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## G = C9×C22⋊Q8order 288 = 25·32

### Direct product of C9 and C22⋊Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C9×C22⋊Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C18 — C2×C36 — Q8×C18 — C9×C22⋊Q8
 Lower central C1 — C22 — C9×C22⋊Q8
 Upper central C1 — C2×C18 — C9×C22⋊Q8

Generators and relations for C9×C22⋊Q8
G = < a,b,c,d,e | a9=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 150 in 111 conjugacy classes, 72 normal (36 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], Q8 [×2], C23, C9, C12 [×2], C12 [×5], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, C18 [×3], C18 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C3×Q8 [×2], C22×C6, C22⋊Q8, C36 [×2], C36 [×5], C2×C18, C2×C18 [×2], C2×C18 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C22×C12, C6×Q8, C2×C36 [×2], C2×C36 [×4], C2×C36 [×2], Q8×C9 [×2], C22×C18, C3×C22⋊Q8, C9×C22⋊C4 [×2], C9×C4⋊C4, C9×C4⋊C4 [×2], C22×C36, Q8×C18, C9×C22⋊Q8
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], Q8 [×2], C23, C9, C2×C6 [×7], C2×D4, C2×Q8, C4○D4, C18 [×7], C3×D4 [×2], C3×Q8 [×2], C22×C6, C22⋊Q8, C2×C18 [×7], C6×D4, C6×Q8, C3×C4○D4, D4×C9 [×2], Q8×C9 [×2], C22×C18, C3×C22⋊Q8, D4×C18, Q8×C18, C9×C4○D4, C9×C22⋊Q8

Smallest permutation representation of C9×C22⋊Q8
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 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 28)(8 29)(9 30)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 19)(17 20)(18 21)(37 52)(38 53)(39 54)(40 46)(41 47)(42 48)(43 49)(44 50)(45 51)(55 81)(56 73)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 86)(65 87)(66 88)(67 89)(68 90)(69 82)(70 83)(71 84)(72 85)(91 104)(92 105)(93 106)(94 107)(95 108)(96 100)(97 101)(98 102)(99 103)(109 118)(110 119)(111 120)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)(127 140)(128 141)(129 142)(130 143)(131 144)(132 136)(133 137)(134 138)(135 139)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 37)(7 38)(8 39)(9 40)(10 138)(11 139)(12 140)(13 141)(14 142)(15 143)(16 144)(17 136)(18 137)(19 131)(20 132)(21 133)(22 134)(23 135)(24 127)(25 128)(26 129)(27 130)(28 53)(29 54)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(55 90)(56 82)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 89)(64 77)(65 78)(66 79)(67 80)(68 81)(69 73)(70 74)(71 75)(72 76)(91 126)(92 118)(93 119)(94 120)(95 121)(96 122)(97 123)(98 124)(99 125)(100 113)(101 114)(102 115)(103 116)(104 117)(105 109)(106 110)(107 111)(108 112)
(1 57 31 74)(2 58 32 75)(3 59 33 76)(4 60 34 77)(5 61 35 78)(6 62 36 79)(7 63 28 80)(8 55 29 81)(9 56 30 73)(10 98 134 115)(11 99 135 116)(12 91 127 117)(13 92 128 109)(14 93 129 110)(15 94 130 111)(16 95 131 112)(17 96 132 113)(18 97 133 114)(19 108 144 121)(20 100 136 122)(21 101 137 123)(22 102 138 124)(23 103 139 125)(24 104 140 126)(25 105 141 118)(26 106 142 119)(27 107 143 120)(37 88 52 66)(38 89 53 67)(39 90 54 68)(40 82 46 69)(41 83 47 70)(42 84 48 71)(43 85 49 72)(44 86 50 64)(45 87 51 65)
(1 93 31 110)(2 94 32 111)(3 95 33 112)(4 96 34 113)(5 97 35 114)(6 98 36 115)(7 99 28 116)(8 91 29 117)(9 92 30 109)(10 79 134 62)(11 80 135 63)(12 81 127 55)(13 73 128 56)(14 74 129 57)(15 75 130 58)(16 76 131 59)(17 77 132 60)(18 78 133 61)(19 85 144 72)(20 86 136 64)(21 87 137 65)(22 88 138 66)(23 89 139 67)(24 90 140 68)(25 82 141 69)(26 83 142 70)(27 84 143 71)(37 124 52 102)(38 125 53 103)(39 126 54 104)(40 118 46 105)(41 119 47 106)(42 120 48 107)(43 121 49 108)(44 122 50 100)(45 123 51 101)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 6J 9A ··· 9F 12A ··· 12H 12I ··· 12P 18A ··· 18R 18S ··· 18AD 36A ··· 36X 36Y ··· 36AV order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 1 1 2 2 1 1 2 2 2 2 4 4 4 4 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 C9 C18 C18 C18 C18 D4 Q8 C4○D4 C3×D4 C3×Q8 C3×C4○D4 D4×C9 Q8×C9 C9×C4○D4 kernel C9×C22⋊Q8 C9×C22⋊C4 C9×C4⋊C4 C22×C36 Q8×C18 C3×C22⋊Q8 C3×C22⋊C4 C3×C4⋊C4 C22×C12 C6×Q8 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C36 C2×C18 C18 C12 C2×C6 C6 C4 C22 C2 # reps 1 2 3 1 1 2 4 6 2 2 6 12 18 6 6 2 2 2 4 4 4 12 12 12

Matrix representation of C9×C22⋊Q8 in GL4(𝔽37) generated by

 16 0 0 0 0 16 0 0 0 0 10 0 0 0 0 10
,
 1 0 0 0 12 36 0 0 0 0 36 24 0 0 0 1
,
 36 0 0 0 0 36 0 0 0 0 36 0 0 0 0 36
,
 1 0 0 0 0 1 0 0 0 0 6 4 0 0 0 31
,
 36 31 0 0 0 1 0 0 0 0 1 13 0 0 34 36
G:=sub<GL(4,GF(37))| [16,0,0,0,0,16,0,0,0,0,10,0,0,0,0,10],[1,12,0,0,0,36,0,0,0,0,36,0,0,0,24,1],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,6,0,0,0,4,31],[36,0,0,0,31,1,0,0,0,0,1,34,0,0,13,36] >;

C9×C22⋊Q8 in GAP, Magma, Sage, TeX

C_9\times C_2^2\rtimes Q_8
% in TeX

G:=Group("C9xC2^2:Q8");
// GroupNames label

G:=SmallGroup(288,172);
// by ID

G=gap.SmallGroup(288,172);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,365,176,1094,360]);
// Polycyclic

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

׿
×
𝔽