Copied to
clipboard

## G = D4×C62order 288 = 25·32

### Direct product of C62 and D4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for D4×C62
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 948 in 708 conjugacy classes, 468 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×8], C3 [×4], C4 [×4], C22 [×15], C22 [×24], C6 [×28], C6 [×32], C2×C4 [×6], D4 [×16], C23, C23 [×12], C23 [×8], C32, C12 [×16], C2×C6 [×60], C2×C6 [×96], C22×C4, C2×D4 [×12], C24 [×2], C3×C6, C3×C6 [×6], C3×C6 [×8], C2×C12 [×24], C3×D4 [×64], C22×C6 [×52], C22×C6 [×32], C22×D4, C3×C12 [×4], C62 [×15], C62 [×24], C22×C12 [×4], C6×D4 [×48], C23×C6 [×8], C6×C12 [×6], D4×C32 [×16], C2×C62, C2×C62 [×12], C2×C62 [×8], D4×C2×C6 [×4], C2×C6×C12, D4×C3×C6 [×12], C22×C62 [×2], D4×C62
Quotients: C1, C2 [×15], C3 [×4], C22 [×35], C6 [×60], D4 [×4], C23 [×15], C32, C2×C6 [×140], C2×D4 [×6], C24, C3×C6 [×15], C3×D4 [×16], C22×C6 [×60], C22×D4, C62 [×35], C6×D4 [×24], C23×C6 [×4], D4×C32 [×4], C2×C62 [×15], D4×C2×C6 [×4], D4×C3×C6 [×6], C22×C62, D4×C62

Smallest permutation representation of D4×C62
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)
(1 106 115 100 109 51)(2 107 116 101 110 52)(3 108 117 102 111 53)(4 103 118 97 112 54)(5 104 119 98 113 49)(6 105 120 99 114 50)(7 140 64 60 13 72)(8 141 65 55 14 67)(9 142 66 56 15 68)(10 143 61 57 16 69)(11 144 62 58 17 70)(12 139 63 59 18 71)(19 33 48 27 42 94)(20 34 43 28 37 95)(21 35 44 29 38 96)(22 36 45 30 39 91)(23 31 46 25 40 92)(24 32 47 26 41 93)(73 87 138 81 132 122)(74 88 133 82 127 123)(75 89 134 83 128 124)(76 90 135 84 129 125)(77 85 136 79 130 126)(78 86 137 80 131 121)
(1 70 25 74)(2 71 26 75)(3 72 27 76)(4 67 28 77)(5 68 29 78)(6 69 30 73)(7 42 90 108)(8 37 85 103)(9 38 86 104)(10 39 87 105)(11 40 88 106)(12 41 89 107)(13 48 125 53)(14 43 126 54)(15 44 121 49)(16 45 122 50)(17 46 123 51)(18 47 124 52)(19 84 102 64)(20 79 97 65)(21 80 98 66)(22 81 99 61)(23 82 100 62)(24 83 101 63)(31 127 109 58)(32 128 110 59)(33 129 111 60)(34 130 112 55)(35 131 113 56)(36 132 114 57)(91 138 120 143)(92 133 115 144)(93 134 116 139)(94 135 117 140)(95 136 118 141)(96 137 119 142)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 73)(7 42)(8 37)(9 38)(10 39)(11 40)(12 41)(13 48)(14 43)(15 44)(16 45)(17 46)(18 47)(19 64)(20 65)(21 66)(22 61)(23 62)(24 63)(25 70)(26 71)(27 72)(28 67)(29 68)(30 69)(31 58)(32 59)(33 60)(34 55)(35 56)(36 57)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(91 143)(92 144)(93 139)(94 140)(95 141)(96 142)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)

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

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

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

180 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A ··· 3H 4A 4B 4C 4D 6A ··· 6BD 6BE ··· 6DP 12A ··· 12AF order 1 2 ··· 2 2 ··· 2 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 ··· 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 C3×D4 kernel D4×C62 C2×C6×C12 D4×C3×C6 C22×C62 D4×C2×C6 C22×C12 C6×D4 C23×C6 C62 C2×C6 # reps 1 1 12 2 8 8 96 16 4 32

Matrix representation of D4×C62 in GL4(𝔽13) generated by

 9 0 0 0 0 10 0 0 0 0 9 0 0 0 0 9
,
 4 0 0 0 0 3 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 12 12 0 0 2 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 11 12
G:=sub<GL(4,GF(13))| [9,0,0,0,0,10,0,0,0,0,9,0,0,0,0,9],[4,0,0,0,0,3,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,2,0,0,12,1],[1,0,0,0,0,1,0,0,0,0,1,11,0,0,0,12] >;

D4×C62 in GAP, Magma, Sage, TeX

D_4\times C_6^2
% in TeX

G:=Group("D4xC6^2");
// GroupNames label

G:=SmallGroup(288,1019);
// by ID

G=gap.SmallGroup(288,1019);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-2,2045]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=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

׿
×
𝔽