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G = C9×C8.C22order 288 = 25·32

Direct product of C9 and C8.C22

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

Aliases: C9×C8.C22, Q162C18, C36.64D4, SD162C18, M4(2)⋊2C18, C36.49C23, C72.13C22, C8.(C2×C18), (C9×Q16)⋊6C2, (C2×Q8)⋊6C18, C4.15(D4×C9), C6.79(C6×D4), C24.13(C2×C6), (Q8×C18)⋊11C2, C4○D4.4C18, (C9×SD16)⋊6C2, D4.3(C2×C18), C2.16(D4×C18), C12.74(C3×D4), C18.79(C2×D4), (C2×C18).26D4, Q8.6(C2×C18), (C6×Q8).21C6, (C3×Q16).5C6, C22.6(D4×C9), (C9×M4(2))⋊6C2, C4.6(C22×C18), (C3×SD16).2C6, (C2×C36).68C22, C12.49(C22×C6), (D4×C9).13C22, (C3×M4(2)).2C6, (Q8×C9).14C22, C3.(C3×C8.C22), (C2×C4).5(C2×C18), (C9×C4○D4).5C2, (C2×C6).30(C3×D4), (C3×C8.C22).C3, (C2×C12).66(C2×C6), (C3×C4○D4).15C6, (C3×D4).15(C2×C6), (C3×Q8).28(C2×C6), SmallGroup(288,187)

Series: Derived Chief Lower central Upper central

C1C4 — C9×C8.C22
C1C2C6C12C36D4×C9C9×SD16 — C9×C8.C22
C1C2C4 — C9×C8.C22
C1C18C2×C36 — C9×C8.C22

Generators and relations for C9×C8.C22
 G = < a,b,c,d | a9=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >

Subgroups: 126 in 90 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×2], Q8, C9, C12 [×2], C12 [×3], C2×C6, C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C18, C18 [×2], C24 [×2], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C3×Q8 [×2], C3×Q8, C8.C22, C36 [×2], C36 [×3], C2×C18, C2×C18, C3×M4(2), C3×SD16 [×2], C3×Q16 [×2], C6×Q8, C3×C4○D4, C72 [×2], C2×C36, C2×C36 [×2], D4×C9, D4×C9, Q8×C9, Q8×C9 [×2], Q8×C9, C3×C8.C22, C9×M4(2), C9×SD16 [×2], C9×Q16 [×2], Q8×C18, C9×C4○D4, C9×C8.C22
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C9, C2×C6 [×7], C2×D4, C18 [×7], C3×D4 [×2], C22×C6, C8.C22, C2×C18 [×7], C6×D4, D4×C9 [×2], C22×C18, C3×C8.C22, D4×C18, C9×C8.C22

Smallest permutation representation of C9×C8.C22
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)
(1 118 59 131 46 112 43 102)(2 119 60 132 47 113 44 103)(3 120 61 133 48 114 45 104)(4 121 62 134 49 115 37 105)(5 122 63 135 50 116 38 106)(6 123 55 127 51 117 39 107)(7 124 56 128 52 109 40 108)(8 125 57 129 53 110 41 100)(9 126 58 130 54 111 42 101)(10 65 36 75 20 94 137 90)(11 66 28 76 21 95 138 82)(12 67 29 77 22 96 139 83)(13 68 30 78 23 97 140 84)(14 69 31 79 24 98 141 85)(15 70 32 80 25 99 142 86)(16 71 33 81 26 91 143 87)(17 72 34 73 27 92 144 88)(18 64 35 74 19 93 136 89)
(10 137)(11 138)(12 139)(13 140)(14 141)(15 142)(16 143)(17 144)(18 136)(19 35)(20 36)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(37 62)(38 63)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(73 88)(74 89)(75 90)(76 82)(77 83)(78 84)(79 85)(80 86)(81 87)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 109)(118 131)(119 132)(120 133)(121 134)(122 135)(123 127)(124 128)(125 129)(126 130)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 73)(8 74)(9 75)(10 126)(11 118)(12 119)(13 120)(14 121)(15 122)(16 123)(17 124)(18 125)(19 110)(20 111)(21 112)(22 113)(23 114)(24 115)(25 116)(26 117)(27 109)(28 131)(29 132)(30 133)(31 134)(32 135)(33 127)(34 128)(35 129)(36 130)(37 69)(38 70)(39 71)(40 72)(41 64)(42 65)(43 66)(44 67)(45 68)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)

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

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

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

99 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E6A6B6C6D6E6F8A8B9A···9F12A12B12C12D12E···12J18A···18F18G···18L18M···18R24A24B24C24D36A···36L36M···36AD72A···72L
order12223344444666666889···91212121212···1218···1818···1818···182424242436···3636···3672···72
size11241122444112244441···122224···41···12···24···444442···24···44···4

99 irreducible representations

dim111111111111111111222222444
type++++++++-
imageC1C2C2C2C2C2C3C6C6C6C6C6C9C18C18C18C18C18D4D4C3×D4C3×D4D4×C9D4×C9C8.C22C3×C8.C22C9×C8.C22
kernelC9×C8.C22C9×M4(2)C9×SD16C9×Q16Q8×C18C9×C4○D4C3×C8.C22C3×M4(2)C3×SD16C3×Q16C6×Q8C3×C4○D4C8.C22M4(2)SD16Q16C2×Q8C4○D4C36C2×C18C12C2×C6C4C22C9C3C1
# reps11221122442266121266112266126

Matrix representation of C9×C8.C22 in GL4(𝔽73) generated by

16000
01600
00160
00016
,
18116218
14431762
19193062
54195955
,
10100
072063
00720
0001
,
10641756
64105617
710639
071963
G:=sub<GL(4,GF(73))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[18,14,19,54,11,43,19,19,62,17,30,59,18,62,62,55],[1,0,0,0,0,72,0,0,10,0,72,0,0,63,0,1],[10,64,71,0,64,10,0,71,17,56,63,9,56,17,9,63] >;

C9×C8.C22 in GAP, Magma, Sage, TeX

C_9\times C_8.C_2^2
% in TeX

G:=Group("C9xC8.C2^2");
// GroupNames label

G:=SmallGroup(288,187);
// by ID

G=gap.SmallGroup(288,187);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,365,1016,3110,192,5884,2951,242]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations

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