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## G = C8.27(C4×D5)  order 320 = 26·5

### 4th non-split extension by C8 of C4×D5 acting via C4×D5/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C8.27(C4×D5)
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — D5×C2×C8 — C8.27(C4×D5)
 Lower central C5 — C10 — C20 — C8.27(C4×D5)
 Upper central C1 — C22 — C2×C4 — C2.D8

Generators and relations for C8.27(C4×D5)
G = < a,b,c,d | a8=b4=c5=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 382 in 114 conjugacy classes, 55 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4.Q8, C2.D8, C2.D8, C42⋊C2, C22×C8, C52C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C23.25D4, C8×D5, C2×C52C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C20.Q8, C405C4, C5×C2.D8, C4⋊C47D5, D5×C2×C8, C8.27(C4×D5)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, C4○D8, C4×D5, C22×D5, C23.25D4, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D83D5, Q8.D10, C8.27(C4×D5)

Smallest permutation representation of C8.27(C4×D5)
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 55 108 155)(2 54 109 154)(3 53 110 153)(4 52 111 160)(5 51 112 159)(6 50 105 158)(7 49 106 157)(8 56 107 156)(9 66 116 24)(10 65 117 23)(11 72 118 22)(12 71 119 21)(13 70 120 20)(14 69 113 19)(15 68 114 18)(16 67 115 17)(25 78 132 128)(26 77 133 127)(27 76 134 126)(28 75 135 125)(29 74 136 124)(30 73 129 123)(31 80 130 122)(32 79 131 121)(33 82 143 61)(34 81 144 60)(35 88 137 59)(36 87 138 58)(37 86 139 57)(38 85 140 64)(39 84 141 63)(40 83 142 62)(41 101 95 145)(42 100 96 152)(43 99 89 151)(44 98 90 150)(45 97 91 149)(46 104 92 148)(47 103 93 147)(48 102 94 146)
(1 23 133 33 97)(2 24 134 34 98)(3 17 135 35 99)(4 18 136 36 100)(5 19 129 37 101)(6 20 130 38 102)(7 21 131 39 103)(8 22 132 40 104)(9 126 81 90 54)(10 127 82 91 55)(11 128 83 92 56)(12 121 84 93 49)(13 122 85 94 50)(14 123 86 95 51)(15 124 87 96 52)(16 125 88 89 53)(25 142 148 107 72)(26 143 149 108 65)(27 144 150 109 66)(28 137 151 110 67)(29 138 152 111 68)(30 139 145 112 69)(31 140 146 105 70)(32 141 147 106 71)(41 159 113 73 57)(42 160 114 74 58)(43 153 115 75 59)(44 154 116 76 60)(45 155 117 77 61)(46 156 118 78 62)(47 157 119 79 63)(48 158 120 80 64)
(1 149)(2 150)(3 151)(4 152)(5 145)(6 146)(7 147)(8 148)(9 64)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 137)(18 138)(19 139)(20 140)(21 141)(22 142)(23 143)(24 144)(25 132)(26 133)(27 134)(28 135)(29 136)(30 129)(31 130)(32 131)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 55)(42 56)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(73 127)(74 128)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(81 120)(82 113)(83 114)(84 115)(85 116)(86 117)(87 118)(88 119)(89 157)(90 158)(91 159)(92 160)(93 153)(94 154)(95 155)(96 156)(97 108)(98 109)(99 110)(100 111)(101 112)(102 105)(103 106)(104 107)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 10 10 2 2 4 4 4 4 5 5 5 5 20 20 20 20 2 2 2 2 2 2 10 10 10 10 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + + + + + - + - + image C1 C2 C2 C2 C2 C2 C4 Q8 D4 D4 D5 D10 D10 C4○D8 C4×D5 Q8×D5 D4×D5 D8⋊3D5 Q8.D10 kernel C8.27(C4×D5) C20.Q8 C40⋊5C4 C5×C2.D8 C4⋊C4⋊7D5 D5×C2×C8 C8×D5 C4×D5 C2×Dic5 C22×D5 C2.D8 C4⋊C4 C2×C8 C10 C8 C4 C22 C2 C2 # reps 1 2 1 1 2 1 8 2 1 1 2 4 2 8 8 2 2 4 4

Matrix representation of C8.27(C4×D5) in GL4(𝔽41) generated by

 3 20 0 0 0 14 0 0 0 0 40 0 0 0 0 40
,
 30 30 0 0 37 11 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 0 40 0 0 1 34
,
 1 15 0 0 0 40 0 0 0 0 34 1 0 0 34 7
G:=sub<GL(4,GF(41))| [3,0,0,0,20,14,0,0,0,0,40,0,0,0,0,40],[30,37,0,0,30,11,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,34],[1,0,0,0,15,40,0,0,0,0,34,34,0,0,1,7] >;

C8.27(C4×D5) in GAP, Magma, Sage, TeX

C_8._{27}(C_4\times D_5)
% in TeX

G:=Group("C8.27(C4xD5)");
// GroupNames label

G:=SmallGroup(320,507);
// by ID

G=gap.SmallGroup(320,507);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^4=c^5=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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