 [×2], C22×D9, C23.8D6, D365C2, D42D9 [×2], C23.8D18

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

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

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

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

54 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E9A9B9C12A12B12C12D18A···18I18J···18O36A···36L
order122223444444444666669991212121218···1818···1836···36
size1111422241818181836362224422244442···24···44···4

54 irreducible representations

dim11111122222222244
type++++++++++++--
imageC1C2C2C2C2C2S3D6D6C4○D4D9D18D18C4○D12D365C2D42S3D42D9
kernelC23.8D18C4×Dic9Dic9⋊C4C4⋊Dic9C18.D4C9×C22⋊C4C3×C22⋊C4C2×C12C22×C6C18C22⋊C4C2×C4C23C6C2C6C2
# reps112121121636341226

Matrix representation of C23.8D18 in GL4(𝔽37) generated by

1400
03600
0010
001136
,
36000
03600
00360
00036
,
1000
0100
00360
00036
,
132900
01700
003128
0006
,
121800
312500
003617
0001
G:=sub<GL(4,GF(37))| [1,0,0,0,4,36,0,0,0,0,1,11,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[13,0,0,0,29,17,0,0,0,0,31,0,0,0,28,6],[12,31,0,0,18,25,0,0,0,0,36,0,0,0,17,1] >;

C23.8D18 in GAP, Magma, Sage, TeX

C_2^3._8D_{18}
% in TeX

G:=Group("C2^3.8D18");
// GroupNames label

G:=SmallGroup(288,89);
// by ID

G=gap.SmallGroup(288,89);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,219,6725,292,9414]);
// Polycyclic

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

׿
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