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## G = C4.Dic10order 160 = 25·5

### 3rd non-split extension by C4 of Dic10 acting via Dic10/Dic5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C4.Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — C4.Dic10
 Lower central C5 — C2×C10 — C4.Dic10
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4.Dic10
G = < a,b,c | a4=b20=1, c2=b10, bab-1=a-1, ac=ca, cbc-1=a2b-1 >

Subgroups: 144 in 56 conjugacy classes, 33 normal (19 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, C10, C42, C4⋊C4, C4⋊C4, Dic5, C20, C20, C2×C10, C42.C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C4.Dic10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, D10, C42.C2, Dic10, C22×D5, C2×Dic10, D42D5, Q82D5, C4.Dic10

Smallest permutation representation of C4.Dic10
Regular action on 160 points
Generators in S160
(1 37 80 129)(2 130 61 38)(3 39 62 131)(4 132 63 40)(5 21 64 133)(6 134 65 22)(7 23 66 135)(8 136 67 24)(9 25 68 137)(10 138 69 26)(11 27 70 139)(12 140 71 28)(13 29 72 121)(14 122 73 30)(15 31 74 123)(16 124 75 32)(17 33 76 125)(18 126 77 34)(19 35 78 127)(20 128 79 36)(41 95 145 115)(42 116 146 96)(43 97 147 117)(44 118 148 98)(45 99 149 119)(46 120 150 100)(47 81 151 101)(48 102 152 82)(49 83 153 103)(50 104 154 84)(51 85 155 105)(52 106 156 86)(53 87 157 107)(54 108 158 88)(55 89 159 109)(56 110 160 90)(57 91 141 111)(58 112 142 92)(59 93 143 113)(60 114 144 94)
(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)
(1 147 11 157)(2 42 12 52)(3 145 13 155)(4 60 14 50)(5 143 15 153)(6 58 16 48)(7 141 17 151)(8 56 18 46)(9 159 19 149)(10 54 20 44)(21 113 31 103)(22 92 32 82)(23 111 33 101)(24 90 34 100)(25 109 35 119)(26 88 36 98)(27 107 37 117)(28 86 38 96)(29 105 39 115)(30 84 40 94)(41 72 51 62)(43 70 53 80)(45 68 55 78)(47 66 57 76)(49 64 59 74)(61 146 71 156)(63 144 73 154)(65 142 75 152)(67 160 77 150)(69 158 79 148)(81 135 91 125)(83 133 93 123)(85 131 95 121)(87 129 97 139)(89 127 99 137)(102 134 112 124)(104 132 114 122)(106 130 116 140)(108 128 118 138)(110 126 120 136)

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 10A ··· 10F 20A ··· 20L order 1 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 4 4 10 10 10 10 20 20 2 2 2 ··· 2 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + - + + - - + image C1 C2 C2 C2 C2 Q8 D5 C4○D4 D10 Dic10 D4⋊2D5 Q8⋊2D5 kernel C4.Dic10 C4×Dic5 C10.D4 C4⋊Dic5 C5×C4⋊C4 C20 C4⋊C4 C10 C2×C4 C4 C2 C2 # reps 1 1 2 3 1 2 2 4 6 8 2 2

Matrix representation of C4.Dic10 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 20 3 0 0 3 21
,
 11 2 0 0 25 27 0 0 0 0 0 1 0 0 40 0
,
 18 27 0 0 32 23 0 0 0 0 32 0 0 0 0 32
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,20,3,0,0,3,21],[11,25,0,0,2,27,0,0,0,0,0,40,0,0,1,0],[18,32,0,0,27,23,0,0,0,0,32,0,0,0,0,32] >;

C4.Dic10 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_{10}
% in TeX

G:=Group("C4.Dic10");
// GroupNames label

G:=SmallGroup(160,111);
// by ID

G=gap.SmallGroup(160,111);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,506,188,50,4613]);
// Polycyclic

G:=Group<a,b,c|a^4=b^20=1,c^2=b^10,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1>;
// generators/relations

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