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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 [×3], C2 [×2], C4 [×8], C4 [×2], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×4], C23, C10 [×3], C10 [×2], C42 [×4], C2×C8 [×4], C22×C4 [×3], C20 [×8], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×C8 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C52C8 [×4], C2×C20 [×6], C2×C20 [×4], C2×C20 [×4], C22×C10, C42.12C4, C2×C52C8 [×4], C4×C20 [×4], C22×C20 [×3], C4×C52C8 [×2], C203C8 [×2], C20.55D4 [×2], C2×C4×C20, C42.6Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, D5, C2×C8 [×6], M4(2) [×2], C22×C4, C4○D4 [×2], Dic5 [×4], D10 [×3], C42⋊C2, C22×C8, C2×M4(2), C52C8 [×4], C2×Dic5 [×6], C22×D5, C42.12C4, C2×C52C8 [×6], C4.Dic5 [×2], C4○D20 [×2], C22×Dic5, C22×C52C8, C2×C4.Dic5, C23.21D10, C42.6Dic5

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

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

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

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

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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