Copied to
clipboard

## G = C42.277D10order 320 = 26·5

### 36th non-split extension by C42 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.277D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — C4×C5⋊D4 — C42.277D10
 Lower central C5 — C2×C10 — C42.277D10
 Upper central C1 — C2×C4 — C2×C42

Generators and relations for C42.277D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=b2c-1 >

Subgroups: 750 in 234 conjugacy classes, 103 normal (51 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C5, C2×C4 [×6], C2×C4 [×16], D4 [×6], Q8 [×2], C23, C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×10], C22×C4 [×3], C22×C4 [×2], C2×D4 [×3], C2×Q8, Dic5 [×6], C20 [×4], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×4], C2×C20 [×6], C2×C20 [×6], C22×D5 [×2], C22×C10, C23.36C23, C4×Dic5 [×2], C10.D4 [×8], C4⋊Dic5 [×2], D10⋊C4 [×8], C23.D5 [×2], C4×C20 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C2×C5⋊D4 [×2], C22×C20 [×3], C4×Dic10, C20.6Q8, C42⋊D5 [×2], C4×D20, C4.D20, C422D5 [×2], C20.48D4, C4×C5⋊D4 [×2], C23.23D10 [×2], C207D4, C2×C4×C20, C42.277D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, C4○D20 [×6], C23×D5, C2×C4○D20 [×3], C42.277D10

Smallest permutation representation of C42.277D10
On 160 points
Generators in S160
```(1 112 97 39)(2 113 98 40)(3 114 99 31)(4 115 100 32)(5 116 91 33)(6 117 92 34)(7 118 93 35)(8 119 94 36)(9 120 95 37)(10 111 96 38)(11 110 42 27)(12 101 43 28)(13 102 44 29)(14 103 45 30)(15 104 46 21)(16 105 47 22)(17 106 48 23)(18 107 49 24)(19 108 50 25)(20 109 41 26)(51 149 124 76)(52 150 125 77)(53 141 126 78)(54 142 127 79)(55 143 128 80)(56 144 129 71)(57 145 130 72)(58 146 121 73)(59 147 122 74)(60 148 123 75)(61 159 134 82)(62 160 135 83)(63 151 136 84)(64 152 137 85)(65 153 138 86)(66 154 139 87)(67 155 140 88)(68 156 131 89)(69 157 132 90)(70 158 133 81)
(1 133 123 24)(2 134 124 25)(3 135 125 26)(4 136 126 27)(5 137 127 28)(6 138 128 29)(7 139 129 30)(8 140 130 21)(9 131 121 22)(10 132 122 23)(11 115 84 78)(12 116 85 79)(13 117 86 80)(14 118 87 71)(15 119 88 72)(16 120 89 73)(17 111 90 74)(18 112 81 75)(19 113 82 76)(20 114 83 77)(31 160 150 41)(32 151 141 42)(33 152 142 43)(34 153 143 44)(35 154 144 45)(36 155 145 46)(37 156 146 47)(38 157 147 48)(39 158 148 49)(40 159 149 50)(51 108 98 61)(52 109 99 62)(53 110 100 63)(54 101 91 64)(55 102 92 65)(56 103 93 66)(57 104 94 67)(58 105 95 68)(59 106 96 69)(60 107 97 70)
(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 10 123 122)(2 121 124 9)(3 8 125 130)(4 129 126 7)(5 6 127 128)(11 45 84 154)(12 153 85 44)(13 43 86 152)(14 151 87 42)(15 41 88 160)(16 159 89 50)(17 49 90 158)(18 157 81 48)(19 47 82 156)(20 155 83 46)(21 62 140 109)(22 108 131 61)(23 70 132 107)(24 106 133 69)(25 68 134 105)(26 104 135 67)(27 66 136 103)(28 102 137 65)(29 64 138 101)(30 110 139 63)(31 145 150 36)(32 35 141 144)(33 143 142 34)(37 149 146 40)(38 39 147 148)(51 95 98 58)(52 57 99 94)(53 93 100 56)(54 55 91 92)(59 97 96 60)(71 115 118 78)(72 77 119 114)(73 113 120 76)(74 75 111 112)(79 117 116 80)```

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

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

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

92 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4N 4O ··· 4T 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 20 20 1 1 1 1 2 ··· 2 20 ··· 20 2 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 C4○D20 C4○D20 kernel C42.277D10 C4×Dic10 C20.6Q8 C42⋊D5 C4×D20 C4.D20 C42⋊2D5 C20.48D4 C4×C5⋊D4 C23.23D10 C20⋊7D4 C2×C4×C20 C2×C42 C20 C2×C10 C42 C22×C4 C4 C22 # reps 1 1 1 2 1 1 2 1 2 2 1 1 2 8 4 8 6 32 16

Matrix representation of C42.277D10 in GL4(𝔽41) generated by

 2 28 0 0 13 39 0 0 0 0 23 6 0 0 35 18
,
 2 28 0 0 13 39 0 0 0 0 32 0 0 0 0 32
,
 18 20 0 0 21 21 0 0 0 0 20 21 0 0 20 23
,
 20 18 0 0 21 21 0 0 0 0 21 20 0 0 23 20
`G:=sub<GL(4,GF(41))| [2,13,0,0,28,39,0,0,0,0,23,35,0,0,6,18],[2,13,0,0,28,39,0,0,0,0,32,0,0,0,0,32],[18,21,0,0,20,21,0,0,0,0,20,20,0,0,21,23],[20,21,0,0,18,21,0,0,0,0,21,23,0,0,20,20] >;`

C42.277D10 in GAP, Magma, Sage, TeX

`C_4^2._{277}D_{10}`
`% in TeX`

`G:=Group("C4^2.277D10");`
`// GroupNames label`

`G:=SmallGroup(320,1151);`
`// by ID`

`G=gap.SmallGroup(320,1151);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,100,675,136,12550]);`
`// Polycyclic`

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

׿
×
𝔽