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## G = C20⋊6SD16order 320 = 26·5

### 6th semidirect product of C20 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C20⋊6SD16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C20⋊4D4 — C20⋊6SD16
 Lower central C5 — C10 — C2×C20 — C20⋊6SD16
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for C206SD16
G = < a,b,c | a20=b8=c2=1, bab-1=a9, cac=a-1, cbc=b3 >

Subgroups: 702 in 142 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, C20, C20, D10, C2×C10, C4×C8, C41D4, C4⋊Q8, C2×SD16, C52C8, D20, C2×C20, C2×C20, C2×C20, C5×Q8, C22×D5, C85D4, C2×C52C8, Q8⋊D5, C4×C20, C5×C4⋊C4, C2×D20, C2×D20, Q8×C10, C4×C52C8, C204D4, C2×Q8⋊D5, C5×C4⋊Q8, C206SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C41D4, C2×SD16, C5⋊D4, C22×D5, C85D4, Q8⋊D5, D4×D5, C2×C5⋊D4, C20⋊D4, C2×Q8⋊D5, C206SD16

Smallest permutation representation of C206SD16
On 160 points
Generators in S160
(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)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 82 123 115 60 35 149 75)(2 91 124 104 41 24 150 64)(3 100 125 113 42 33 151 73)(4 89 126 102 43 22 152 62)(5 98 127 111 44 31 153 71)(6 87 128 120 45 40 154 80)(7 96 129 109 46 29 155 69)(8 85 130 118 47 38 156 78)(9 94 131 107 48 27 157 67)(10 83 132 116 49 36 158 76)(11 92 133 105 50 25 159 65)(12 81 134 114 51 34 160 74)(13 90 135 103 52 23 141 63)(14 99 136 112 53 32 142 72)(15 88 137 101 54 21 143 61)(16 97 138 110 55 30 144 70)(17 86 139 119 56 39 145 79)(18 95 140 108 57 28 146 68)(19 84 121 117 58 37 147 77)(20 93 122 106 59 26 148 66)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 63)(28 62)(29 61)(30 80)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 70)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(81 116)(82 115)(83 114)(84 113)(85 112)(86 111)(87 110)(88 109)(89 108)(90 107)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 120)(98 119)(99 118)(100 117)(121 151)(122 150)(123 149)(124 148)(125 147)(126 146)(127 145)(128 144)(129 143)(130 142)(131 141)(132 160)(133 159)(134 158)(135 157)(136 156)(137 155)(138 154)(139 153)(140 152)

G:=sub<Sym(160)| (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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,82,123,115,60,35,149,75)(2,91,124,104,41,24,150,64)(3,100,125,113,42,33,151,73)(4,89,126,102,43,22,152,62)(5,98,127,111,44,31,153,71)(6,87,128,120,45,40,154,80)(7,96,129,109,46,29,155,69)(8,85,130,118,47,38,156,78)(9,94,131,107,48,27,157,67)(10,83,132,116,49,36,158,76)(11,92,133,105,50,25,159,65)(12,81,134,114,51,34,160,74)(13,90,135,103,52,23,141,63)(14,99,136,112,53,32,142,72)(15,88,137,101,54,21,143,61)(16,97,138,110,55,30,144,70)(17,86,139,119,56,39,145,79)(18,95,140,108,57,28,146,68)(19,84,121,117,58,37,147,77)(20,93,122,106,59,26,148,66), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(81,116)(82,115)(83,114)(84,113)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,120)(98,119)(99,118)(100,117)(121,151)(122,150)(123,149)(124,148)(125,147)(126,146)(127,145)(128,144)(129,143)(130,142)(131,141)(132,160)(133,159)(134,158)(135,157)(136,156)(137,155)(138,154)(139,153)(140,152)>;

