Copied to
clipboard

## G = Dic18.C6order 432 = 24·33

### 1st non-split extension by Dic18 of C6 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — Dic18.C6
 Chief series C1 — C3 — C9 — C18 — C36 — C4×3- 1+2 — C36.C6 — Dic18.C6
 Lower central C9 — C18 — C36 — Dic18.C6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Dic18.C6
G = < a,b,c | a36=1, b2=c6=a18, bab-1=a-1, cac-1=a7, cbc-1=a9b >

Subgroups: 190 in 60 conjugacy classes, 26 normal (all characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, Q8, C9, C9, C32, Dic3, C12, C12, Q16, C18, C18, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C3×Q8, 3- 1+2, Dic9, C36, C36, C3×Dic3, C3×C12, C3×C12, C3⋊Q16, C3×Q16, C2×3- 1+2, C9⋊C8, Dic18, Q8×C9, Q8×C9, C3×C3⋊C8, C3×Dic6, Q8×C32, C9⋊C12, C4×3- 1+2, C4×3- 1+2, C9⋊Q16, C3×C3⋊Q16, C9⋊C24, C36.C6, Q8×3- 1+2, Dic18.C6
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, C3×S3, C3⋊D4, C3×D4, S3×C6, C3⋊Q16, C3×Q16, C9⋊C6, C3×C3⋊D4, C2×C9⋊C6, C3×C3⋊Q16, Dic9⋊C6, Dic18.C6

Smallest permutation representation of Dic18.C6
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)
(1 101 19 83)(2 100 20 82)(3 99 21 81)(4 98 22 80)(5 97 23 79)(6 96 24 78)(7 95 25 77)(8 94 26 76)(9 93 27 75)(10 92 28 74)(11 91 29 73)(12 90 30 108)(13 89 31 107)(14 88 32 106)(15 87 33 105)(16 86 34 104)(17 85 35 103)(18 84 36 102)(37 135 55 117)(38 134 56 116)(39 133 57 115)(40 132 58 114)(41 131 59 113)(42 130 60 112)(43 129 61 111)(44 128 62 110)(45 127 63 109)(46 126 64 144)(47 125 65 143)(48 124 66 142)(49 123 67 141)(50 122 68 140)(51 121 69 139)(52 120 70 138)(53 119 71 137)(54 118 72 136)
(1 72 19 54)(2 67 8 37 14 43 20 49 26 55 32 61)(3 62 33 56 27 50 21 44 15 38 9 68)(4 57 22 39)(5 52 11 58 17 64 23 70 29 40 35 46)(6 47 36 41 30 71 24 65 18 59 12 53)(7 42 25 60)(10 63 28 45)(13 48 31 66)(16 69 34 51)(73 123 103 117 97 111 91 141 85 135 79 129)(74 118 92 136)(75 113 81 119 87 125 93 131 99 137 105 143)(76 144 106 138 100 132 94 126 88 120 82 114)(77 139 95 121)(78 134 84 140 90 110 96 116 102 122 108 128)(80 124 98 142)(83 109 101 127)(86 130 104 112)(89 115 107 133)

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

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

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

41 conjugacy classes

 class 1 2 3A 3B 3C 4A 4B 4C 6A 6B 6C 8A 8B 9A 9B 9C 12A 12B 12C 12D 12E 12F 12G 12H 12I 18A 18B 18C 24A 24B 24C 24D 36A ··· 36I order 1 2 3 3 3 4 4 4 6 6 6 8 8 9 9 9 12 12 12 12 12 12 12 12 12 18 18 18 24 24 24 24 36 ··· 36 size 1 1 2 3 3 2 4 36 2 3 3 18 18 6 6 6 4 4 4 6 6 12 12 36 36 6 6 6 18 18 18 18 12 ··· 12

41 irreducible representations

 dim 1 1 1 1 1 1 1 1 12 2 2 2 2 2 2 2 2 2 2 4 4 6 6 6 type + + + + - + + + - - + + image C1 C2 C2 C2 C3 C6 C6 C6 Dic18.C6 S3 D4 D6 Q16 C3×S3 C3×D4 C3⋊D4 S3×C6 C3×Q16 C3×C3⋊D4 C3⋊Q16 C3×C3⋊Q16 C9⋊C6 C2×C9⋊C6 Dic9⋊C6 kernel Dic18.C6 C9⋊C24 C36.C6 Q8×3- 1+2 C9⋊Q16 C9⋊C8 Dic18 Q8×C9 C1 Q8×C32 C2×3- 1+2 C3×C12 3- 1+2 C3×Q8 C18 C3×C6 C12 C9 C6 C32 C3 Q8 C4 C2 # reps 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 4 4 1 2 1 1 2

Matrix representation of Dic18.C6 in GL10(𝔽73)

 0 0 1 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 0 0 0 72 72 2 1 0 0 0 0 0 0 72 72 1 2 0 0 0 0 0 0 0 0 1 1 72 72 0 0 0 0 72 72 1 1 1 0 0 0 0 0 0 72 1 1 0 0 0 0 0 0 72 0 1 1 0 0
,
 41 45 22 1 0 0 0 0 0 0 13 32 23 51 0 0 0 0 0 0 22 1 32 28 0 0 0 0 0 0 23 51 60 41 0 0 0 0 0 0 0 0 0 0 62 71 0 0 0 0 0 0 0 0 60 11 0 0 0 0 0 0 0 0 0 11 0 0 62 60 0 0 0 0 62 0 0 0 71 11 0 0 0 0 0 11 62 60 0 0 0 0 0 0 62 0 71 11 0 0
,
 65 16 58 30 0 0 0 0 0 0 57 8 43 15 0 0 0 0 0 0 58 30 8 57 0 0 0 0 0 0 43 15 16 65 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 72 72 0 0 0 0 0 0 1 1 0 0 72 72 0 0 0 0 0 0 0 0 1 0

G:=sub<GL(10,GF(73))| [0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,72,72,0,72,0,72,0,0,0,0,72,72,0,72,72,0,0,0,0,0,2,1,1,1,1,1,0,0,0,0,1,2,1,1,1,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0],[41,13,22,23,0,0,0,0,0,0,45,32,1,51,0,0,0,0,0,0,22,23,32,60,0,0,0,0,0,0,1,51,28,41,0,0,0,0,0,0,0,0,0,0,62,60,0,62,0,62,0,0,0,0,71,11,11,0,11,0,0,0,0,0,0,0,0,0,62,71,0,0,0,0,0,0,0,0,60,11,0,0,0,0,0,0,62,71,0,0,0,0,0,0,0,0,60,11,0,0],[65,57,58,43,0,0,0,0,0,0,16,8,30,15,0,0,0,0,0,0,58,43,8,16,0,0,0,0,0,0,30,15,57,65,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0] >;

Dic18.C6 in GAP, Magma, Sage, TeX

{\rm Dic}_{18}.C_6
% in TeX

G:=Group("Dic18.C6");
// GroupNames label

G:=SmallGroup(432,162);
// by ID

G=gap.SmallGroup(432,162);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,176,1011,514,80,10085,2035,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^36=1,b^2=c^6=a^18,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b>;
// generators/relations

׿
×
𝔽