Copied to
clipboard

G = C23.16D18order 288 = 25·32

1st non-split extension by C23 of D18 acting via D18/D9=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.16D18, Dic9⋊C47C2, (C2×Dic9)⋊3C4, (C4×Dic9)⋊9C2, (C2×C4).24D18, C22⋊C4.3D9, C22.6(C4×D9), C92(C42⋊C2), (C2×C12).174D6, C18.5(C22×C4), Dic9.6(C2×C4), (C22×C6).37D6, C18.20(C4○D4), C2.1(D42D9), (C2×C36).53C22, (C2×C18).18C23, C6.72(D42S3), C18.D4.1C2, (C22×C18).7C22, C3.(C23.16D6), (C22×Dic9).2C2, C22.12(C22×D9), (C2×Dic9).23C22, C2.7(C2×C4×D9), C6.44(S3×C2×C4), (C2×C6).5(C4×S3), (C2×C18).4(C2×C4), (C9×C22⋊C4).3C2, (C3×C22⋊C4).11S3, (C2×C6).175(C22×S3), SmallGroup(288,87)

Series: Derived Chief Lower central Upper central

C1C18 — C23.16D18
C1C3C9C18C2×C18C2×Dic9C22×Dic9 — C23.16D18
C9C18 — C23.16D18
C1C22C22⋊C4

Generators and relations for C23.16D18
 G = < a,b,c,d,e | a2=b2=c2=1, d18=b, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d17 >

Subgroups: 368 in 114 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×8], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C23, C9, Dic3 [×6], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4, C22⋊C4, C4⋊C4 [×2], C22×C4, C18, C18 [×2], C18 [×2], C2×Dic3 [×8], C2×C12 [×2], C22×C6, C42⋊C2, Dic9 [×4], Dic9 [×2], C36 [×2], C2×C18, C2×C18 [×2], C2×C18 [×2], C4×Dic3 [×2], Dic3⋊C4 [×2], C6.D4, C3×C22⋊C4, C22×Dic3, C2×Dic9 [×2], C2×Dic9 [×6], C2×C36 [×2], C22×C18, C23.16D6, C4×Dic9 [×2], Dic9⋊C4 [×2], C18.D4, C9×C22⋊C4, C22×Dic9, C23.16D18
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, C4○D4 [×2], D9, C4×S3 [×2], C22×S3, C42⋊C2, D18 [×3], S3×C2×C4, D42S3 [×2], C4×D9 [×2], C22×D9, C23.16D6, C2×C4×D9, D42D9 [×2], C23.16D18

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

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I···4N6A6B6C6D6E9A9B9C12A12B12C12D18A···18I18J···18O36A···36L
order1222223444444444···4666669991212121218···1818···1836···36
size11112222222999918···182224422244442···24···44···4

60 irreducible representations

dim111111122222222244
type++++++++++++--
imageC1C2C2C2C2C2C4S3D6D6C4○D4D9C4×S3D18D18C4×D9D42S3D42D9
kernelC23.16D18C4×Dic9Dic9⋊C4C18.D4C9×C22⋊C4C22×Dic9C2×Dic9C3×C22⋊C4C2×C12C22×C6C18C22⋊C4C2×C6C2×C4C23C22C6C2
# reps1221118121434631226

Matrix representation of C23.16D18 in GL5(𝔽37)

360000
01000
00100
00011
000036
,
360000
01000
00100
00010
00001
,
10000
01000
00100
000360
000036
,
60000
0112000
0173100
00010
0003536
,
60000
0123300
082500
000310
000126

G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1,36],[36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36],[6,0,0,0,0,0,11,17,0,0,0,20,31,0,0,0,0,0,1,35,0,0,0,0,36],[6,0,0,0,0,0,12,8,0,0,0,33,25,0,0,0,0,0,31,12,0,0,0,0,6] >;

C23.16D18 in GAP, Magma, Sage, TeX

C_2^3._{16}D_{18}
% in TeX

G:=Group("C2^3.16D18");
// GroupNames label

G:=SmallGroup(288,87);
// by ID

G=gap.SmallGroup(288,87);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,219,58,6725,292,9414]);
// Polycyclic

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

׿
×
𝔽