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## G = Dic5.C42order 320 = 26·5

### 1st non-split extension by Dic5 of C42 acting via C42/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Dic5.C42
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C22×C5⋊C8 — Dic5.C42
 Lower central C5 — C10 — Dic5.C42
 Upper central C1 — C22 — C22⋊C4

Generators and relations for Dic5.C42
G = < a,b,c,d | a10=d4=1, b2=c4=a5, bab-1=a-1, cac-1=a3, ad=da, bc=cb, dbd-1=a5b, cd=dc >

Subgroups: 346 in 130 conjugacy classes, 68 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C5, C8 [×8], C2×C4 [×2], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4, C22⋊C4, C4⋊C4 [×2], C2×C8 [×8], M4(2) [×4], C22×C4, Dic5 [×2], Dic5 [×2], Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×C8 [×2], C8⋊C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C5⋊C8 [×8], C2×Dic5 [×2], C2×Dic5 [×6], C2×C20 [×2], C22×C10, C82M4(2), C4×Dic5 [×2], C10.D4 [×2], C23.D5, C5×C22⋊C4, C2×C5⋊C8 [×2], C2×C5⋊C8 [×6], C22.F5 [×4], C22×Dic5, C4×C5⋊C8 [×2], C10.C42 [×2], C23.11D10, C22×C5⋊C8, C2×C22.F5, Dic5.C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], C22×C4 [×3], F5, C2×C42, C8○D4 [×2], C2×F5 [×3], C82M4(2), C4×F5 [×2], C22×F5, C2×C4×F5, D4.F5 [×2], Dic5.C42

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

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 5 8A ··· 8H 8I ··· 8T 10A 10B 10C 10D 10E 20A 20B 20C 20D order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 5 8 ··· 8 8 ··· 8 10 10 10 10 10 20 20 20 20 size 1 1 1 1 2 2 2 2 2 2 5 5 5 5 10 ··· 10 4 5 ··· 5 10 ··· 10 4 4 4 8 8 8 8 8 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 4 4 4 4 8 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 C4 C8○D4 F5 C2×F5 C2×F5 C4×F5 D4.F5 kernel Dic5.C42 C4×C5⋊C8 C10.C42 C23.11D10 C22×C5⋊C8 C2×C22.F5 C10.D4 C23.D5 C5×C22⋊C4 C2×C5⋊C8 C22.F5 C10 C22⋊C4 C2×C4 C23 C22 C2 # reps 1 2 2 1 1 1 4 2 2 8 8 8 1 2 1 4 2

Matrix representation of Dic5.C42 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 1 0 0 0 0 40 0 1 0 0 0 40 0 0 1 0 0 40 0 0 0
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 29 14 29 17 0 0 2 2 5 10 0 0 31 19 39 39 0 0 7 12 27 12
,
 27 0 0 0 0 0 0 27 0 0 0 0 0 0 21 29 14 22 0 0 35 10 10 2 0 0 31 31 39 16 0 0 19 4 20 12
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,29,2,31,7,0,0,14,2,19,12,0,0,29,5,39,27,0,0,17,10,39,12],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,21,35,31,19,0,0,29,10,31,4,0,0,14,10,39,20,0,0,22,2,16,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;

Dic5.C42 in GAP, Magma, Sage, TeX

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

G:=Group("Dic5.C4^2");
// GroupNames label

G:=SmallGroup(320,1029);
// by ID

G=gap.SmallGroup(320,1029);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^10=d^4=1,b^2=c^4=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^3,a*d=d*a,b*c=c*b,d*b*d^-1=a^5*b,c*d=d*c>;
// generators/relations

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