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G = C23.26D18order 288 = 25·32

2nd non-split extension by C23 of D18 acting via D18/C18=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.26D18, (C2×C36)⋊6C4, (C2×C4)⋊4Dic9, C36.39(C2×C4), C4⋊Dic917C2, (C22×C4).9D9, C94(C42⋊C2), (C4×Dic9)⋊15C2, (C2×C12).374D6, (C2×C4).102D18, C4.15(C2×Dic9), C18.16(C4○D4), C6.86(C4○D12), C18.24(C22×C4), (C22×C12).27S3, (C22×C36).10C2, (C2×C18).44C23, C12.45(C2×Dic3), (C2×C12).18Dic3, (C22×C6).138D6, C2.4(D365C2), C2.5(C22×Dic9), C22.5(C2×Dic9), (C2×C36).112C22, C18.D4.5C2, C3.(C23.26D6), C6.25(C22×Dic3), C22.22(C22×D9), (C22×C18).36C22, (C2×Dic9).38C22, (C2×C18).35(C2×C4), (C2×C6).40(C2×Dic3), (C2×C6).201(C22×S3), SmallGroup(288,136)

Series: Derived Chief Lower central Upper central

C1C18 — C23.26D18
C1C3C9C18C2×C18C2×Dic9C4×Dic9 — C23.26D18
C9C18 — C23.26D18
C1C2×C4C22×C4

Generators and relations for C23.26D18
 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=d17 >

Subgroups: 328 in 114 conjugacy classes, 68 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C9, Dic3 [×4], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C18, C18 [×2], C18 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C22×C6, C42⋊C2, Dic9 [×4], C36 [×4], C2×C18, C2×C18 [×2], C2×C18 [×2], C4×Dic3 [×2], C4⋊Dic3 [×2], C6.D4 [×2], C22×C12, C2×Dic9 [×4], C2×C36 [×2], C2×C36 [×4], C22×C18, C23.26D6, C4×Dic9 [×2], C4⋊Dic9 [×2], C18.D4 [×2], C22×C36, C23.26D18
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], C22×C4, C4○D4 [×2], D9, C2×Dic3 [×6], C22×S3, C42⋊C2, Dic9 [×4], D18 [×3], C4○D12 [×2], C22×Dic3, C2×Dic9 [×6], C22×D9, C23.26D6, D365C2 [×2], C22×Dic9, C23.26D18

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

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G···4N6A···6G9A9B9C12A···12H18A···18U36A···36X
order12222234444444···46···699912···1218···1836···36
size111122211112218···182···22222···22···22···2

84 irreducible representations

dim11111122222222222
type++++++-+++-++
imageC1C2C2C2C2C4S3Dic3D6D6C4○D4D9Dic9D18D18C4○D12D365C2
kernelC23.26D18C4×Dic9C4⋊Dic9C18.D4C22×C36C2×C36C22×C12C2×C12C2×C12C22×C6C18C22×C4C2×C4C2×C4C23C6C2
# reps1222181421431263824

Matrix representation of C23.26D18 in GL4(𝔽37) generated by

1000
0100
0010
00036
,
36000
03600
0010
0001
,
1000
0100
00360
00036
,
33000
30900
00240
00020
,
282200
3900
00016
00300
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[33,30,0,0,0,9,0,0,0,0,24,0,0,0,0,20],[28,3,0,0,22,9,0,0,0,0,0,30,0,0,16,0] >;

C23.26D18 in GAP, Magma, Sage, TeX

C_2^3._{26}D_{18}
% in TeX

G:=Group("C2^3.26D18");
// GroupNames label

G:=SmallGroup(288,136);
// by ID

G=gap.SmallGroup(288,136);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,120,422,6725,292,9414]);
// 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=d^17>;
// generators/relations

׿
×
𝔽