Copied to
clipboard

## G = D20.7D4order 320 = 26·5

### 7th non-split extension by D20 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.7D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D20.2C4 — D20.7D4
 Lower central C5 — C10 — C2×C20 — D20.7D4
 Upper central C1 — C2 — C2×C4 — C4.10D4

Generators and relations for D20.7D4
G = < a,b,c,d | a20=b2=1, c4=a10, d2=a5, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, dbd-1=a15b, dcd-1=a5c3 >

Subgroups: 382 in 100 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C4.10D4, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C52C8, C52C8, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×Q8, D4.5D4, C8×D5, C8⋊D5, C40⋊C2, Dic20, C2×C52C8, C4.Dic5, Q8⋊D5, C5⋊Q16, C5×M4(2), C2×Dic10, C4○D20, Q8×C10, C20.53D4, C4.12D20, C5×C4.10D4, D20.2C4, C8.D10, C20.C23, C2×C5⋊Q16, D20.7D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C22×D5, D4.5D4, C4○D20, D4×D5, D10⋊D4, D20.7D4

Smallest permutation representation of D20.7D4
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 105)(102 104)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(121 122)(123 140)(124 139)(125 138)(126 137)(127 136)(128 135)(129 134)(130 133)(131 132)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)
(1 81 132 79 11 91 122 69)(2 92 133 70 12 82 123 80)(3 83 134 61 13 93 124 71)(4 94 135 72 14 84 125 62)(5 85 136 63 15 95 126 73)(6 96 137 74 16 86 127 64)(7 87 138 65 17 97 128 75)(8 98 139 76 18 88 129 66)(9 89 140 67 19 99 130 77)(10 100 121 78 20 90 131 68)(21 158 106 59 31 148 116 49)(22 149 107 50 32 159 117 60)(23 160 108 41 33 150 118 51)(24 151 109 52 34 141 119 42)(25 142 110 43 35 152 120 53)(26 153 111 54 36 143 101 44)(27 144 112 45 37 154 102 55)(28 155 113 56 38 145 103 46)(29 146 114 47 39 156 104 57)(30 157 115 58 40 147 105 48)
(1 101 6 106 11 111 16 116)(2 102 7 107 12 112 17 117)(3 103 8 108 13 113 18 118)(4 104 9 109 14 114 19 119)(5 105 10 110 15 115 20 120)(21 122 26 127 31 132 36 137)(22 123 27 128 32 133 37 138)(23 124 28 129 33 134 38 139)(24 125 29 130 34 135 39 140)(25 126 30 131 35 136 40 121)(41 76 46 61 51 66 56 71)(42 77 47 62 52 67 57 72)(43 78 48 63 53 68 58 73)(44 79 49 64 54 69 59 74)(45 80 50 65 55 70 60 75)(81 158 86 143 91 148 96 153)(82 159 87 144 92 149 97 154)(83 160 88 145 93 150 98 155)(84 141 89 146 94 151 99 156)(85 142 90 147 95 152 100 157)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 8E 8F 8G 10A 10B 10C 10D 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 2 4 4 4 4 4 5 5 8 8 8 8 8 8 8 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 2 20 2 2 8 20 40 2 2 4 4 8 10 10 20 40 2 2 4 4 4 4 4 4 8 8 8 8 8 ··· 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 8 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D10 C4○D20 D4.5D4 D4×D5 D20.7D4 kernel D20.7D4 C20.53D4 C4.12D20 C5×C4.10D4 D20.2C4 C8.D10 C20.C23 C2×C5⋊Q16 C5⋊2C8 Dic10 D20 C4.10D4 C2×C10 M4(2) C2×Q8 C22 C5 C4 C1 # reps 1 1 1 1 1 1 1 1 2 1 1 2 2 4 2 8 2 4 2

Matrix representation of D20.7D4 in GL8(𝔽41)

 0 40 0 0 0 0 0 0 1 7 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 1 7 0 0 0 0 0 0 0 0 40 18 0 39 0 0 0 0 0 40 9 32 0 0 0 0 1 0 0 1 0 0 0 0 1 23 40 2
,
 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 18 0 39 0 0 0 0 32 40 32 32 0 0 0 0 1 0 0 1 0 0 0 0 1 23 1 2
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 17 15 15 0 0 0 0 12 0 24 0 0 0 0 0 29 30 24 0
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 33 36 4 33 0 0 0 0 10 4 23 23 0 0 0 0 6 10 2 6 0 0 0 0 10 36 39 2

`G:=sub<GL(8,GF(41))| [0,1,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,40,0,1,1,0,0,0,0,18,40,0,23,0,0,0,0,0,9,0,40,0,0,0,0,39,32,1,2],[0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,32,1,1,0,0,0,0,18,40,0,23,0,0,0,0,0,32,0,1,0,0,0,0,39,32,1,2],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,29,0,0,0,0,0,17,0,30,0,0,0,0,17,15,24,24,0,0,0,0,0,15,0,0],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,33,10,6,10,0,0,0,0,36,4,10,36,0,0,0,0,4,23,2,39,0,0,0,0,33,23,6,2] >;`

D20.7D4 in GAP, Magma, Sage, TeX

`D_{20}._7D_4`
`% in TeX`

`G:=Group("D20.7D4");`
`// GroupNames label`

`G:=SmallGroup(320,382);`
`// by ID`

`G=gap.SmallGroup(320,382);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,184,297,136,1684,851,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^10*b,d*b*d^-1=a^15*b,d*c*d^-1=a^5*c^3>;`
`// generators/relations`

׿
×
𝔽