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G = C20.47(C4⋊C4)  order 320 = 26·5

16th non-split extension by C20 of C4⋊C4 acting via C4⋊C4/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.47(C4⋊C4), (C2×C20).16Q8, C20.63(C2×Q8), C4⋊C4.225D10, (C2×C20).132D4, C4.Dic513C4, C10.D824C2, C20.Q824C2, C4.28(C2×Dic10), (C2×C4).11Dic10, (C22×C4).92D10, C10.82(C8⋊C22), C54(M4(2)⋊C4), C20.117(C22×C4), (C2×C20).317C23, (C22×C10).182D4, C23.75(C5⋊D4), C2.1(D4.D10), C4.16(C10.D4), C2.1(C20.C23), C10.82(C8.C22), C4⋊Dic5.323C22, (C22×C20).132C22, C23.21D10.12C2, C22.15(C10.D4), C4.86(C2×C4×D5), C52C86(C2×C4), (C2×C4⋊C4).6D5, (C10×C4⋊C4).5C2, C10.57(C2×C4⋊C4), (C2×C4).38(C4×D5), (C2×C10).74(C4⋊C4), (C2×C20).249(C2×C4), (C2×C10).437(C2×D4), C2.9(C2×C10.D4), C22.56(C2×C5⋊D4), (C2×C4).181(C5⋊D4), (C5×C4⋊C4).256C22, (C2×C52C8).79C22, (C2×C4).417(C22×D5), (C2×C4.Dic5).15C2, SmallGroup(320,591)

Series: Derived Chief Lower central Upper central

C1C20 — C20.47(C4⋊C4)
C1C5C10C2×C10C2×C20C2×C52C8C2×C4.Dic5 — C20.47(C4⋊C4)
C5C10C20 — C20.47(C4⋊C4)
C1C22C22×C4C2×C4⋊C4

Generators and relations for C20.47(C4⋊C4)
 G = < a,b,c | a20=c4=1, b4=a10, bab-1=a-1, cac-1=a11, cbc-1=a10b3 >

Subgroups: 318 in 118 conjugacy classes, 63 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4, Dic5 [×2], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C2×M4(2), C52C8 [×4], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×4], C22×C10, M4(2)⋊C4, C2×C52C8 [×2], C4.Dic5 [×4], C4×Dic5, C4⋊Dic5 [×2], C23.D5, C5×C4⋊C4 [×2], C5×C4⋊C4, C22×C20, C22×C20, C10.D8 [×2], C20.Q8 [×2], C2×C4.Dic5, C23.21D10, C10×C4⋊C4, C20.47(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C8⋊C22, C8.C22, Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, M4(2)⋊C4, C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×C10.D4, D4.D10, C20.C23, C20.47(C4⋊C4)

Smallest permutation representation of C20.47(C4⋊C4)
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)
(1 115 159 83 11 105 149 93)(2 114 160 82 12 104 150 92)(3 113 141 81 13 103 151 91)(4 112 142 100 14 102 152 90)(5 111 143 99 15 101 153 89)(6 110 144 98 16 120 154 88)(7 109 145 97 17 119 155 87)(8 108 146 96 18 118 156 86)(9 107 147 95 19 117 157 85)(10 106 148 94 20 116 158 84)(21 127 51 71 31 137 41 61)(22 126 52 70 32 136 42 80)(23 125 53 69 33 135 43 79)(24 124 54 68 34 134 44 78)(25 123 55 67 35 133 45 77)(26 122 56 66 36 132 46 76)(27 121 57 65 37 131 47 75)(28 140 58 64 38 130 48 74)(29 139 59 63 39 129 49 73)(30 138 60 62 40 128 50 72)
(1 136 154 75)(2 127 155 66)(3 138 156 77)(4 129 157 68)(5 140 158 79)(6 131 159 70)(7 122 160 61)(8 133 141 72)(9 124 142 63)(10 135 143 74)(11 126 144 65)(12 137 145 76)(13 128 146 67)(14 139 147 78)(15 130 148 69)(16 121 149 80)(17 132 150 71)(18 123 151 62)(19 134 152 73)(20 125 153 64)(21 87 56 114)(22 98 57 105)(23 89 58 116)(24 100 59 107)(25 91 60 118)(26 82 41 109)(27 93 42 120)(28 84 43 111)(29 95 44 102)(30 86 45 113)(31 97 46 104)(32 88 47 115)(33 99 48 106)(34 90 49 117)(35 81 50 108)(36 92 51 119)(37 83 52 110)(38 94 53 101)(39 85 54 112)(40 96 55 103)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D10A···10N20A···20X
order12222244444444444455888810···1020···20
size111122222244442020202022202020202···24···4

62 irreducible representations

dim111111122222222224444
type+++++++-++++-+-
imageC1C2C2C2C2C2C4D4Q8D4D5D10D10Dic10C4×D5C5⋊D4C5⋊D4C8⋊C22C8.C22D4.D10C20.C23
kernelC20.47(C4⋊C4)C10.D8C20.Q8C2×C4.Dic5C23.21D10C10×C4⋊C4C4.Dic5C2×C20C2×C20C22×C10C2×C4⋊C4C4⋊C4C22×C4C2×C4C2×C4C2×C4C23C10C10C2C2
# reps122111812124288441144

Matrix representation of C20.47(C4⋊C4) in GL6(𝔽41)

1000000
16370000
0003700
004000
0000031
0000100
,
9280000
3320000
000010
000001
000100
0040000
,
900000
090000
0031900
00193800
00001938
00003822

G:=sub<GL(6,GF(41))| [10,16,0,0,0,0,0,37,0,0,0,0,0,0,0,4,0,0,0,0,37,0,0,0,0,0,0,0,0,10,0,0,0,0,31,0],[9,3,0,0,0,0,28,32,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,19,0,0,0,0,19,38,0,0,0,0,0,0,19,38,0,0,0,0,38,22] >;

C20.47(C4⋊C4) in GAP, Magma, Sage, TeX

C_{20}._{47}(C_4\rtimes C_4)
% in TeX

G:=Group("C20.47(C4:C4)");
// GroupNames label

G:=SmallGroup(320,591);
// by ID

G=gap.SmallGroup(320,591);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,422,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=c^4=1,b^4=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b^3>;
// generators/relations

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