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G = D8xC3xC6order 288 = 25·32

Direct product of C3xC6 and D8

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

Aliases: D8xC3xC6, C8:2C62, D4:1C62, C62.144D4, (C2xC24):8C6, (C6xC24):13C2, C24:10(C2xC6), (C6xD4):13C6, C6.91(C6xD4), C12.50(C3xD4), C4.1(C2xC62), C4.6(D4xC32), (C3xC24):28C22, (C3xC12).147D4, (C2xC4).25C62, C12.55(C22xC6), (C3xC12).185C23, (C6xC12).374C22, (D4xC32):28C22, C22.14(D4xC32), (C2xC8):3(C3xC6), (D4xC3xC6):22C2, C2.11(D4xC3xC6), (C2xD4):4(C3xC6), (C3xD4):10(C2xC6), (C2xC6).72(C3xD4), (C3xC6).308(C2xD4), (C2xC12).161(C2xC6), SmallGroup(288,829)

Series: Derived Chief Lower central Upper central

C1C4 — D8xC3xC6
C1C2C4C12C3xC12D4xC32C32xD8 — D8xC3xC6
C1C2C4 — D8xC3xC6
C1C62C6xC12 — D8xC3xC6

Generators and relations for D8xC3xC6
 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, C2, C3, C4, C22, C22, C6, C6, C8, C2xC4, D4, D4, C23, C32, C12, C2xC6, C2xC6, C2xC8, D8, C2xD4, C3xC6, C3xC6, C3xC6, C24, C2xC12, C3xD4, C3xD4, C22xC6, C2xD8, C3xC12, C62, C62, C2xC24, C3xD8, C6xD4, C3xC24, C6xC12, D4xC32, D4xC32, C2xC62, C6xD8, C6xC24, C32xD8, D4xC3xC6, D8xC3xC6
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2xC6, D8, C2xD4, C3xC6, C3xD4, C22xC6, C2xD8, C62, C3xD8, C6xD4, D4xC32, C2xC62, C6xD8, C32xD8, D4xC3xC6, D8xC3xC6

Smallest permutation representation of D8xC3xC6
On 144 points
Generators in S144
(1 32 24)(2 25 17)(3 26 18)(4 27 19)(5 28 20)(6 29 21)(7 30 22)(8 31 23)(9 35 130)(10 36 131)(11 37 132)(12 38 133)(13 39 134)(14 40 135)(15 33 136)(16 34 129)(41 92 49)(42 93 50)(43 94 51)(44 95 52)(45 96 53)(46 89 54)(47 90 55)(48 91 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)(81 127 119)(82 128 120)(83 121 113)(84 122 114)(85 123 115)(86 124 116)(87 125 117)(88 126 118)(97 143 105)(98 144 106)(99 137 107)(100 138 108)(101 139 109)(102 140 110)(103 141 111)(104 142 112)
(1 101 81 58 46 131)(2 102 82 59 47 132)(3 103 83 60 48 133)(4 104 84 61 41 134)(5 97 85 62 42 135)(6 98 86 63 43 136)(7 99 87 64 44 129)(8 100 88 57 45 130)(9 31 138 126 78 96)(10 32 139 127 79 89)(11 25 140 128 80 90)(12 26 141 121 73 91)(13 27 142 122 74 92)(14 28 143 123 75 93)(15 29 144 124 76 94)(16 30 137 125 77 95)(17 110 120 67 55 37)(18 111 113 68 56 38)(19 112 114 69 49 39)(20 105 115 70 50 40)(21 106 116 71 51 33)(22 107 117 72 52 34)(23 108 118 65 53 35)(24 109 119 66 54 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)(9 123)(10 122)(11 121)(12 128)(13 127)(14 126)(15 125)(16 124)(17 68)(18 67)(19 66)(20 65)(21 72)(22 71)(23 70)(24 69)(25 73)(26 80)(27 79)(28 78)(29 77)(30 76)(31 75)(32 74)(33 117)(34 116)(35 115)(36 114)(37 113)(38 120)(39 119)(40 118)(41 101)(42 100)(43 99)(44 98)(45 97)(46 104)(47 103)(48 102)(49 109)(50 108)(51 107)(52 106)(53 105)(54 112)(55 111)(56 110)(81 134)(82 133)(83 132)(84 131)(85 130)(86 129)(87 136)(88 135)(89 142)(90 141)(91 140)(92 139)(93 138)(94 137)(95 144)(96 143)

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

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

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

126 conjugacy classes

class 1 2A2B2C2D2E2F2G3A···3H4A4B6A···6X6Y···6BD8A8B8C8D12A···12P24A···24AF
order122222223···3446···66···6888812···1224···24
size111144441···1221···14···422222···22···2

126 irreducible representations

dim11111111222222
type+++++++
imageC1C2C2C2C3C6C6C6D4D4D8C3xD4C3xD4C3xD8
kernelD8xC3xC6C6xC24C32xD8D4xC3xC6C6xD8C2xC24C3xD8C6xD4C3xC12C62C3xC6C12C2xC6C6
# reps11428832161148832

Matrix representation of D8xC3xC6 in GL4(F73) generated by

1000
0100
0080
0008
,
8000
07200
0090
0009
,
72000
07200
005716
005757
,
1000
07200
005716
001616
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] >;

D8xC3xC6 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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