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## G = Q16×3- 1+2order 432 = 24·33

### Direct product of Q16 and 3- 1+2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Q16×3- 1+2
 Chief series C1 — C2 — C6 — C12 — C3×C12 — C4×3- 1+2 — Q8×3- 1+2 — Q16×3- 1+2
 Lower central C1 — C2 — C12 — Q16×3- 1+2
 Upper central C1 — C6 — C4×3- 1+2 — Q16×3- 1+2

Generators and relations for Q16×3- 1+2
G = < a,b,c,d | a8=c9=d3=1, b2=a4, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 110 in 72 conjugacy classes, 49 normal (20 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C9, C32, C12, C12, Q16, C18, C3×C6, C24, C24, C3×Q8, C3×Q8, 3- 1+2, C36, C36, C3×C12, C3×C12, C3×Q16, C3×Q16, C2×3- 1+2, C72, Q8×C9, C3×C24, Q8×C32, C4×3- 1+2, C4×3- 1+2, C9×Q16, C32×Q16, C8×3- 1+2, Q8×3- 1+2, Q16×3- 1+2
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, Q16, C3×C6, C3×D4, 3- 1+2, C62, C3×Q16, C2×3- 1+2, D4×C32, C22×3- 1+2, C32×Q16, D4×3- 1+2, Q16×3- 1+2

Smallest permutation representation of Q16×3- 1+2
On 144 points
Generators in S144
(1 138 54 15 42 21 63 31)(2 139 46 16 43 22 55 32)(3 140 47 17 44 23 56 33)(4 141 48 18 45 24 57 34)(5 142 49 10 37 25 58 35)(6 143 50 11 38 26 59 36)(7 144 51 12 39 27 60 28)(8 136 52 13 40 19 61 29)(9 137 53 14 41 20 62 30)(64 112 82 100 97 118 76 133)(65 113 83 101 98 119 77 134)(66 114 84 102 99 120 78 135)(67 115 85 103 91 121 79 127)(68 116 86 104 92 122 80 128)(69 117 87 105 93 123 81 129)(70 109 88 106 94 124 73 130)(71 110 89 107 95 125 74 131)(72 111 90 108 96 126 75 132)
(1 78 42 84)(2 79 43 85)(3 80 44 86)(4 81 45 87)(5 73 37 88)(6 74 38 89)(7 75 39 90)(8 76 40 82)(9 77 41 83)(10 106 35 130)(11 107 36 131)(12 108 28 132)(13 100 29 133)(14 101 30 134)(15 102 31 135)(16 103 32 127)(17 104 33 128)(18 105 34 129)(19 112 136 118)(20 113 137 119)(21 114 138 120)(22 115 139 121)(23 116 140 122)(24 117 141 123)(25 109 142 124)(26 110 143 125)(27 111 144 126)(46 91 55 67)(47 92 56 68)(48 93 57 69)(49 94 58 70)(50 95 59 71)(51 96 60 72)(52 97 61 64)(53 98 62 65)(54 99 63 66)
(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 8 5)(3 6 9)(10 16 13)(11 14 17)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(37 43 40)(38 41 44)(46 52 49)(47 50 53)(55 61 58)(56 59 62)(64 70 67)(65 68 71)(73 79 76)(74 77 80)(82 88 85)(83 86 89)(91 97 94)(92 95 98)(100 106 103)(101 104 107)(109 115 112)(110 113 116)(118 124 121)(119 122 125)(127 133 130)(128 131 134)(136 142 139)(137 140 143)

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

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

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

77 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 4C 6A 6B 6C 6D 8A 8B 9A ··· 9F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 18A ··· 18F 24A 24B 24C 24D 24E 24F 24G 24H 36A ··· 36F 36G ··· 36R 72A ··· 72L order 1 2 3 3 3 3 4 4 4 6 6 6 6 8 8 9 ··· 9 12 12 12 12 12 12 12 12 12 12 12 12 18 ··· 18 24 24 24 24 24 24 24 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 1 1 3 3 2 4 4 1 1 3 3 2 2 3 ··· 3 2 2 4 4 4 4 6 6 12 12 12 12 3 ··· 3 2 2 2 2 6 6 6 6 6 ··· 6 12 ··· 12 6 ··· 6

77 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 6 6 type + + + + - image C1 C2 C2 C3 C3 C6 C6 C6 C6 D4 Q16 C3×D4 C3×D4 C3×Q16 C3×Q16 3- 1+2 C2×3- 1+2 C2×3- 1+2 D4×3- 1+2 Q16×3- 1+2 kernel Q16×3- 1+2 C8×3- 1+2 Q8×3- 1+2 C9×Q16 C32×Q16 C72 Q8×C9 C3×C24 Q8×C32 C2×3- 1+2 3- 1+2 C18 C3×C6 C9 C32 Q16 C8 Q8 C2 C1 # reps 1 1 2 6 2 6 12 2 4 1 2 6 2 12 4 2 2 4 2 4

Matrix representation of Q16×3- 1+2 in GL5(𝔽73)

 57 16 0 0 0 57 57 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 64 8 0 0 0 8 9 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 64 0 0 0 0 0 64 0 0 0 0 0 8 7 0 0 0 64 65 8 0 0 0 9 0
,
 64 0 0 0 0 0 64 0 0 0 0 0 1 0 0 0 0 8 8 0 0 0 65 0 64

G:=sub<GL(5,GF(73))| [57,57,0,0,0,16,57,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[64,8,0,0,0,8,9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[64,0,0,0,0,0,64,0,0,0,0,0,8,64,0,0,0,7,65,9,0,0,0,8,0],[64,0,0,0,0,0,64,0,0,0,0,0,1,8,65,0,0,0,8,0,0,0,0,0,64] >;

Q16×3- 1+2 in GAP, Magma, Sage, TeX

Q_{16}\times 3_-^{1+2}
% in TeX

G:=Group("Q16xES-(3,1)");
// GroupNames label

G:=SmallGroup(432,223);
// by ID

G=gap.SmallGroup(432,223);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,1512,533,1520,394,605,8824,4421,242]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^9=d^3=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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