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G = C23.28D18order 288 = 25·32

4th non-split extension by C23 of D18 acting via D18/C18=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.28D18, D18⋊C42C2, (C22×C4)⋊5D9, Dic9⋊C43C2, (C22×C36)⋊2C2, (C2×C4).69D18, C18.43(C2×D4), (C2×C18).37D4, (C2×C12).345D6, C18.18(C4○D4), C6.88(C4○D12), C18.D46C2, (C2×C18).47C23, (C22×C12).12S3, (C2×C36).77C22, (C22×C6).141D6, C94(C22.D4), C22.9(C9⋊D4), C3.(C23.28D6), C2.18(D365C2), (C22×D9).9C22, C22.55(C22×D9), (C22×C18).39C22, (C2×Dic9).13C22, C2.6(C2×C9⋊D4), (C2×C9⋊D4).6C2, C6.90(C2×C3⋊D4), (C2×C6).76(C3⋊D4), (C2×C6).204(C22×S3), SmallGroup(288,139)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C23.28D18
C1C3C9C18C2×C18C22×D9C2×C9⋊D4 — C23.28D18
C9C2×C18 — C23.28D18
C1C22C22×C4

Generators and relations for C23.28D18
 G = < a,b,c,d,e | a2=b2=c2=1, d18=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd17 >

Subgroups: 484 in 117 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], S3, C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C9, Dic3 [×3], C12 [×2], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, D9, C18, C18 [×2], C18 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×2], C22×S3, C22×C6, C22.D4, Dic9 [×3], C36 [×2], D18 [×3], C2×C18, C2×C18 [×2], C2×C18 [×2], Dic3⋊C4 [×2], D6⋊C4 [×2], C6.D4, C2×C3⋊D4, C22×C12, C2×Dic9, C2×Dic9 [×2], C9⋊D4 [×2], C2×C36 [×2], C2×C36 [×2], C22×D9, C22×C18, C23.28D6, Dic9⋊C4 [×2], D18⋊C4 [×2], C18.D4, C2×C9⋊D4, C22×C36, C23.28D18
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], C4○D12 [×2], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C23.28D6, D365C2 [×2], C2×C9⋊D4, C23.28D18

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

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

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

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

78 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G6A···6G9A9B9C12A···12H18A···18U36A···36X
order1222222344444446···699912···1218···1836···36
size11112236222223636362···22222···22···22···2

78 irreducible representations

dim111111222222222222
type+++++++++++++
imageC1C2C2C2C2C2S3D4D6D6C4○D4D9C3⋊D4D18D18C4○D12C9⋊D4D365C2
kernelC23.28D18Dic9⋊C4D18⋊C4C18.D4C2×C9⋊D4C22×C36C22×C12C2×C18C2×C12C22×C6C18C22×C4C2×C6C2×C4C23C6C22C2
# reps12211112214346381224

Matrix representation of C23.28D18 in GL4(𝔽37) generated by

36000
03600
003014
00237
,
36000
03600
00360
00036
,
1000
0100
00360
00036
,
303000
72300
002933
00425
,
301400
7700
00254
002912
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,30,23,0,0,14,7],[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],[30,7,0,0,30,23,0,0,0,0,29,4,0,0,33,25],[30,7,0,0,14,7,0,0,0,0,25,29,0,0,4,12] >;

C23.28D18 in GAP, Magma, Sage, TeX

C_2^3._{28}D_{18}
% in TeX

G:=Group("C2^3.28D18");
// GroupNames label

G:=SmallGroup(288,139);
// by ID

G=gap.SmallGroup(288,139);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,100,6725,292,9414]);
// Polycyclic

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

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