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## G = D5×Q32order 320 = 26·5

### Direct product of D5 and Q32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D5×Q32
 Chief series C1 — C5 — C10 — C20 — C40 — C8×D5 — D5×Q16 — D5×Q32
 Lower central C5 — C10 — C20 — C40 — D5×Q32
 Upper central C1 — C2 — C4 — C8 — Q32

Generators and relations for D5×Q32
G = < a,b,c,d | a5=b2=c16=1, d2=c8, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 358 in 82 conjugacy classes, 33 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, Q8, D5, C10, C16, C16, C2×C8, Q16, Q16, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C16, Q32, Q32, C2×Q16, C52C8, C40, Dic10, C4×D5, C4×D5, C5×Q8, C2×Q32, C52C16, C80, C8×D5, Dic20, C5⋊Q16, C5×Q16, Q8×D5, D5×C16, Dic40, C5⋊Q32, C5×Q32, D5×Q16, D5×Q32
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, Q32, C2×D8, C22×D5, C2×Q32, D4×D5, D5×D8, D5×Q32

Smallest permutation representation of D5×Q32
On 160 points
Generators in S160
(1 70 120 35 58)(2 71 121 36 59)(3 72 122 37 60)(4 73 123 38 61)(5 74 124 39 62)(6 75 125 40 63)(7 76 126 41 64)(8 77 127 42 49)(9 78 128 43 50)(10 79 113 44 51)(11 80 114 45 52)(12 65 115 46 53)(13 66 116 47 54)(14 67 117 48 55)(15 68 118 33 56)(16 69 119 34 57)(17 83 153 130 105)(18 84 154 131 106)(19 85 155 132 107)(20 86 156 133 108)(21 87 157 134 109)(22 88 158 135 110)(23 89 159 136 111)(24 90 160 137 112)(25 91 145 138 97)(26 92 146 139 98)(27 93 147 140 99)(28 94 148 141 100)(29 95 149 142 101)(30 96 150 143 102)(31 81 151 144 103)(32 82 152 129 104)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 49)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 76)(34 77)(35 78)(36 79)(37 80)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(81 111)(82 112)(83 97)(84 98)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)(129 160)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)(137 152)(138 153)(139 154)(140 155)(141 156)(142 157)(143 158)(144 159)
(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)
(1 141 9 133)(2 140 10 132)(3 139 11 131)(4 138 12 130)(5 137 13 129)(6 136 14 144)(7 135 15 143)(8 134 16 142)(17 123 25 115)(18 122 26 114)(19 121 27 113)(20 120 28 128)(21 119 29 127)(22 118 30 126)(23 117 31 125)(24 116 32 124)(33 96 41 88)(34 95 42 87)(35 94 43 86)(36 93 44 85)(37 92 45 84)(38 91 46 83)(39 90 47 82)(40 89 48 81)(49 157 57 149)(50 156 58 148)(51 155 59 147)(52 154 60 146)(53 153 61 145)(54 152 62 160)(55 151 63 159)(56 150 64 158)(65 105 73 97)(66 104 74 112)(67 103 75 111)(68 102 76 110)(69 101 77 109)(70 100 78 108)(71 99 79 107)(72 98 80 106)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A 10B 16A 16B 16C 16D 16E 16F 16G 16H 20A 20B 20C 20D 20E 20F 40A 40B 40C 40D 80A ··· 80H order 1 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 10 16 16 16 16 16 16 16 16 20 20 20 20 20 20 40 40 40 40 80 ··· 80 size 1 1 5 5 2 8 8 10 40 40 2 2 2 2 10 10 2 2 2 2 2 2 10 10 10 10 4 4 16 16 16 16 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 Q32 D4×D5 D5×D8 D5×Q32 kernel D5×Q32 D5×C16 Dic40 C5⋊Q32 C5×Q32 D5×Q16 C5⋊2C8 C4×D5 Q32 Dic5 D10 C16 Q16 D5 C4 C2 C1 # reps 1 1 1 2 1 2 1 1 2 2 2 2 4 8 2 4 8

Matrix representation of D5×Q32 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 189 1 0 0 240 0
,
 240 0 0 0 0 240 0 0 0 0 1 189 0 0 0 240
,
 58 57 0 0 34 129 0 0 0 0 240 0 0 0 0 240
,
 5 237 0 0 127 236 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,189,240,0,0,1,0],[240,0,0,0,0,240,0,0,0,0,1,0,0,0,189,240],[58,34,0,0,57,129,0,0,0,0,240,0,0,0,0,240],[5,127,0,0,237,236,0,0,0,0,1,0,0,0,0,1] >;

D5×Q32 in GAP, Magma, Sage, TeX

D_5\times Q_{32}
% in TeX

G:=Group("D5xQ32");
// GroupNames label

G:=SmallGroup(320,544);
// by ID

G=gap.SmallGroup(320,544);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,135,184,346,185,192,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^16=1,d^2=c^8,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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