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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 [×3], C4, C4 [×3], C22 [×3], C5, C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, C10, C10 [×3], C16, C16 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C20, C20 [×3], C2×C10 [×3], C2×C16 [×3], M5(2) [×3], C8○D4, C40, C40 [×3], C2×C20 [×3], C5×D4 [×3], C5×Q8, D4○C16, C80, C80 [×3], C2×C40 [×3], C5×M4(2) [×3], C5×C4○D4, C2×C80 [×3], C5×M5(2) [×3], C5×C8○D4, C5×D4○C16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C8 [×4], C2×C4 [×6], C23, C10 [×7], C2×C8 [×6], C22×C4, C20 [×4], C2×C10 [×7], C22×C8, C40 [×4], C2×C20 [×6], C22×C10, D4○C16, C2×C40 [×6], C22×C20, C22×C40, C5×D4○C16

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