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## G = C2×C5⋊SD32order 320 = 26·5

### Direct product of C2 and C5⋊SD32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C2×C5⋊SD32
 Chief series C1 — C5 — C10 — C20 — C40 — D40 — C2×D40 — C2×C5⋊SD32
 Lower central C5 — C10 — C20 — C40 — C2×C5⋊SD32
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×Q16

Generators and relations for C2×C5⋊SD32
G = < a,b,c,d | a2=b5=c16=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c7 >

Subgroups: 478 in 90 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C16, C2×C8, D8, Q16, Q16, C2×D4, C2×Q8, C20, C20, D10, C2×C10, C2×C16, SD32, C2×D8, C2×Q16, C40, D20, C2×C20, C2×C20, C5×Q8, C22×D5, C2×SD32, C52C16, D40, D40, C2×C40, C5×Q16, C5×Q16, C2×D20, Q8×C10, C2×C52C16, C5⋊SD32, C2×D40, C10×Q16, C2×C5⋊SD32
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, SD32, C2×D8, C5⋊D4, C22×D5, C2×SD32, D4⋊D5, C2×C5⋊D4, C5⋊SD32, C2×D4⋊D5, C2×C5⋊SD32

Smallest permutation representation of C2×C5⋊SD32
On 160 points
Generators in S160
(1 134)(2 135)(3 136)(4 137)(5 138)(6 139)(7 140)(8 141)(9 142)(10 143)(11 144)(12 129)(13 130)(14 131)(15 132)(16 133)(17 78)(18 79)(19 80)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(81 126)(82 127)(83 128)(84 113)(85 114)(86 115)(87 116)(88 117)(89 118)(90 119)(91 120)(92 121)(93 122)(94 123)(95 124)(96 125)(97 153)(98 154)(99 155)(100 156)(101 157)(102 158)(103 159)(104 160)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)
(1 58 155 91 24)(2 25 92 156 59)(3 60 157 93 26)(4 27 94 158 61)(5 62 159 95 28)(6 29 96 160 63)(7 64 145 81 30)(8 31 82 146 49)(9 50 147 83 32)(10 17 84 148 51)(11 52 149 85 18)(12 19 86 150 53)(13 54 151 87 20)(14 21 88 152 55)(15 56 153 89 22)(16 23 90 154 57)(33 135 70 121 100)(34 101 122 71 136)(35 137 72 123 102)(36 103 124 73 138)(37 139 74 125 104)(38 105 126 75 140)(39 141 76 127 106)(40 107 128 77 142)(41 143 78 113 108)(42 109 114 79 144)(43 129 80 115 110)(44 111 116 65 130)(45 131 66 117 112)(46 97 118 67 132)(47 133 68 119 98)(48 99 120 69 134)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 57)(18 64)(19 55)(20 62)(21 53)(22 60)(23 51)(24 58)(25 49)(26 56)(27 63)(28 54)(29 61)(30 52)(31 59)(32 50)(33 76)(34 67)(35 74)(36 65)(37 72)(38 79)(39 70)(40 77)(41 68)(42 75)(43 66)(44 73)(45 80)(46 71)(47 78)(48 69)(81 149)(82 156)(83 147)(84 154)(85 145)(86 152)(87 159)(88 150)(89 157)(90 148)(91 155)(92 146)(93 153)(94 160)(95 151)(96 158)(97 122)(98 113)(99 120)(100 127)(101 118)(102 125)(103 116)(104 123)(105 114)(106 121)(107 128)(108 119)(109 126)(110 117)(111 124)(112 115)(129 131)(130 138)(132 136)(133 143)(135 141)(137 139)(140 144)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 10A ··· 10F 16A ··· 16H 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 2 2 4 4 4 4 5 5 8 8 8 8 10 ··· 10 16 ··· 16 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 40 40 2 2 8 8 2 2 2 2 2 2 2 ··· 2 10 ··· 10 4 4 4 4 8 ··· 8 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 SD32 C5⋊D4 C5⋊D4 D4⋊D5 D4⋊D5 C5⋊SD32 kernel C2×C5⋊SD32 C2×C5⋊2C16 C5⋊SD32 C2×D40 C10×Q16 C40 C2×C20 C2×Q16 C20 C2×C10 C2×C8 Q16 C10 C8 C2×C4 C4 C22 C2 # reps 1 1 4 1 1 1 1 2 2 2 2 4 8 4 4 2 2 8

Matrix representation of C2×C5⋊SD32 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 240 0 0 0 0 240
,
 1 0 0 0 0 1 0 0 0 0 189 240 0 0 1 0
,
 97 160 0 0 223 179 0 0 0 0 240 0 0 0 52 1
,
 1 0 0 0 118 240 0 0 0 0 1 0 0 0 189 240
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,189,1,0,0,240,0],[97,223,0,0,160,179,0,0,0,0,240,52,0,0,0,1],[1,118,0,0,0,240,0,0,0,0,1,189,0,0,0,240] >;

C2×C5⋊SD32 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes {\rm SD}_{32}
% in TeX

G:=Group("C2xC5:SD32");
// GroupNames label

G:=SmallGroup(320,805);
// by ID

G=gap.SmallGroup(320,805);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,184,675,185,192,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^7>;
// generators/relations

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