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## G = C2×Q8×D9order 288 = 25·32

### Direct product of C2, Q8 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C2×Q8×D9
 Chief series C1 — C3 — C9 — C18 — D18 — C22×D9 — C2×C4×D9 — C2×Q8×D9
 Lower central C9 — C18 — C2×Q8×D9
 Upper central C1 — C22 — C2×Q8

Generators and relations for C2×Q8×D9
G = < a,b,c,d,e | a2=b4=d9=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 824 in 234 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×6], C4 [×6], C22, C22 [×6], S3 [×4], C6, C6 [×2], C2×C4 [×3], C2×C4 [×15], Q8 [×4], Q8 [×12], C23, C9, Dic3 [×6], C12 [×6], D6 [×6], C2×C6, C22×C4 [×3], C2×Q8, C2×Q8 [×11], D9 [×4], C18, C18 [×2], Dic6 [×12], C4×S3 [×12], C2×Dic3 [×3], C2×C12 [×3], C3×Q8 [×4], C22×S3, C22×Q8, Dic9 [×6], C36 [×6], D18 [×6], C2×C18, C2×Dic6 [×3], S3×C2×C4 [×3], S3×Q8 [×8], C6×Q8, Dic18 [×12], C4×D9 [×12], C2×Dic9 [×3], C2×C36 [×3], Q8×C9 [×4], C22×D9, C2×S3×Q8, C2×Dic18 [×3], C2×C4×D9 [×3], Q8×D9 [×8], Q8×C18, C2×Q8×D9
Quotients: C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C24, D9, C22×S3 [×7], C22×Q8, D18 [×7], S3×Q8 [×2], S3×C23, C22×D9 [×7], C2×S3×Q8, Q8×D9 [×2], C23×D9, C2×Q8×D9

Smallest permutation representation of C2×Q8×D9
On 144 points
Generators in S144
(1 77)(2 78)(3 79)(4 80)(5 81)(6 73)(7 74)(8 75)(9 76)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(1 68 14 59)(2 69 15 60)(3 70 16 61)(4 71 17 62)(5 72 18 63)(6 64 10 55)(7 65 11 56)(8 66 12 57)(9 67 13 58)(19 37 28 46)(20 38 29 47)(21 39 30 48)(22 40 31 49)(23 41 32 50)(24 42 33 51)(25 43 34 52)(26 44 35 53)(27 45 36 54)(73 136 82 127)(74 137 83 128)(75 138 84 129)(76 139 85 130)(77 140 86 131)(78 141 87 132)(79 142 88 133)(80 143 89 134)(81 144 90 135)(91 109 100 118)(92 110 101 119)(93 111 102 120)(94 112 103 121)(95 113 104 122)(96 114 105 123)(97 115 106 124)(98 116 107 125)(99 117 108 126)
(1 104 14 95)(2 105 15 96)(3 106 16 97)(4 107 17 98)(5 108 18 99)(6 100 10 91)(7 101 11 92)(8 102 12 93)(9 103 13 94)(19 73 28 82)(20 74 29 83)(21 75 30 84)(22 76 31 85)(23 77 32 86)(24 78 33 87)(25 79 34 88)(26 80 35 89)(27 81 36 90)(37 127 46 136)(38 128 47 137)(39 129 48 138)(40 130 49 139)(41 131 50 140)(42 132 51 141)(43 133 52 142)(44 134 53 143)(45 135 54 144)(55 118 64 109)(56 119 65 110)(57 120 66 111)(58 121 67 112)(59 122 68 113)(60 123 69 114)(61 124 70 115)(62 125 71 116)(63 126 72 117)
(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 76)(2 75)(3 74)(4 73)(5 81)(6 80)(7 79)(8 78)(9 77)(10 89)(11 88)(12 87)(13 86)(14 85)(15 84)(16 83)(17 82)(18 90)(19 98)(20 97)(21 96)(22 95)(23 94)(24 93)(25 92)(26 91)(27 99)(28 107)(29 106)(30 105)(31 104)(32 103)(33 102)(34 101)(35 100)(36 108)(37 116)(38 115)(39 114)(40 113)(41 112)(42 111)(43 110)(44 109)(45 117)(46 125)(47 124)(48 123)(49 122)(50 121)(51 120)(52 119)(53 118)(54 126)(55 134)(56 133)(57 132)(58 131)(59 130)(60 129)(61 128)(62 127)(63 135)(64 143)(65 142)(66 141)(67 140)(68 139)(69 138)(70 137)(71 136)(72 144)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A ··· 4F 4G ··· 4L 6A 6B 6C 9A 9B 9C 12A ··· 12F 18A ··· 18I 36A ··· 36R order 1 2 2 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 6 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 9 9 9 9 2 2 ··· 2 18 ··· 18 2 2 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + - + + + + + - - image C1 C2 C2 C2 C2 S3 Q8 D6 D6 D9 D18 D18 S3×Q8 Q8×D9 kernel C2×Q8×D9 C2×Dic18 C2×C4×D9 Q8×D9 Q8×C18 C6×Q8 D18 C2×C12 C3×Q8 C2×Q8 C2×C4 Q8 C6 C2 # reps 1 3 3 8 1 1 4 3 4 3 9 12 2 6

Matrix representation of C2×Q8×D9 in GL4(𝔽37) generated by

 36 0 0 0 0 36 0 0 0 0 1 0 0 0 0 1
,
 36 0 0 0 0 36 0 0 0 0 9 28 0 0 5 28
,
 36 0 0 0 0 36 0 0 0 0 36 35 0 0 1 1
,
 26 6 0 0 31 20 0 0 0 0 1 0 0 0 0 1
,
 11 31 0 0 20 26 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,9,5,0,0,28,28],[36,0,0,0,0,36,0,0,0,0,36,1,0,0,35,1],[26,31,0,0,6,20,0,0,0,0,1,0,0,0,0,1],[11,20,0,0,31,26,0,0,0,0,1,0,0,0,0,1] >;

C2×Q8×D9 in GAP, Magma, Sage, TeX

C_2\times Q_8\times D_9
% in TeX

G:=Group("C2xQ8xD9");
// GroupNames label

G:=SmallGroup(288,359);
// by ID

G=gap.SmallGroup(288,359);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,185,80,6725,292,9414]);
// Polycyclic

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

׿
×
𝔽