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## G = C42.7Dic5order 320 = 26·5

### 4th non-split extension by C42 of Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.7Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C20.55D4 — C42.7Dic5
 Lower central C5 — C2×C10 — C42.7Dic5
 Upper central C1 — C2×C4 — C2×C42

Generators and relations for C42.7Dic5
G = < a,b,c,d | a4=b4=1, c10=a2, d2=c5, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, dbd-1=a2b, dcd-1=c9 >

Subgroups: 222 in 110 conjugacy classes, 63 normal (41 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×6], C2×C4 [×6], C23, C10 [×3], C10 [×2], C42 [×4], C2×C8 [×4], C22×C4 [×3], C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C8⋊C4 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C52C8 [×4], C2×C20 [×6], C2×C20 [×6], C22×C10, C42.6C4, C2×C52C8 [×4], C4×C20 [×4], C22×C20 [×3], C42.D5 [×2], C203C8 [×2], C20.55D4 [×2], C2×C4×C20, C42.7Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, M4(2) [×4], C22×C4, C4○D4 [×2], Dic5 [×4], D10 [×3], C42⋊C2, C2×M4(2) [×2], C2×Dic5 [×6], C22×D5, C42.6C4, C4.Dic5 [×4], C4○D20 [×2], C22×Dic5, C2×C4.Dic5 [×2], C23.21D10, C42.7Dic5

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

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

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

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

92 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 5A 5B 8A ··· 8H 10A ··· 10N 20A ··· 20AV order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 2 2 20 ··· 20 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - + image C1 C2 C2 C2 C2 C4 C4 D5 M4(2) C4○D4 M4(2) Dic5 D10 Dic5 D10 C4.Dic5 C4○D20 C4.Dic5 kernel C42.7Dic5 C42.D5 C20⋊3C8 C20.55D4 C2×C4×C20 C4×C20 C22×C20 C2×C42 C20 C20 C2×C10 C42 C42 C22×C4 C22×C4 C4 C4 C22 # reps 1 2 2 2 1 4 4 2 4 4 4 4 4 4 2 16 16 16

Matrix representation of C42.7Dic5 in GL4(𝔽41) generated by

 32 0 0 0 0 9 0 0 0 0 40 0 0 0 38 1
,
 1 0 0 0 0 40 0 0 0 0 9 0 0 0 0 9
,
 32 0 0 0 0 32 0 0 0 0 31 0 0 0 20 4
,
 0 1 0 0 32 0 0 0 0 0 14 18 0 0 5 27
G:=sub<GL(4,GF(41))| [32,0,0,0,0,9,0,0,0,0,40,38,0,0,0,1],[1,0,0,0,0,40,0,0,0,0,9,0,0,0,0,9],[32,0,0,0,0,32,0,0,0,0,31,20,0,0,0,4],[0,32,0,0,1,0,0,0,0,0,14,5,0,0,18,27] >;

C42.7Dic5 in GAP, Magma, Sage, TeX

C_4^2._7{\rm Dic}_5
% in TeX

G:=Group("C4^2.7Dic5");
// GroupNames label

G:=SmallGroup(320,553);
// by ID

G=gap.SmallGroup(320,553);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,758,100,136,12550]);
// Polycyclic

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

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