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## G = C5×D4○C16order 320 = 26·5

### Direct product of C5 and D4○C16

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×D4○C16
 Chief series C1 — C2 — C4 — C8 — C40 — C80 — C2×C80 — C5×D4○C16
 Lower central C1 — C2 — C5×D4○C16
 Upper central C1 — C80 — C5×D4○C16

Generators and relations for C5×D4○C16
G = < a,b,c,d | a5=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >

Subgroups: 90 in 84 conjugacy classes, 78 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, C10, C10, C16, C16, C2×C8, M4(2), C4○D4, C20, C20, C2×C10, C2×C16, M5(2), C8○D4, C40, C40, C2×C20, C5×D4, C5×Q8, D4○C16, C80, C80, C2×C40, C5×M4(2), C5×C4○D4, C2×C80, C5×M5(2), C5×C8○D4, C5×D4○C16
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, C23, C10, C2×C8, C22×C4, C20, C2×C10, C22×C8, C40, C2×C20, C22×C10, D4○C16, C2×C40, C22×C20, C22×C40, C5×D4○C16

Smallest permutation representation of C5×D4○C16
On 160 points
Generators in S160
(1 106 130 43 55)(2 107 131 44 56)(3 108 132 45 57)(4 109 133 46 58)(5 110 134 47 59)(6 111 135 48 60)(7 112 136 33 61)(8 97 137 34 62)(9 98 138 35 63)(10 99 139 36 64)(11 100 140 37 49)(12 101 141 38 50)(13 102 142 39 51)(14 103 143 40 52)(15 104 144 41 53)(16 105 129 42 54)(17 94 128 69 153)(18 95 113 70 154)(19 96 114 71 155)(20 81 115 72 156)(21 82 116 73 157)(22 83 117 74 158)(23 84 118 75 159)(24 85 119 76 160)(25 86 120 77 145)(26 87 121 78 146)(27 88 122 79 147)(28 89 123 80 148)(29 90 124 65 149)(30 91 125 66 150)(31 92 126 67 151)(32 93 127 68 152)
(1 123 9 115)(2 124 10 116)(3 125 11 117)(4 126 12 118)(5 127 13 119)(6 128 14 120)(7 113 15 121)(8 114 16 122)(17 40 25 48)(18 41 26 33)(19 42 27 34)(20 43 28 35)(21 44 29 36)(22 45 30 37)(23 46 31 38)(24 47 32 39)(49 83 57 91)(50 84 58 92)(51 85 59 93)(52 86 60 94)(53 87 61 95)(54 88 62 96)(55 89 63 81)(56 90 64 82)(65 99 73 107)(66 100 74 108)(67 101 75 109)(68 102 76 110)(69 103 77 111)(70 104 78 112)(71 105 79 97)(72 106 80 98)(129 147 137 155)(130 148 138 156)(131 149 139 157)(132 150 140 158)(133 151 141 159)(134 152 142 160)(135 153 143 145)(136 154 144 146)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 113)(16 114)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 89)(64 90)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 97)(80 98)(129 155)(130 156)(131 157)(132 158)(133 159)(134 160)(135 145)(136 146)(137 147)(138 148)(139 149)(140 150)(141 151)(142 152)(143 153)(144 154)
(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)

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

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

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

200 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 5C 5D 8A 8B 8C 8D 8E ··· 8J 10A 10B 10C 10D 10E ··· 10P 16A ··· 16H 16I ··· 16T 20A ··· 20H 20I ··· 20T 40A ··· 40P 40Q ··· 40AN 80A ··· 80AF 80AG ··· 80CB order 1 2 2 2 2 4 4 4 4 4 5 5 5 5 8 8 8 8 8 ··· 8 10 10 10 10 10 ··· 10 16 ··· 16 16 ··· 16 20 ··· 20 20 ··· 20 40 ··· 40 40 ··· 40 80 ··· 80 80 ··· 80 size 1 1 2 2 2 1 1 2 2 2 1 1 1 1 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

200 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C4 C4 C5 C8 C8 C10 C10 C10 C20 C20 C40 C40 D4○C16 C5×D4○C16 kernel C5×D4○C16 C2×C80 C5×M5(2) C5×C8○D4 C5×M4(2) C5×C4○D4 D4○C16 C5×D4 C5×Q8 C2×C16 M5(2) C8○D4 M4(2) C4○D4 D4 Q8 C5 C1 # reps 1 3 3 1 6 2 4 12 4 12 12 4 24 8 48 16 8 32

Matrix representation of C5×D4○C16 in GL3(𝔽241) generated by

 91 0 0 0 1 0 0 0 1
,
 240 0 0 0 0 1 0 240 0
,
 1 0 0 0 0 1 0 1 0
,
 240 0 0 0 111 0 0 0 111
G:=sub<GL(3,GF(241))| [91,0,0,0,1,0,0,0,1],[240,0,0,0,0,240,0,1,0],[1,0,0,0,0,1,0,1,0],[240,0,0,0,111,0,0,0,111] >;

C5×D4○C16 in GAP, Magma, Sage, TeX

C_5\times D_4\circ C_{16}
% in TeX

G:=Group("C5xD4oC16");
// GroupNames label

G:=SmallGroup(320,1005);
// by ID

G=gap.SmallGroup(320,1005);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,1731,102,124]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=c^2=1,d^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations

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