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G = C20.65(C4⋊C4)  order 320 = 26·5

12nd non-split extension by C20 of C4⋊C4 acting via C4⋊C4/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20.65(C4⋊C4)
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — C23.21D10 — C20.65(C4⋊C4)
 Lower central C5 — C2×C10 — C20.65(C4⋊C4)
 Upper central C1 — C2×C4 — C22×C8

Generators and relations for C20.65(C4⋊C4)
G = < a,b,c | a20=b4=1, c4=a10, bab-1=a-1, ac=ca, cbc-1=a10b-1 >

Subgroups: 286 in 114 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C52C8, C40, C2×Dic5, C2×C20, C2×C20, C22×C10, C42.6C22, C2×C52C8, C4.Dic5, C4×Dic5, C4⋊Dic5, C23.D5, C2×C40, C2×C40, C22×C20, C20.8Q8, C2×C4.Dic5, C23.21D10, C22×C40, C20.65(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, C8○D4, Dic10, C4×D5, C5⋊D4, C22×D5, C42.6C22, C10.D4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, D20.3C4, C2×C10.D4, C20.65(C4⋊C4)

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

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

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

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

92 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 8A ··· 8H 8I 8J 8K 8L 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 8 ··· 8 8 8 8 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 1 1 1 1 2 2 20 20 20 20 2 2 2 ··· 2 20 20 20 20 2 ··· 2 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 D4 Q8 D5 D10 D10 C8○D4 Dic10 C4×D5 C5⋊D4 C4×D5 D20.3C4 kernel C20.65(C4⋊C4) C20.8Q8 C2×C4.Dic5 C23.21D10 C22×C40 C4⋊Dic5 C23.D5 C2×C20 C2×C20 C22×C8 C2×C8 C22×C4 C10 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 4 4 2 2 2 4 2 8 8 4 8 4 32

Matrix representation of C20.65(C4⋊C4) in GL4(𝔽41) generated by

 20 0 0 0 0 39 0 0 0 0 21 37 0 0 0 2
,
 0 32 0 0 32 0 0 0 0 0 32 34 0 0 29 9
,
 27 0 0 0 0 27 0 0 0 0 38 25 0 0 0 3
G:=sub<GL(4,GF(41))| [20,0,0,0,0,39,0,0,0,0,21,0,0,0,37,2],[0,32,0,0,32,0,0,0,0,0,32,29,0,0,34,9],[27,0,0,0,0,27,0,0,0,0,38,0,0,0,25,3] >;

C20.65(C4⋊C4) in GAP, Magma, Sage, TeX

C_{20}._{65}(C_4\rtimes C_4)
% in TeX

G:=Group("C20.65(C4:C4)");
// GroupNames label

G:=SmallGroup(320,729);
// by ID

G=gap.SmallGroup(320,729);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,422,58,136,12550]);
// Polycyclic

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

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