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## G = D8×C3×C6order 288 = 25·32

### Direct product of C3×C6 and D8

direct product, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — D8×C3×C6
 Chief series C1 — C2 — C4 — C12 — C3×C12 — D4×C32 — C32×D8 — D8×C3×C6
 Lower central C1 — C2 — C4 — D8×C3×C6
 Upper central C1 — C62 — C6×C12 — D8×C3×C6

Generators and relations for D8×C3×C6
G = < a,b,c,d | a3=b6=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 420 in 228 conjugacy classes, 132 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×4], C4 [×2], C22, C22 [×8], C6 [×12], C6 [×16], C8 [×2], C2×C4, D4 [×4], D4 [×2], C23 [×2], C32, C12 [×8], C2×C6 [×4], C2×C6 [×32], C2×C8, D8 [×4], C2×D4 [×2], C3×C6, C3×C6 [×2], C3×C6 [×4], C24 [×8], C2×C12 [×4], C3×D4 [×16], C3×D4 [×8], C22×C6 [×8], C2×D8, C3×C12 [×2], C62, C62 [×8], C2×C24 [×4], C3×D8 [×16], C6×D4 [×8], C3×C24 [×2], C6×C12, D4×C32 [×4], D4×C32 [×2], C2×C62 [×2], C6×D8 [×4], C6×C24, C32×D8 [×4], D4×C3×C6 [×2], D8×C3×C6
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], D4 [×2], C23, C32, C2×C6 [×28], D8 [×2], C2×D4, C3×C6 [×7], C3×D4 [×8], C22×C6 [×4], C2×D8, C62 [×7], C3×D8 [×8], C6×D4 [×4], D4×C32 [×2], C2×C62, C6×D8 [×4], C32×D8 [×2], D4×C3×C6, D8×C3×C6

Smallest permutation representation of D8×C3×C6
On 144 points
Generators in S144
(1 28 20)(2 29 21)(3 30 22)(4 31 23)(5 32 24)(6 25 17)(7 26 18)(8 27 19)(9 33 132)(10 34 133)(11 35 134)(12 36 135)(13 37 136)(14 38 129)(15 39 130)(16 40 131)(41 96 49)(42 89 50)(43 90 51)(44 91 52)(45 92 53)(46 93 54)(47 94 55)(48 95 56)(57 78 65)(58 79 66)(59 80 67)(60 73 68)(61 74 69)(62 75 70)(63 76 71)(64 77 72)(81 123 115)(82 124 116)(83 125 117)(84 126 118)(85 127 119)(86 128 120)(87 121 113)(88 122 114)(97 139 105)(98 140 106)(99 141 107)(100 142 108)(101 143 109)(102 144 110)(103 137 111)(104 138 112)
(1 97 85 58 46 131)(2 98 86 59 47 132)(3 99 87 60 48 133)(4 100 88 61 41 134)(5 101 81 62 42 135)(6 102 82 63 43 136)(7 103 83 64 44 129)(8 104 84 57 45 130)(9 29 140 128 80 94)(10 30 141 121 73 95)(11 31 142 122 74 96)(12 32 143 123 75 89)(13 25 144 124 76 90)(14 26 137 125 77 91)(15 27 138 126 78 92)(16 28 139 127 79 93)(17 110 116 71 51 37)(18 111 117 72 52 38)(19 112 118 65 53 39)(20 105 119 66 54 40)(21 106 120 67 55 33)(22 107 113 68 56 34)(23 108 114 69 49 35)(24 109 115 70 50 36)
(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 61)(2 60)(3 59)(4 58)(5 57)(6 64)(7 63)(8 62)(9 121)(10 128)(11 127)(12 126)(13 125)(14 124)(15 123)(16 122)(17 72)(18 71)(19 70)(20 69)(21 68)(22 67)(23 66)(24 65)(25 77)(26 76)(27 75)(28 74)(29 73)(30 80)(31 79)(32 78)(33 113)(34 120)(35 119)(36 118)(37 117)(38 116)(39 115)(40 114)(41 97)(42 104)(43 103)(44 102)(45 101)(46 100)(47 99)(48 98)(49 105)(50 112)(51 111)(52 110)(53 109)(54 108)(55 107)(56 106)(81 130)(82 129)(83 136)(84 135)(85 134)(86 133)(87 132)(88 131)(89 138)(90 137)(91 144)(92 143)(93 142)(94 141)(95 140)(96 139)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3H 4A 4B 6A ··· 6X 6Y ··· 6BD 8A 8B 8C 8D 12A ··· 12P 24A ··· 24AF order 1 2 2 2 2 2 2 2 3 ··· 3 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 1 1 4 4 4 4 1 ··· 1 2 2 1 ··· 1 4 ··· 4 2 2 2 2 2 ··· 2 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D4 D8 C3×D4 C3×D4 C3×D8 kernel D8×C3×C6 C6×C24 C32×D8 D4×C3×C6 C6×D8 C2×C24 C3×D8 C6×D4 C3×C12 C62 C3×C6 C12 C2×C6 C6 # reps 1 1 4 2 8 8 32 16 1 1 4 8 8 32

Matrix representation of D8×C3×C6 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 8 0 0 0 0 8
,
 8 0 0 0 0 72 0 0 0 0 9 0 0 0 0 9
,
 72 0 0 0 0 72 0 0 0 0 57 16 0 0 57 57
,
 1 0 0 0 0 72 0 0 0 0 57 16 0 0 16 16
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,0,72,0,0,0,0,9,0,0,0,0,9],[72,0,0,0,0,72,0,0,0,0,57,57,0,0,16,57],[1,0,0,0,0,72,0,0,0,0,57,16,0,0,16,16] >;

D8×C3×C6 in GAP, Magma, Sage, TeX

D_8\times C_3\times C_6
% in TeX

G:=Group("D8xC3xC6");
// GroupNames label

G:=SmallGroup(288,829);
// by ID

G=gap.SmallGroup(288,829);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,9077,4548,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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