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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,82,123,115,60,35,149,75)(2,91,124,104,41,24,150,64)(3,100,125,113,42,33,151,73)(4,89,126,102,43,22,152,62)(5,98,127,111,44,31,153,71)(6,87,128,120,45,40,154,80)(7,96,129,109,46,29,155,69)(8,85,130,118,47,38,156,78)(9,94,131,107,48,27,157,67)(10,83,132,116,49,36,158,76)(11,92,133,105,50,25,159,65)(12,81,134,114,51,34,160,74)(13,90,135,103,52,23,141,63)(14,99,136,112,53,32,142,72)(15,88,137,101,54,21,143,61)(16,97,138,110,55,30,144,70)(17,86,139,119,56,39,145,79)(18,95,140,108,57,28,146,68)(19,84,121,117,58,37,147,77)(20,93,122,106,59,26,148,66), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(81,116)(82,115)(83,114)(84,113)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,120)(98,119)(99,118)(100,117)(121,151)(122,150)(123,149)(124,148)(125,147)(126,146)(127,145)(128,144)(129,143)(130,142)(131,141)(132,160)(133,159)(134,158)(135,157)(136,156)(137,155)(138,154)(139,153)(140,152) );

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),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,82,123,115,60,35,149,75),(2,91,124,104,41,24,150,64),(3,100,125,113,42,33,151,73),(4,89,126,102,43,22,152,62),(5,98,127,111,44,31,153,71),(6,87,128,120,45,40,154,80),(7,96,129,109,46,29,155,69),(8,85,130,118,47,38,156,78),(9,94,131,107,48,27,157,67),(10,83,132,116,49,36,158,76),(11,92,133,105,50,25,159,65),(12,81,134,114,51,34,160,74),(13,90,135,103,52,23,141,63),(14,99,136,112,53,32,142,72),(15,88,137,101,54,21,143,61),(16,97,138,110,55,30,144,70),(17,86,139,119,56,39,145,79),(18,95,140,108,57,28,146,68),(19,84,121,117,58,37,147,77),(20,93,122,106,59,26,148,66)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,69),(22,68),(23,67),(24,66),(25,65),(26,64),(27,63),(28,62),(29,61),(30,80),(31,79),(32,78),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71),(40,70),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(81,116),(82,115),(83,114),(84,113),(85,112),(86,111),(87,110),(88,109),(89,108),(90,107),(91,106),(92,105),(93,104),(94,103),(95,102),(96,101),(97,120),(98,119),(99,118),(100,117),(121,151),(122,150),(123,149),(124,148),(125,147),(126,146),(127,145),(128,144),(129,143),(130,142),(131,141),(132,160),(133,159),(134,158),(135,157),(136,156),(137,155),(138,154),(139,153),(140,152)]])

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4F 4G 4H 5A 5B 8A ··· 8H 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 2 2 2 4 ··· 4 4 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 40 40 2 ··· 2 8 8 2 2 10 ··· 10 2 ··· 2 4 ··· 4 8 ··· 8

50 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D5 SD16 D10 D10 C5⋊D4 Q8⋊D5 D4×D5 kernel C20⋊6SD16 C4×C5⋊2C8 C20⋊4D4 C2×Q8⋊D5 C5×C4⋊Q8 C5⋊2C8 C2×C20 C4⋊Q8 C20 C42 C2×Q8 C2×C4 C4 C4 # reps 1 1 1 4 1 4 2 2 8 2 4 8 8 4

Matrix representation of C206SD16 in GL6(𝔽41)

 40 39 0 0 0 0 1 1 0 0 0 0 0 0 22 10 0 0 0 0 13 19 0 0 0 0 0 0 0 40 0 0 0 0 1 34
,
 0 11 0 0 0 0 15 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 34 0 0 0 0 1 34
,
 1 0 0 0 0 0 40 40 0 0 0 0 0 0 1 0 0 0 0 0 12 40 0 0 0 0 0 0 34 7 0 0 0 0 40 7

G:=sub<GL(6,GF(41))| [40,1,0,0,0,0,39,1,0,0,0,0,0,0,22,13,0,0,0,0,10,19,0,0,0,0,0,0,0,1,0,0,0,0,40,34],[0,15,0,0,0,0,11,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,1,0,0,0,0,34,34],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,1,12,0,0,0,0,0,40,0,0,0,0,0,0,34,40,0,0,0,0,7,7] >;

C206SD16 in GAP, Magma, Sage, TeX

C_{20}\rtimes_6{\rm SD}_{16}
% in TeX

G:=Group("C20:6SD16");
// GroupNames label

G:=SmallGroup(320,712);
// by ID

G=gap.SmallGroup(320,712);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,219,184,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=b^3>;
// generators/relations

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