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## G = C5×Q32⋊C2order 320 = 26·5

### Direct product of C5 and Q32⋊C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C5×Q32⋊C2
 Chief series C1 — C2 — C4 — C8 — C40 — C5×D8 — C5×SD32 — C5×Q32⋊C2
 Lower central C1 — C2 — C4 — C8 — C5×Q32⋊C2
 Upper central C1 — C10 — C2×C20 — C2×C40 — C5×Q32⋊C2

Generators and relations for C5×Q32⋊C2
G = < a,b,c,d | a5=b16=d2=1, c2=b8, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, cd=dc >

Subgroups: 162 in 82 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C10, C10, C16, C2×C8, D8, SD16, Q16, Q16, Q16, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, M5(2), SD32, Q32, C2×Q16, C4○D8, C40, C2×C20, C2×C20, C5×D4, C5×Q8, Q32⋊C2, C80, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×Q16, C5×Q16, Q8×C10, C5×C4○D4, C5×M5(2), C5×SD32, C5×Q32, C10×Q16, C5×C4○D8, C5×Q32⋊C2
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C2×C10, C2×D8, C5×D4, C22×C10, Q32⋊C2, C5×D8, D4×C10, C10×D8, C5×Q32⋊C2

Smallest permutation representation of C5×Q32⋊C2
On 160 points
Generators in S160
(1 119 91 145 17)(2 120 92 146 18)(3 121 93 147 19)(4 122 94 148 20)(5 123 95 149 21)(6 124 96 150 22)(7 125 81 151 23)(8 126 82 152 24)(9 127 83 153 25)(10 128 84 154 26)(11 113 85 155 27)(12 114 86 156 28)(13 115 87 157 29)(14 116 88 158 30)(15 117 89 159 31)(16 118 90 160 32)(33 80 131 49 107)(34 65 132 50 108)(35 66 133 51 109)(36 67 134 52 110)(37 68 135 53 111)(38 69 136 54 112)(39 70 137 55 97)(40 71 138 56 98)(41 72 139 57 99)(42 73 140 58 100)(43 74 141 59 101)(44 75 142 60 102)(45 76 143 61 103)(46 77 144 62 104)(47 78 129 63 105)(48 79 130 64 106)
(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 43 9 35)(2 42 10 34)(3 41 11 33)(4 40 12 48)(5 39 13 47)(6 38 14 46)(7 37 15 45)(8 36 16 44)(17 101 25 109)(18 100 26 108)(19 99 27 107)(20 98 28 106)(21 97 29 105)(22 112 30 104)(23 111 31 103)(24 110 32 102)(49 147 57 155)(50 146 58 154)(51 145 59 153)(52 160 60 152)(53 159 61 151)(54 158 62 150)(55 157 63 149)(56 156 64 148)(65 120 73 128)(66 119 74 127)(67 118 75 126)(68 117 76 125)(69 116 77 124)(70 115 78 123)(71 114 79 122)(72 113 80 121)(81 135 89 143)(82 134 90 142)(83 133 91 141)(84 132 92 140)(85 131 93 139)(86 130 94 138)(87 129 95 137)(88 144 96 136)
(1 34)(2 43)(3 36)(4 45)(5 38)(6 47)(7 40)(8 33)(9 42)(10 35)(11 44)(12 37)(13 46)(14 39)(15 48)(16 41)(17 108)(18 101)(19 110)(20 103)(21 112)(22 105)(23 98)(24 107)(25 100)(26 109)(27 102)(28 111)(29 104)(30 97)(31 106)(32 99)(49 152)(50 145)(51 154)(52 147)(53 156)(54 149)(55 158)(56 151)(57 160)(58 153)(59 146)(60 155)(61 148)(62 157)(63 150)(64 159)(65 119)(66 128)(67 121)(68 114)(69 123)(70 116)(71 125)(72 118)(73 127)(74 120)(75 113)(76 122)(77 115)(78 124)(79 117)(80 126)(81 138)(82 131)(83 140)(84 133)(85 142)(86 135)(87 144)(88 137)(89 130)(90 139)(91 132)(92 141)(93 134)(94 143)(95 136)(96 129)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 5A 5B 5C 5D 8A 8B 8C 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 16A 16B 16C 16D 20A ··· 20H 20I ··· 20T 40A ··· 40H 40I 40J 40K 40L 80A ··· 80P order 1 2 2 2 4 4 4 4 4 5 5 5 5 8 8 8 10 10 10 10 10 10 10 10 10 10 10 10 16 16 16 16 20 ··· 20 20 ··· 20 40 ··· 40 40 40 40 40 80 ··· 80 size 1 1 2 8 2 2 8 8 8 1 1 1 1 2 2 4 1 1 1 1 2 2 2 2 8 8 8 8 4 4 4 4 2 ··· 2 8 ··· 8 2 ··· 2 4 4 4 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 D8 D8 C5×D4 C5×D4 C5×D8 C5×D8 Q32⋊C2 C5×Q32⋊C2 kernel C5×Q32⋊C2 C5×M5(2) C5×SD32 C5×Q32 C10×Q16 C5×C4○D8 Q32⋊C2 M5(2) SD32 Q32 C2×Q16 C4○D8 C40 C2×C20 C20 C2×C10 C8 C2×C4 C4 C22 C5 C1 # reps 1 1 2 2 1 1 4 4 8 8 4 4 1 1 2 2 4 4 8 8 2 8

Matrix representation of C5×Q32⋊C2 in GL6(𝔽241)

 205 0 0 0 0 0 0 205 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 11 11 0 0 0 0 230 11 0 0 0 0 0 0 55 239 96 106 0 0 114 35 202 184 0 0 58 224 233 119 0 0 212 42 34 159
,
 230 230 0 0 0 0 230 11 0 0 0 0 0 0 104 80 40 84 0 0 148 199 117 164 0 0 46 229 78 85 0 0 67 27 238 101
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 80 0 1 0 0 0 23 138 1 239 0 0 108 0 161 0 0 0 151 2 161 103

G:=sub<GL(6,GF(241))| [205,0,0,0,0,0,0,205,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,230,0,0,0,0,11,11,0,0,0,0,0,0,55,114,58,212,0,0,239,35,224,42,0,0,96,202,233,34,0,0,106,184,119,159],[230,230,0,0,0,0,230,11,0,0,0,0,0,0,104,148,46,67,0,0,80,199,229,27,0,0,40,117,78,238,0,0,84,164,85,101],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,80,23,108,151,0,0,0,138,0,2,0,0,1,1,161,161,0,0,0,239,0,103] >;

C5×Q32⋊C2 in GAP, Magma, Sage, TeX

C_5\times Q_{32}\rtimes C_2
% in TeX

G:=Group("C5xQ32:C2");
// GroupNames label

G:=SmallGroup(320,1011);
// by ID

G=gap.SmallGroup(320,1011);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1128,3446,4204,2111,242,10085,5052,124]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^16=d^2=1,c^2=b^8,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^9,c*d=d*c>;
// generators/relations

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