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## G = C42.180D10order 320 = 26·5

### 180th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.180D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10⋊3Q8 — C42.180D10
 Lower central C5 — C2×C10 — C42.180D10
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for C42.180D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b, dcd-1=c9 >

Subgroups: 678 in 196 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4, Q8 [×3], C23 [×2], D5 [×2], C10, C10 [×2], C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×2], C2×D4, C2×Q8 [×2], C2×Q8, Dic5 [×6], C20 [×7], D10 [×6], C2×C10, C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C422C2 [×4], C4⋊Q8, C4⋊Q8, Dic10, C4×D5 [×2], D20, C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×2], C22.57C24, C4×Dic5 [×2], C10.D4 [×10], C4⋊Dic5 [×2], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, Q8×C10 [×2], C422D5 [×2], Dic5.Q8 [×2], D10.13D4 [×2], D10⋊Q8 [×2], C4⋊C4⋊D5 [×2], Dic5⋊Q8, D103Q8 [×2], C20.23D4, C5×C4⋊Q8, C42.180D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4, 2- 1+4 [×2], C22×D5 [×7], C22.57C24, C23×D5, D46D10, Q8.10D10 [×2], C42.180D10

Smallest permutation representation of C42.180D10
On 160 points
Generators in S160
```(1 50 11 60)(2 41 12 51)(3 52 13 42)(4 43 14 53)(5 54 15 44)(6 45 16 55)(7 56 17 46)(8 47 18 57)(9 58 19 48)(10 49 20 59)(21 121 31 131)(22 132 32 122)(23 123 33 133)(24 134 34 124)(25 125 35 135)(26 136 36 126)(27 127 37 137)(28 138 38 128)(29 129 39 139)(30 140 40 130)(61 81 71 91)(62 92 72 82)(63 83 73 93)(64 94 74 84)(65 85 75 95)(66 96 76 86)(67 87 77 97)(68 98 78 88)(69 89 79 99)(70 100 80 90)(101 147 111 157)(102 158 112 148)(103 149 113 159)(104 160 114 150)(105 151 115 141)(106 142 116 152)(107 153 117 143)(108 144 118 154)(109 155 119 145)(110 146 120 156)
(1 25 84 112)(2 113 85 26)(3 27 86 114)(4 115 87 28)(5 29 88 116)(6 117 89 30)(7 31 90 118)(8 119 91 32)(9 33 92 120)(10 101 93 34)(11 35 94 102)(12 103 95 36)(13 37 96 104)(14 105 97 38)(15 39 98 106)(16 107 99 40)(17 21 100 108)(18 109 81 22)(19 23 82 110)(20 111 83 24)(41 159 75 136)(42 137 76 160)(43 141 77 138)(44 139 78 142)(45 143 79 140)(46 121 80 144)(47 145 61 122)(48 123 62 146)(49 147 63 124)(50 125 64 148)(51 149 65 126)(52 127 66 150)(53 151 67 128)(54 129 68 152)(55 153 69 130)(56 131 70 154)(57 155 71 132)(58 133 72 156)(59 157 73 134)(60 135 74 158)
(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 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 115 31 105)(22 104 32 114)(23 113 33 103)(24 102 34 112)(25 111 35 101)(26 120 36 110)(27 109 37 119)(28 118 38 108)(29 107 39 117)(30 116 40 106)(41 72 51 62)(42 61 52 71)(43 70 53 80)(44 79 54 69)(45 68 55 78)(46 77 56 67)(47 66 57 76)(48 75 58 65)(49 64 59 74)(50 73 60 63)(81 86 91 96)(82 95 92 85)(83 84 93 94)(87 100 97 90)(88 89 98 99)(121 128 131 138)(122 137 132 127)(123 126 133 136)(124 135 134 125)(129 140 139 130)(141 144 151 154)(142 153 152 143)(145 160 155 150)(146 149 156 159)(147 158 157 148)```

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 D10 D10 D10 2+ 1+4 2- 1+4 D4⋊6D10 Q8.10D10 kernel C42.180D10 C42⋊2D5 Dic5.Q8 D10.13D4 D10⋊Q8 C4⋊C4⋊D5 Dic5⋊Q8 D10⋊3Q8 C20.23D4 C5×C4⋊Q8 C4⋊Q8 C42 C4⋊C4 C2×Q8 C10 C10 C2 C2 # reps 1 2 2 2 2 2 1 2 1 1 2 2 8 4 1 2 4 8

Matrix representation of C42.180D10 in GL10(𝔽41)

 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 0
,
 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 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 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 0 35 0 0 0 0 0 0 0 0 7 34 0 0 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 32 0 0
,
 7 35 0 0 0 0 0 0 0 0 8 34 0 0 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 9 0 0

`G:=sub<GL(10,GF(41))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,7,0,0,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,32,0,0],[7,8,0,0,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,9,0,0] >;`

C42.180D10 in GAP, Magma, Sage, TeX

`C_4^2._{180}D_{10}`
`% in TeX`

`G:=Group("C4^2.180D10");`
`// GroupNames label`

`G:=SmallGroup(320,1407);`
`// by ID`

`G=gap.SmallGroup(320,1407);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,100,1571,570,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;`
`// generators/relations`

׿
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