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## G = C42.6Dic5order 320 = 26·5

### 3rd non-split extension by C42 of Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42.6Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C4×C5⋊2C8 — C42.6Dic5
 Lower central C5 — C10 — C42.6Dic5
 Upper central C1 — C42 — C2×C42

Generators and relations for C42.6Dic5
G = < a,b,c,d | a4=b4=1, c10=b2, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c9 >

Subgroups: 222 in 118 conjugacy classes, 79 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C52C8, C2×C20, C2×C20, C2×C20, C22×C10, C42.12C4, C2×C52C8, C4×C20, C22×C20, C4×C52C8, C203C8, C20.55D4, C2×C4×C20, C42.6Dic5
Quotients:

Smallest permutation representation of C42.6Dic5
On 160 points
Generators in S160
(1 78 50 146)(2 79 51 147)(3 80 52 148)(4 61 53 149)(5 62 54 150)(6 63 55 151)(7 64 56 152)(8 65 57 153)(9 66 58 154)(10 67 59 155)(11 68 60 156)(12 69 41 157)(13 70 42 158)(14 71 43 159)(15 72 44 160)(16 73 45 141)(17 74 46 142)(18 75 47 143)(19 76 48 144)(20 77 49 145)(21 93 111 134)(22 94 112 135)(23 95 113 136)(24 96 114 137)(25 97 115 138)(26 98 116 139)(27 99 117 140)(28 100 118 121)(29 81 119 122)(30 82 120 123)(31 83 101 124)(32 84 102 125)(33 85 103 126)(34 86 104 127)(35 87 105 128)(36 88 106 129)(37 89 107 130)(38 90 108 131)(39 91 109 132)(40 92 110 133)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 116 31 106)(22 117 32 107)(23 118 33 108)(24 119 34 109)(25 120 35 110)(26 101 36 111)(27 102 37 112)(28 103 38 113)(29 104 39 114)(30 105 40 115)(41 46 51 56)(42 47 52 57)(43 48 53 58)(44 49 54 59)(45 50 55 60)(61 66 71 76)(62 67 72 77)(63 68 73 78)(64 69 74 79)(65 70 75 80)(81 127 91 137)(82 128 92 138)(83 129 93 139)(84 130 94 140)(85 131 95 121)(86 132 96 122)(87 133 97 123)(88 134 98 124)(89 135 99 125)(90 136 100 126)(141 146 151 156)(142 147 152 157)(143 148 153 158)(144 149 154 159)(145 150 155 160)
(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)
(1 100 6 85 11 90 16 95)(2 89 7 94 12 99 17 84)(3 98 8 83 13 88 18 93)(4 87 9 92 14 97 19 82)(5 96 10 81 15 86 20 91)(21 148 26 153 31 158 36 143)(22 157 27 142 32 147 37 152)(23 146 28 151 33 156 38 141)(24 155 29 160 34 145 39 150)(25 144 30 149 35 154 40 159)(41 140 46 125 51 130 56 135)(42 129 47 134 52 139 57 124)(43 138 48 123 53 128 58 133)(44 127 49 132 54 137 59 122)(45 136 50 121 55 126 60 131)(61 105 66 110 71 115 76 120)(62 114 67 119 72 104 77 109)(63 103 68 108 73 113 78 118)(64 112 69 117 74 102 79 107)(65 101 70 106 75 111 80 116)

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

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

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

104 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4L 4M ··· 4R 5A 5B 8A ··· 8P 10A ··· 10N 20A ··· 20AV order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 1 ··· 1 2 ··· 2 2 2 10 ··· 10 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - + image C1 C2 C2 C2 C2 C4 C4 C8 D5 M4(2) C4○D4 Dic5 D10 Dic5 D10 C5⋊2C8 C4.Dic5 C4○D20 kernel C42.6Dic5 C4×C5⋊2C8 C20⋊3C8 C20.55D4 C2×C4×C20 C4×C20 C22×C20 C2×C20 C2×C42 C20 C20 C42 C42 C22×C4 C22×C4 C2×C4 C4 C4 # reps 1 2 2 2 1 4 4 16 2 4 4 4 4 4 2 16 16 16

Matrix representation of C42.6Dic5 in GL3(𝔽41) generated by

 1 0 0 0 9 0 0 0 9
,
 9 0 0 0 9 7 0 0 32
,
 32 0 0 0 8 8 0 0 5
,
 27 0 0 0 7 25 0 23 34
G:=sub<GL(3,GF(41))| [1,0,0,0,9,0,0,0,9],[9,0,0,0,9,0,0,7,32],[32,0,0,0,8,0,0,8,5],[27,0,0,0,7,23,0,25,34] >;

C42.6Dic5 in GAP, Magma, Sage, TeX

C_4^2._6{\rm Dic}_5
% in TeX

G:=Group("C4^2.6Dic5");
// GroupNames label

G:=SmallGroup(320,552);
// by ID

G=gap.SmallGroup(320,552);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations

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