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G = M4(2)×C18order 288 = 25·32

Direct product of C18 and M4(2)

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: M4(2)×C18, C7214C22, C23.4C36, C36.54C23, C84(C2×C18), (C2×C8)⋊6C18, (C2×C72)⋊14C2, (C2×C4).6C36, C4.9(C2×C36), C3.(C6×M4(2)), (C2×C36).15C4, (C2×C24).30C6, C36.47(C2×C4), C24.38(C2×C6), (C6×M4(2)).C3, C12.54(C2×C12), (C2×C12).22C12, C22.6(C2×C36), (C22×C4).8C18, (C22×C18).4C4, C2.6(C22×C36), (C22×C36).16C2, C18.34(C22×C4), (C22×C12).30C6, C12.65(C22×C6), (C22×C6).11C12, C4.11(C22×C18), C6.34(C22×C12), C6.10(C3×M4(2)), (C2×C36).136C22, (C3×M4(2)).11C6, (C2×C18).23(C2×C4), (C2×C4).24(C2×C18), (C2×C6).28(C2×C12), (C2×C12).142(C2×C6), SmallGroup(288,180)

Series: Derived Chief Lower central Upper central

C1C2 — M4(2)×C18
C1C2C6C12C36C72C9×M4(2) — M4(2)×C18
C1C2 — M4(2)×C18
C1C2×C36 — M4(2)×C18

Generators and relations for M4(2)×C18
 G = < a,b,c | a18=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Subgroups: 114 in 102 conjugacy classes, 90 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C9, C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C18, C18 [×2], C18 [×2], C24 [×4], C2×C12 [×2], C2×C12 [×4], C22×C6, C2×M4(2), C36 [×2], C36 [×2], C2×C18, C2×C18 [×2], C2×C18 [×2], C2×C24 [×2], C3×M4(2) [×4], C22×C12, C72 [×4], C2×C36 [×2], C2×C36 [×4], C22×C18, C6×M4(2), C2×C72 [×2], C9×M4(2) [×4], C22×C36, M4(2)×C18
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, C9, C12 [×4], C2×C6 [×7], M4(2) [×2], C22×C4, C18 [×7], C2×C12 [×6], C22×C6, C2×M4(2), C36 [×4], C2×C18 [×7], C3×M4(2) [×2], C22×C12, C2×C36 [×6], C22×C18, C6×M4(2), C9×M4(2) [×2], C22×C36, M4(2)×C18

Smallest permutation representation of M4(2)×C18
On 144 points
Generators in S144
(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)
(1 128 106 76 39 118 64 29)(2 129 107 77 40 119 65 30)(3 130 108 78 41 120 66 31)(4 131 91 79 42 121 67 32)(5 132 92 80 43 122 68 33)(6 133 93 81 44 123 69 34)(7 134 94 82 45 124 70 35)(8 135 95 83 46 125 71 36)(9 136 96 84 47 126 72 19)(10 137 97 85 48 109 55 20)(11 138 98 86 49 110 56 21)(12 139 99 87 50 111 57 22)(13 140 100 88 51 112 58 23)(14 141 101 89 52 113 59 24)(15 142 102 90 53 114 60 25)(16 143 103 73 54 115 61 26)(17 144 104 74 37 116 62 27)(18 127 105 75 38 117 63 28)
(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 81)(35 82)(36 83)(109 137)(110 138)(111 139)(112 140)(113 141)(114 142)(115 143)(116 144)(117 127)(118 128)(119 129)(120 130)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)

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

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

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

180 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F6A···6F6G6H6I6J8A···8H9A···9F12A···12H12I12J12K12L18A···18R18S···18AD24A···24P36A···36X36Y···36AJ72A···72AV
order122222334444446···666668···89···912···121212121218···1818···1824···2436···3636···3672···72
size111122111111221···122222···21···11···122221···12···22···21···12···22···2

180 irreducible representations

dim111111111111111111222
type++++
imageC1C2C2C2C3C4C4C6C6C6C9C12C12C18C18C18C36C36M4(2)C3×M4(2)C9×M4(2)
kernelM4(2)×C18C2×C72C9×M4(2)C22×C36C6×M4(2)C2×C36C22×C18C2×C24C3×M4(2)C22×C12C2×M4(2)C2×C12C22×C6C2×C8M4(2)C22×C4C2×C4C23C18C6C2
# reps124126248261241224636124824

Matrix representation of M4(2)×C18 in GL3(𝔽73) generated by

7200
040
004
,
7200
002
0230
,
7200
010
0072
G:=sub<GL(3,GF(73))| [72,0,0,0,4,0,0,0,4],[72,0,0,0,0,23,0,2,0],[72,0,0,0,1,0,0,0,72] >;

M4(2)×C18 in GAP, Magma, Sage, TeX

M_4(2)\times C_{18}
% in TeX

G:=Group("M4(2)xC18");
// GroupNames label

G:=SmallGroup(288,180);
// by ID

G=gap.SmallGroup(288,180);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,168,2045,192,242]);
// Polycyclic

G:=Group<a,b,c|a^18=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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