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## G = C5×C8⋊D4order 320 = 26·5

### Direct product of C5 and C8⋊D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C8⋊D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — D4×C10 — C10×SD16 — C5×C8⋊D4
 Lower central C1 — C2 — C2×C4 — C5×C8⋊D4
 Upper central C1 — C2×C10 — C22×C20 — C5×C8⋊D4

Generators and relations for C5×C8⋊D4
G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b3, dcd=c-1 >

Subgroups: 234 in 120 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C40, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C8⋊D4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C5×M4(2), C5×SD16, C22×C20, D4×C10, D4×C10, Q8×C10, C5×D4⋊C4, C5×Q8⋊C4, C5×C2.D8, C5×C4⋊D4, C5×C22⋊Q8, C10×M4(2), C10×SD16, C5×C8⋊D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4⋊D4, C8⋊C22, C8.C22, C5×D4, C22×C10, C8⋊D4, D4×C10, C5×C4○D4, C5×C4⋊D4, C5×C8⋊C22, C5×C8.C22, C5×C8⋊D4

Smallest permutation representation of C5×C8⋊D4
On 160 points
Generators in S160
(1 143 61 135 53)(2 144 62 136 54)(3 137 63 129 55)(4 138 64 130 56)(5 139 57 131 49)(6 140 58 132 50)(7 141 59 133 51)(8 142 60 134 52)(9 33 20 94 99)(10 34 21 95 100)(11 35 22 96 101)(12 36 23 89 102)(13 37 24 90 103)(14 38 17 91 104)(15 39 18 92 97)(16 40 19 93 98)(25 41 126 118 107)(26 42 127 119 108)(27 43 128 120 109)(28 44 121 113 110)(29 45 122 114 111)(30 46 123 115 112)(31 47 124 116 105)(32 48 125 117 106)(65 83 155 73 147)(66 84 156 74 148)(67 85 157 75 149)(68 86 158 76 150)(69 87 159 77 151)(70 88 160 78 152)(71 81 153 79 145)(72 82 154 80 146)
(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 69 9 28)(2 68 10 27)(3 67 11 26)(4 66 12 25)(5 65 13 32)(6 72 14 31)(7 71 15 30)(8 70 16 29)(17 124 58 154)(18 123 59 153)(19 122 60 160)(20 121 61 159)(21 128 62 158)(22 127 63 157)(23 126 64 156)(24 125 57 155)(33 44 143 87)(34 43 144 86)(35 42 137 85)(36 41 138 84)(37 48 139 83)(38 47 140 82)(39 46 141 81)(40 45 142 88)(49 147 103 106)(50 146 104 105)(51 145 97 112)(52 152 98 111)(53 151 99 110)(54 150 100 109)(55 149 101 108)(56 148 102 107)(73 90 117 131)(74 89 118 130)(75 96 119 129)(76 95 120 136)(77 94 113 135)(78 93 114 134)(79 92 115 133)(80 91 116 132)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 19)(18 22)(21 23)(25 68)(26 71)(27 66)(28 69)(29 72)(30 67)(31 70)(32 65)(34 36)(35 39)(38 40)(41 86)(42 81)(43 84)(44 87)(45 82)(46 85)(47 88)(48 83)(50 52)(51 55)(54 56)(58 60)(59 63)(62 64)(73 117)(74 120)(75 115)(76 118)(77 113)(78 116)(79 119)(80 114)(89 95)(91 93)(92 96)(97 101)(98 104)(100 102)(105 152)(106 147)(107 150)(108 145)(109 148)(110 151)(111 146)(112 149)(121 159)(122 154)(123 157)(124 160)(125 155)(126 158)(127 153)(128 156)(129 133)(130 136)(132 134)(137 141)(138 144)(140 142)

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

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

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

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 C10 D4 D4 D4 C4○D4 C5×D4 C5×D4 C5×D4 C5×C4○D4 C8⋊C22 C8.C22 C5×C8⋊C22 C5×C8.C22 kernel C5×C8⋊D4 C5×D4⋊C4 C5×Q8⋊C4 C5×C2.D8 C5×C4⋊D4 C5×C22⋊Q8 C10×M4(2) C10×SD16 C8⋊D4 D4⋊C4 Q8⋊C4 C2.D8 C4⋊D4 C22⋊Q8 C2×M4(2) C2×SD16 C40 C2×C20 C22×C10 C20 C8 C2×C4 C23 C4 C10 C10 C2 C2 # reps 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 2 1 1 2 8 4 4 8 1 1 4 4

Matrix representation of C5×C8⋊D4 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 37 0 0 0 0 0 0 37 0 0 0 0 0 0 37 0 0 0 0 0 0 37
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 12 29 40 1 0 0 12 12 40 40 0 0 1 40 29 12 0 0 1 1 29 29
,
 40 2 0 0 0 0 40 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 1 0 0 0 0 0 0 40 0 0
,
 1 0 0 0 0 0 1 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,12,12,1,1,0,0,29,12,40,1,0,0,40,40,29,29,0,0,1,40,12,29],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,40,0,0],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

C5×C8⋊D4 in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes D_4
% in TeX

G:=Group("C5xC8:D4");
// GroupNames label

G:=SmallGroup(320,969);
// by ID

G=gap.SmallGroup(320,969);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,288,1766,1731,7004,172]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^3,d*c*d=c^-1>;
// generators/relations

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