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## G = C4.(C2×D20)  order 320 = 26·5

### 15th non-split extension by C4 of C2×D20 acting via C2×D20/C22×D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C4.(C2×D20)
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×Dic10 — C22×Dic10 — C4.(C2×D20)
 Lower central C5 — C10 — C20 — C4.(C2×D20)
 Upper central C1 — C22 — C22×C4 — C42⋊C2

Generators and relations for C4.(C2×D20)
G = < a,b,c,d | a4=c20=1, b2=a2, d2=cac-1=a-1, ab=ba, ad=da, bc=cb, dbd-1=a2b, dcd-1=ac-1 >

Subgroups: 494 in 150 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, Dic5, C20, C20, C20, C2×C10, C2×C10, C2×C10, Q8⋊C4, C42⋊C2, C2×M4(2), C22×Q8, C52C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C2×C20, C22×C10, C23.38D4, C2×C52C8, C4.Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C10.Q16, C2×C4.Dic5, C5×C42⋊C2, C22×Dic10, C4.(C2×D20)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C8.C22, C4×D5, D20, C5⋊D4, C22×D5, C23.38D4, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, D4.9D10, C4.(C2×D20)

Smallest permutation representation of C4.(C2×D20)
On 160 points
Generators in S160
(1 97 142 54)(2 55 143 98)(3 99 144 56)(4 57 145 100)(5 81 146 58)(6 59 147 82)(7 83 148 60)(8 41 149 84)(9 85 150 42)(10 43 151 86)(11 87 152 44)(12 45 153 88)(13 89 154 46)(14 47 155 90)(15 91 156 48)(16 49 157 92)(17 93 158 50)(18 51 159 94)(19 95 160 52)(20 53 141 96)(21 67 104 126)(22 127 105 68)(23 69 106 128)(24 129 107 70)(25 71 108 130)(26 131 109 72)(27 73 110 132)(28 133 111 74)(29 75 112 134)(30 135 113 76)(31 77 114 136)(32 137 115 78)(33 79 116 138)(34 139 117 80)(35 61 118 140)(36 121 119 62)(37 63 120 122)(38 123 101 64)(39 65 102 124)(40 125 103 66)
(1 87 142 44)(2 88 143 45)(3 89 144 46)(4 90 145 47)(5 91 146 48)(6 92 147 49)(7 93 148 50)(8 94 149 51)(9 95 150 52)(10 96 151 53)(11 97 152 54)(12 98 153 55)(13 99 154 56)(14 100 155 57)(15 81 156 58)(16 82 157 59)(17 83 158 60)(18 84 159 41)(19 85 160 42)(20 86 141 43)(21 77 104 136)(22 78 105 137)(23 79 106 138)(24 80 107 139)(25 61 108 140)(26 62 109 121)(27 63 110 122)(28 64 111 123)(29 65 112 124)(30 66 113 125)(31 67 114 126)(32 68 115 127)(33 69 116 128)(34 70 117 129)(35 71 118 130)(36 72 119 131)(37 73 120 132)(38 74 101 133)(39 75 102 134)(40 76 103 135)
(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)
(1 107 54 129 142 24 97 70)(2 128 98 106 143 69 55 23)(3 105 56 127 144 22 99 68)(4 126 100 104 145 67 57 21)(5 103 58 125 146 40 81 66)(6 124 82 102 147 65 59 39)(7 101 60 123 148 38 83 64)(8 122 84 120 149 63 41 37)(9 119 42 121 150 36 85 62)(10 140 86 118 151 61 43 35)(11 117 44 139 152 34 87 80)(12 138 88 116 153 79 45 33)(13 115 46 137 154 32 89 78)(14 136 90 114 155 77 47 31)(15 113 48 135 156 30 91 76)(16 134 92 112 157 75 49 29)(17 111 50 133 158 28 93 74)(18 132 94 110 159 73 51 27)(19 109 52 131 160 26 95 72)(20 130 96 108 141 71 53 25)

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C4 D4 D4 D5 D10 D10 C4×D5 D20 C5⋊D4 C5⋊D4 C8.C22 D4.9D10 kernel C4.(C2×D20) C10.Q16 C2×C4.Dic5 C5×C42⋊C2 C22×Dic10 C2×Dic10 C2×C20 C22×C10 C42⋊C2 C4⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C23 C10 C2 # reps 1 4 1 1 1 8 3 1 2 4 2 8 8 4 4 2 8

Matrix representation of C4.(C2×D20) in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 39 28 0 0 0 0 13 2 0 0 0 0 26 18 30 13 0 0 8 29 19 11
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 13 0 0 0 0 28 39 0 0 0 0 0 0 30 13 0 0 0 0 19 11
,
 9 30 0 0 0 0 11 14 0 0 0 0 0 0 40 12 17 18 0 0 29 30 3 20 0 0 19 15 13 29 0 0 4 1 14 40
,
 40 0 0 0 0 0 34 1 0 0 0 0 0 0 13 36 1 40 0 0 32 28 40 0 0 0 13 11 4 9 0 0 20 39 22 37

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,39,13,26,8,0,0,28,2,18,29,0,0,0,0,30,19,0,0,0,0,13,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,30,19,0,0,0,0,13,11],[9,11,0,0,0,0,30,14,0,0,0,0,0,0,40,29,19,4,0,0,12,30,15,1,0,0,17,3,13,14,0,0,18,20,29,40],[40,34,0,0,0,0,0,1,0,0,0,0,0,0,13,32,13,20,0,0,36,28,11,39,0,0,1,40,4,22,0,0,40,0,9,37] >;

C4.(C2×D20) in GAP, Magma, Sage, TeX

C_4.(C_2\times D_{20})
% in TeX

G:=Group("C4.(C2xD20)");
// GroupNames label

G:=SmallGroup(320,631);
// by ID

G=gap.SmallGroup(320,631);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,422,387,58,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^20=1,b^2=a^2,d^2=c*a*c^-1=a^-1,a*b=b*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=a*c^-1>;
// generators/relations

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