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## G = C4⋊C4.197D10order 320 = 26·5

### 70th non-split extension by C4⋊C4 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C4⋊C4.197D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C4×Dic5 — C4⋊C4.197D10
 Lower central C5 — C2×C10 — C4⋊C4.197D10
 Upper central C1 — C22 — C22.D4

Generators and relations for C4⋊C4.197D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, bab-1=a-1, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 734 in 234 conjugacy classes, 99 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×14], C22, C22 [×2], C22 [×8], C5, C2×C4 [×5], C2×C4 [×17], D4 [×6], Q8 [×2], C23 [×2], C23, D5, C10 [×3], C10 [×3], C42 [×6], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, Dic5 [×4], Dic5 [×5], C20 [×5], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C42.C2, C422C2 [×2], Dic10 [×2], C4×D5 [×2], C2×Dic5 [×7], C2×Dic5 [×6], C5⋊D4 [×4], C2×C20 [×5], C2×C20 [×2], C5×D4 [×2], C22×D5, C22×C10 [×2], C23.36C23, C4×Dic5 [×6], C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×3], C5×C22⋊C4 [×3], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5, C22×Dic5 [×3], C2×C5⋊D4 [×2], C22×C20, D4×C10, C23.11D10, Dic5.14D4, C23.D10, Dic54D4 [×2], Dic5.5D4, Dic53Q8, Dic5.Q8, C4⋊C47D5, C4⋊C4⋊D5, C2×C4×Dic5, C23.23D10, D4×Dic5, Dic5⋊D4, C5×C22.D4, C4⋊C4.197D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, D42D5 [×2], C23×D5, C2×D42D5, D5×C4○D4 [×2], C4⋊C4.197D10

Smallest permutation representation of C4⋊C4.197D10
On 160 points
Generators in S160
```(1 17 123 81)(2 48 124 159)(3 19 125 83)(4 50 126 151)(5 11 127 85)(6 42 128 153)(7 13 129 87)(8 44 130 155)(9 15 121 89)(10 46 122 157)(12 55 86 100)(14 57 88 92)(16 59 90 94)(18 51 82 96)(20 53 84 98)(21 36 132 147)(22 120 133 75)(23 38 134 149)(24 112 135 77)(25 40 136 141)(26 114 137 79)(27 32 138 143)(28 116 139 71)(29 34 140 145)(30 118 131 73)(31 64 142 109)(33 66 144 101)(35 68 146 103)(37 70 148 105)(39 62 150 107)(41 54 152 99)(43 56 154 91)(45 58 156 93)(47 60 158 95)(49 52 160 97)(61 76 106 111)(63 78 108 113)(65 80 110 115)(67 72 102 117)(69 74 104 119)
(1 115 95 32)(2 116 96 33)(3 117 97 34)(4 118 98 35)(5 119 99 36)(6 120 100 37)(7 111 91 38)(8 112 92 39)(9 113 93 40)(10 114 94 31)(11 104 41 21)(12 105 42 22)(13 106 43 23)(14 107 44 24)(15 108 45 25)(16 109 46 26)(17 110 47 27)(18 101 48 28)(19 102 49 29)(20 103 50 30)(51 144 124 71)(52 145 125 72)(53 146 126 73)(54 147 127 74)(55 148 128 75)(56 149 129 76)(57 150 130 77)(58 141 121 78)(59 142 122 79)(60 143 123 80)(61 154 134 87)(62 155 135 88)(63 156 136 89)(64 157 137 90)(65 158 138 81)(66 159 139 82)(67 160 140 83)(68 151 131 84)(69 152 132 85)(70 153 133 86)
(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 46 123 157)(2 45 124 156)(3 44 125 155)(4 43 126 154)(5 42 127 153)(6 41 128 152)(7 50 129 151)(8 49 130 160)(9 48 121 159)(10 47 122 158)(11 55 85 100)(12 54 86 99)(13 53 87 98)(14 52 88 97)(15 51 89 96)(16 60 90 95)(17 59 81 94)(18 58 82 93)(19 57 83 92)(20 56 84 91)(21 75 132 120)(22 74 133 119)(23 73 134 118)(24 72 135 117)(25 71 136 116)(26 80 137 115)(27 79 138 114)(28 78 139 113)(29 77 140 112)(30 76 131 111)(31 110 142 65)(32 109 143 64)(33 108 144 63)(34 107 145 62)(35 106 146 61)(36 105 147 70)(37 104 148 69)(38 103 149 68)(39 102 150 67)(40 101 141 66)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L ··· 4Q 4R 4S 4T 5A 5B 10A ··· 10F 10G 10H 10I 10J 10K 10L 20A ··· 20H 20I ··· 20N order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 5 5 10 ··· 10 10 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 20 2 2 2 2 4 4 4 5 5 5 5 10 ··· 10 20 20 20 2 2 2 ··· 2 4 4 4 4 8 8 4 ··· 4 8 ··· 8

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D10 D4⋊2D5 D5×C4○D4 kernel C4⋊C4.197D10 C23.11D10 Dic5.14D4 C23.D10 Dic5⋊4D4 Dic5.5D4 Dic5⋊3Q8 Dic5.Q8 C4⋊C4⋊7D5 C4⋊C4⋊D5 C2×C4×Dic5 C23.23D10 D4×Dic5 Dic5⋊D4 C5×C22.D4 C22.D4 Dic5 C2×C10 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C22 C2 # reps 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 8 4 6 4 2 2 4 8

Matrix representation of C4⋊C4.197D10 in GL6(𝔽41)

 9 0 0 0 0 0 16 32 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 23 40
,
 9 36 0 0 0 0 16 32 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 34 34 0 0 0 0 7 1 0 0 0 0 0 0 9 1 0 0 0 0 2 32
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 7 7 0 0 0 0 40 34 0 0 0 0 0 0 32 40 0 0 0 0 39 9

`G:=sub<GL(6,GF(41))| [9,16,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,23,0,0,0,0,0,40],[9,16,0,0,0,0,36,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,9,2,0,0,0,0,1,32],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,32,39,0,0,0,0,40,9] >;`

C4⋊C4.197D10 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{197}D_{10}`
`% in TeX`

`G:=Group("C4:C4.197D10");`
`// GroupNames label`

`G:=SmallGroup(320,1321);`
`// by ID`

`G=gap.SmallGroup(320,1321);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,346,297,12550]);`
`// Polycyclic`

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

׿
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