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## G = C5×D8.C4order 320 = 26·5

### Direct product of C5 and D8.C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C5×D8.C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C40 — C5×C8.C4 — C5×D8.C4
 Lower central C1 — C2 — C4 — C8 — C5×D8.C4
 Upper central C1 — C20 — C2×C20 — C2×C40 — C5×D8.C4

Generators and relations for C5×D8.C4
G = < a,b,c,d | a5=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=b5c >

Smallest permutation representation of C5×D8.C4
On 160 points
Generators in S160
(1 78 62 46 30)(2 79 63 47 31)(3 80 64 48 32)(4 73 57 41 25)(5 74 58 42 26)(6 75 59 43 27)(7 76 60 44 28)(8 77 61 45 29)(9 148 132 116 100)(10 149 133 117 101)(11 150 134 118 102)(12 151 135 119 103)(13 152 136 120 104)(14 145 129 113 97)(15 146 130 114 98)(16 147 131 115 99)(17 87 65 49 33)(18 88 66 50 34)(19 81 67 51 35)(20 82 68 52 36)(21 83 69 53 37)(22 84 70 54 38)(23 85 71 55 39)(24 86 72 56 40)(89 153 137 121 105)(90 154 138 122 106)(91 155 139 123 107)(92 156 140 124 108)(93 157 141 125 109)(94 158 142 126 110)(95 159 143 127 111)(96 160 144 128 112)
(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 8)(2 7)(3 6)(4 5)(9 13)(10 12)(14 16)(17 20)(18 19)(21 24)(22 23)(25 26)(27 32)(28 31)(29 30)(33 36)(34 35)(37 40)(38 39)(41 42)(43 48)(44 47)(45 46)(49 52)(50 51)(53 56)(54 55)(57 58)(59 64)(60 63)(61 62)(65 68)(66 67)(69 72)(70 71)(73 74)(75 80)(76 79)(77 78)(81 88)(82 87)(83 86)(84 85)(89 93)(90 92)(94 96)(97 99)(100 104)(101 103)(105 109)(106 108)(110 112)(113 115)(116 120)(117 119)(121 125)(122 124)(126 128)(129 131)(132 136)(133 135)(137 141)(138 140)(142 144)(145 147)(148 152)(149 151)(153 157)(154 156)(158 160)
(1 103 23 92 5 99 19 96)(2 102 24 91 6 98 20 95)(3 101 17 90 7 97 21 94)(4 100 18 89 8 104 22 93)(9 88 153 77 13 84 157 73)(10 87 154 76 14 83 158 80)(11 86 155 75 15 82 159 79)(12 85 156 74 16 81 160 78)(25 116 34 105 29 120 38 109)(26 115 35 112 30 119 39 108)(27 114 36 111 31 118 40 107)(28 113 37 110 32 117 33 106)(41 132 50 121 45 136 54 125)(42 131 51 128 46 135 55 124)(43 130 52 127 47 134 56 123)(44 129 53 126 48 133 49 122)(57 148 66 137 61 152 70 141)(58 147 67 144 62 151 71 140)(59 146 68 143 63 150 72 139)(60 145 69 142 64 149 65 138)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 8A 8B 8C 8D 8E 8F 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 16A ··· 16H 20A ··· 20H 20I 20J 20K 20L 20M 20N 20O 20P 40A ··· 40P 40Q ··· 40X 80A ··· 80AF order 1 2 2 2 4 4 4 4 5 5 5 5 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 10 10 10 16 ··· 16 20 ··· 20 20 20 20 20 20 20 20 20 40 ··· 40 40 ··· 40 80 ··· 80 size 1 1 2 8 1 1 2 8 1 1 1 1 2 2 2 2 8 8 1 1 1 1 2 2 2 2 8 8 8 8 2 ··· 2 1 ··· 1 2 2 2 2 8 8 8 8 2 ··· 2 8 ··· 8 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 D4 D4 D8 SD16 C5×D4 C5×D4 D8.C4 C5×D8 C5×SD16 C5×D8.C4 kernel C5×D8.C4 C5×C8.C4 C2×C80 C5×C4○D8 C5×D8 C5×Q16 D8.C4 C8.C4 C2×C16 C4○D8 D8 Q16 C40 C2×C20 C20 C2×C10 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 2 2 4 4 4 4 8 8 1 1 2 2 4 4 8 8 8 32

Matrix representation of C5×D8.C4 in GL2(𝔽241) generated by

 98 0 0 98
,
 11 230 11 11
,
 11 230 230 230
,
 25 43 43 216
G:=sub<GL(2,GF(241))| [98,0,0,98],[11,11,230,11],[11,230,230,230],[25,43,43,216] >;

C5×D8.C4 in GAP, Magma, Sage, TeX

C_5\times D_8.C_4
% in TeX

G:=Group("C5xD8.C4");
// GroupNames label

G:=SmallGroup(320,164);
// by ID

G=gap.SmallGroup(320,164);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,1410,360,172,10085,5052,124]);
// Polycyclic

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

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