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## G = C9×2- 1+4order 288 = 25·32

### Direct product of C9 and 2- 1+4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C9×2- 1+4
 Chief series C1 — C3 — C6 — C18 — C2×C18 — D4×C9 — C9×C4○D4 — C9×2- 1+4
 Lower central C1 — C2 — C9×2- 1+4
 Upper central C1 — C18 — C9×2- 1+4

Generators and relations for C9×2- 1+4
G = < a,b,c,d,e | a9=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 234 in 219 conjugacy classes, 204 normal (9 characteristic)
C1, C2, C2 [×5], C3, C4 [×10], C22 [×5], C6, C6 [×5], C2×C4 [×15], D4 [×10], Q8 [×10], C9, C12 [×10], C2×C6 [×5], C2×Q8 [×5], C4○D4 [×10], C18, C18 [×5], C2×C12 [×15], C3×D4 [×10], C3×Q8 [×10], 2- 1+4, C36 [×10], C2×C18 [×5], C6×Q8 [×5], C3×C4○D4 [×10], C2×C36 [×15], D4×C9 [×10], Q8×C9 [×10], C3×2- 1+4, Q8×C18 [×5], C9×C4○D4 [×10], C9×2- 1+4
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], C9, C2×C6 [×35], C24, C18 [×15], C22×C6 [×15], 2- 1+4, C2×C18 [×35], C23×C6, C22×C18 [×15], C3×2- 1+4, C23×C18, C9×2- 1+4

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

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

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

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

153 conjugacy classes

 class 1 2A 2B ··· 2F 3A 3B 4A ··· 4J 6A 6B 6C ··· 6L 9A ··· 9F 12A ··· 12T 18A ··· 18F 18G ··· 18AJ 36A ··· 36BH order 1 2 2 ··· 2 3 3 4 ··· 4 6 6 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 2 ··· 2 1 1 2 ··· 2 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

153 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 4 4 4 type + + + - image C1 C2 C2 C3 C6 C6 C9 C18 C18 2- 1+4 C3×2- 1+4 C9×2- 1+4 kernel C9×2- 1+4 Q8×C18 C9×C4○D4 C3×2- 1+4 C6×Q8 C3×C4○D4 2- 1+4 C2×Q8 C4○D4 C9 C3 C1 # reps 1 5 10 2 10 20 6 30 60 1 2 6

Matrix representation of C9×2- 1+4 in GL4(𝔽37) generated by

 33 0 0 0 0 33 0 0 0 0 33 0 0 0 0 33
,
 27 0 19 0 3 25 27 22 20 0 10 0 2 22 15 12
,
 33 35 0 0 26 4 0 0 0 23 1 14 26 5 0 36
,
 33 0 21 35 26 0 3 4 18 23 36 23 31 1 3 5
,
 12 30 30 13 33 14 36 11 26 8 3 5 32 24 36 8
G:=sub<GL(4,GF(37))| [33,0,0,0,0,33,0,0,0,0,33,0,0,0,0,33],[27,3,20,2,0,25,0,22,19,27,10,15,0,22,0,12],[33,26,0,26,35,4,23,5,0,0,1,0,0,0,14,36],[33,26,18,31,0,0,23,1,21,3,36,3,35,4,23,5],[12,33,26,32,30,14,8,24,30,36,3,36,13,11,5,8] >;

C9×2- 1+4 in GAP, Magma, Sage, TeX

C_9\times 2_-^{1+4}
% in TeX

G:=Group("C9xES-(2,2)");
// GroupNames label

G:=SmallGroup(288,372);
// by ID

G=gap.SmallGroup(288,372);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,701,344,555,268,1571,242]);
// Polycyclic

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

׿
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