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## G = C20⋊4SD16order 320 = 26·5

### 4th semidirect product of C20 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C20⋊4SD16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×Dic10 — C20⋊2Q8 — C20⋊4SD16
 Lower central C5 — C10 — C2×C20 — C20⋊4SD16
 Upper central C1 — C22 — C42 — C4⋊1D4

Generators and relations for C204SD16
G = < a,b,c | a20=b8=c2=1, bab-1=a9, cac=a11, cbc=b3 >

Subgroups: 510 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, C10, C10, C10, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C4×C8, C41D4, C4⋊Q8, C2×SD16, C52C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C85D4, C2×C52C8, C4⋊Dic5, D4.D5, C4×C20, C2×Dic10, D4×C10, D4×C10, C4×C52C8, C202Q8, C2×D4.D5, C5×C41D4, C204SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C41D4, C2×SD16, C5⋊D4, C22×D5, C85D4, D4.D5, D4×D5, C2×C5⋊D4, C2×D4.D5, C20⋊D4, C204SD16

Smallest permutation representation of C204SD16
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)
(1 120 28 81 70 127 154 48)(2 109 29 90 71 136 155 57)(3 118 30 99 72 125 156 46)(4 107 31 88 73 134 157 55)(5 116 32 97 74 123 158 44)(6 105 33 86 75 132 159 53)(7 114 34 95 76 121 160 42)(8 103 35 84 77 130 141 51)(9 112 36 93 78 139 142 60)(10 101 37 82 79 128 143 49)(11 110 38 91 80 137 144 58)(12 119 39 100 61 126 145 47)(13 108 40 89 62 135 146 56)(14 117 21 98 63 124 147 45)(15 106 22 87 64 133 148 54)(16 115 23 96 65 122 149 43)(17 104 24 85 66 131 150 52)(18 113 25 94 67 140 151 41)(19 102 26 83 68 129 152 50)(20 111 27 92 69 138 153 59)
(1 75)(2 66)(3 77)(4 68)(5 79)(6 70)(7 61)(8 72)(9 63)(10 74)(11 65)(12 76)(13 67)(14 78)(15 69)(16 80)(17 71)(18 62)(19 73)(20 64)(21 36)(22 27)(23 38)(24 29)(25 40)(26 31)(28 33)(30 35)(32 37)(34 39)(41 108)(42 119)(43 110)(44 101)(45 112)(46 103)(47 114)(48 105)(49 116)(50 107)(51 118)(52 109)(53 120)(54 111)(55 102)(56 113)(57 104)(58 115)(59 106)(60 117)(81 132)(82 123)(83 134)(84 125)(85 136)(86 127)(87 138)(88 129)(89 140)(90 131)(91 122)(92 133)(93 124)(94 135)(95 126)(96 137)(97 128)(98 139)(99 130)(100 121)(141 156)(142 147)(143 158)(144 149)(145 160)(146 151)(148 153)(150 155)(152 157)(154 159)

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

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

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

50 conjugacy classes

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

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 D4.D5 D4×D5 kernel C20⋊4SD16 C4×C5⋊2C8 C20⋊2Q8 C2×D4.D5 C5×C4⋊1D4 C5⋊2C8 C2×C20 C4⋊1D4 C20 C42 C2×D4 C2×C4 C4 C4 # reps 1 1 1 4 1 4 2 2 8 2 4 8 8 4

Matrix representation of C204SD16 in GL6(𝔽41)

 37 30 0 0 0 0 0 10 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 2 0 0 0 0 40 1
,
 40 16 0 0 0 0 5 1 0 0 0 0 0 0 0 32 0 0 0 0 9 30 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 25 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 24 40 0 0 0 0 0 0 1 39 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [37,0,0,0,0,0,30,10,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,40,0,0,0,0,2,1],[40,5,0,0,0,0,16,1,0,0,0,0,0,0,0,9,0,0,0,0,32,30,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,25,40,0,0,0,0,0,0,1,24,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,39,40] >;

C204SD16 in GAP, Magma, Sage, TeX

C_{20}\rtimes_4{\rm SD}_{16}
% in TeX

G:=Group("C20:4SD16");
// GroupNames label

G:=SmallGroup(320,703);
// by ID

G=gap.SmallGroup(320,703);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,219,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^9,c*a*c=a^11,c*b*c=b^3>;
// generators/relations

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