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## G = C2×C32⋊9SD16order 288 = 25·32

### Direct product of C2 and C32⋊9SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C2×C32⋊9SD16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32⋊4Q8 — C2×C32⋊4Q8 — C2×C32⋊9SD16
 Lower central C32 — C3×C6 — C3×C12 — C2×C32⋊9SD16
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for C2×C329SD16
G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=c-1, ce=ec, ede=d3 >

Subgroups: 628 in 204 conjugacy classes, 77 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×4], C6 [×12], C6 [×8], C8 [×2], C2×C4, C2×C4, D4 [×2], D4, Q8 [×3], C23, C32, Dic3 [×8], C12 [×8], C2×C6 [×4], C2×C6 [×16], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×C6, C3×C6 [×2], C3×C6 [×2], C3⋊C8 [×8], Dic6 [×12], C2×Dic3 [×4], C2×C12 [×4], C3×D4 [×8], C3×D4 [×4], C22×C6 [×4], C2×SD16, C3⋊Dic3 [×2], C3×C12 [×2], C62, C62 [×4], C2×C3⋊C8 [×4], D4.S3 [×16], C2×Dic6 [×4], C6×D4 [×4], C324C8 [×2], C324Q8 [×2], C324Q8, C2×C3⋊Dic3, C6×C12, D4×C32 [×2], D4×C32, C2×C62, C2×D4.S3 [×4], C2×C324C8, C329SD16 [×4], C2×C324Q8, D4×C3×C6, C2×C329SD16
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], SD16 [×2], C2×D4, C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C2×SD16, C2×C3⋊S3 [×3], D4.S3 [×8], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, C2×D4.S3 [×4], C329SD16 [×2], C2×C327D4, C2×C329SD16

Smallest permutation representation of C2×C329SD16
On 144 points
Generators in S144
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(25 71)(26 72)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)(41 88)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 105)(80 106)(89 124)(90 125)(91 126)(92 127)(93 128)(94 121)(95 122)(96 123)(97 135)(98 136)(99 129)(100 130)(101 131)(102 132)(103 133)(104 134)(113 142)(114 143)(115 144)(116 137)(117 138)(118 139)(119 140)(120 141)
(1 141 49)(2 50 142)(3 143 51)(4 52 144)(5 137 53)(6 54 138)(7 139 55)(8 56 140)(9 24 118)(10 119 17)(11 18 120)(12 113 19)(13 20 114)(14 115 21)(15 22 116)(16 117 23)(25 40 48)(26 41 33)(27 34 42)(28 43 35)(29 36 44)(30 45 37)(31 38 46)(32 47 39)(57 70 86)(58 87 71)(59 72 88)(60 81 65)(61 66 82)(62 83 67)(63 68 84)(64 85 69)(73 132 93)(74 94 133)(75 134 95)(76 96 135)(77 136 89)(78 90 129)(79 130 91)(80 92 131)(97 110 123)(98 124 111)(99 112 125)(100 126 105)(101 106 127)(102 128 107)(103 108 121)(104 122 109)
(1 112 69)(2 70 105)(3 106 71)(4 72 107)(5 108 65)(6 66 109)(7 110 67)(8 68 111)(9 135 44)(10 45 136)(11 129 46)(12 47 130)(13 131 48)(14 41 132)(15 133 42)(16 43 134)(17 30 77)(18 78 31)(19 32 79)(20 80 25)(21 26 73)(22 74 27)(23 28 75)(24 76 29)(33 93 115)(34 116 94)(35 95 117)(36 118 96)(37 89 119)(38 120 90)(39 91 113)(40 114 92)(49 99 85)(50 86 100)(51 101 87)(52 88 102)(53 103 81)(54 82 104)(55 97 83)(56 84 98)(57 126 142)(58 143 127)(59 128 144)(60 137 121)(61 122 138)(62 139 123)(63 124 140)(64 141 125)
(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)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 23)(19 21)(20 24)(25 29)(26 32)(28 30)(33 39)(35 37)(36 40)(41 47)(43 45)(44 48)(50 52)(51 55)(54 56)(57 59)(58 62)(61 63)(66 68)(67 71)(70 72)(73 79)(75 77)(76 80)(82 84)(83 87)(86 88)(89 95)(91 93)(92 96)(97 101)(98 104)(100 102)(105 107)(106 110)(109 111)(113 115)(114 118)(117 119)(122 124)(123 127)(126 128)(130 132)(131 135)(134 136)(138 140)(139 143)(142 144)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6L 6M ··· 6AB 8A 8B 8C 8D 12A ··· 12H order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 1 1 4 4 2 2 2 2 2 2 36 36 2 ··· 2 4 ··· 4 18 18 18 18 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 SD16 C3⋊D4 C3⋊D4 D4.S3 kernel C2×C32⋊9SD16 C2×C32⋊4C8 C32⋊9SD16 C2×C32⋊4Q8 D4×C3×C6 C6×D4 C3×C12 C62 C2×C12 C3×D4 C3×C6 C12 C2×C6 C6 # reps 1 1 4 1 1 4 1 1 4 8 4 8 8 8

Matrix representation of C2×C329SD16 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 8 0 0 0 0 0 0 64 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 40 64
,
 64 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 12 60 0 0 0 0 28 0 0 0 0 0 0 0 28 10 0 0 0 0 31 45
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 10 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 9 72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[8,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,40,0,0,0,0,0,64],[64,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,12,28,0,0,0,0,60,0,0,0,0,0,0,0,28,31,0,0,0,0,10,45],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,10,72,0,0,0,0,0,0,1,9,0,0,0,0,0,72] >;

C2×C329SD16 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_9{\rm SD}_{16}
% in TeX

G:=Group("C2xC3^2:9SD16");
// GroupNames label

G:=SmallGroup(288,790);
// by ID

G=gap.SmallGroup(288,790);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,675,185,80,2693,9414]);
// Polycyclic

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

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