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G = C9×C8.C4order 288 = 25·32

Direct product of C9 and C8.C4

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C9×C8.C4, C8.1C36, C72.5C4, C24.8C12, C36.68D4, M4(2).2C18, C4.8(C2×C36), (C2×C8).5C18, C22.(Q8×C9), C4.19(D4×C9), (C2×C18).2Q8, C36.45(C2×C4), (C2×C24).18C6, (C2×C72).15C2, C12.85(C3×D4), C18.14(C4⋊C4), C12.51(C2×C12), (C3×M4(2)).6C6, (C9×M4(2)).4C2, (C2×C36).118C22, C2.5(C9×C4⋊C4), C6.14(C3×C4⋊C4), C3.(C3×C8.C4), (C2×C6).3(C3×Q8), (C3×C8.C4).C3, (C2×C4).17(C2×C18), (C2×C12).138(C2×C6), SmallGroup(288,58)

Series: Derived Chief Lower central Upper central

C1C4 — C9×C8.C4
C1C2C6C12C2×C12C2×C36C9×M4(2) — C9×C8.C4
C1C2C4 — C9×C8.C4
C1C36C2×C36 — C9×C8.C4

Generators and relations for C9×C8.C4
 G = < a,b,c | a9=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >

2C2
2C6
2C8
2C8
2C18
2C24
2C24
2C72
2C72

Smallest permutation representation of C9×C8.C4
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 89 59 95 53 78 41 66)(2 90 60 96 54 79 42 67)(3 82 61 97 46 80 43 68)(4 83 62 98 47 81 44 69)(5 84 63 99 48 73 45 70)(6 85 55 91 49 74 37 71)(7 86 56 92 50 75 38 72)(8 87 57 93 51 76 39 64)(9 88 58 94 52 77 40 65)(10 109 32 135 27 120 142 106)(11 110 33 127 19 121 143 107)(12 111 34 128 20 122 144 108)(13 112 35 129 21 123 136 100)(14 113 36 130 22 124 137 101)(15 114 28 131 23 125 138 102)(16 115 29 132 24 126 139 103)(17 116 30 133 25 118 140 104)(18 117 31 134 26 119 141 105)
(1 131 59 114 53 102 41 125)(2 132 60 115 54 103 42 126)(3 133 61 116 46 104 43 118)(4 134 62 117 47 105 44 119)(5 135 63 109 48 106 45 120)(6 127 55 110 49 107 37 121)(7 128 56 111 50 108 38 122)(8 129 57 112 51 100 39 123)(9 130 58 113 52 101 40 124)(10 73 142 70 27 84 32 99)(11 74 143 71 19 85 33 91)(12 75 144 72 20 86 34 92)(13 76 136 64 21 87 35 93)(14 77 137 65 22 88 36 94)(15 78 138 66 23 89 28 95)(16 79 139 67 24 90 29 96)(17 80 140 68 25 82 30 97)(18 81 141 69 26 83 31 98)

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

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

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

126 conjugacy classes

class 1 2A2B3A3B4A4B4C6A6B6C6D8A8B8C8D8E8F8G8H9A···9F12A12B12C12D12E12F18A···18F18G···18L24A···24H24I···24P36A···36L36M···36R72A···72X72Y···72AV
order122334446666888888889···912121212121218···1818···1824···2424···2436···3636···3672···7272···72
size112111121122222244441···11111221···12···22···24···41···12···22···24···4

126 irreducible representations

dim111111111111222222222
type++++-
imageC1C2C2C3C4C6C6C9C12C18C18C36D4Q8C3×D4C3×Q8C8.C4D4×C9Q8×C9C3×C8.C4C9×C8.C4
kernelC9×C8.C4C2×C72C9×M4(2)C3×C8.C4C72C2×C24C3×M4(2)C8.C4C24C2×C8M4(2)C8C36C2×C18C12C2×C6C9C4C22C3C1
# reps112242468612241122466824

Matrix representation of C9×C8.C4 in GL3(𝔽73) generated by

5500
010
001
,
7200
0630
0051
,
2700
001
0270
G:=sub<GL(3,GF(73))| [55,0,0,0,1,0,0,0,1],[72,0,0,0,63,0,0,0,51],[27,0,0,0,0,27,0,1,0] >;

C9×C8.C4 in GAP, Magma, Sage, TeX

C_9\times C_8.C_4
% in TeX

G:=Group("C9xC8.C4");
// GroupNames label

G:=SmallGroup(288,58);
// by ID

G=gap.SmallGroup(288,58);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,92,268,4371,360,242]);
// Polycyclic

G:=Group<a,b,c|a^9=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C9×C8.C4 in TeX

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