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## G = C42.89D10order 320 = 26·5

### 89th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.89D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C2×Dic10 — C4×Dic10 — C42.89D10
 Lower central C5 — C2×C10 — C42.89D10
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C42.89D10
G = < a,b,c,d | a4=b4=1, c10=a2, d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c9 >

Subgroups: 542 in 192 conjugacy classes, 95 normal (23 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×4], C23, C10, C10 [×2], C10, C42 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×18], C22×C4, C2×Q8 [×2], Dic5 [×8], C20 [×2], C20 [×5], C2×C10, C2×C10 [×3], C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×5], C422C2 [×4], C4⋊Q8, Dic10 [×4], C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×C10, C22.35C24, C4×Dic5 [×4], C10.D4 [×12], C4⋊Dic5 [×6], C23.D5 [×4], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×2], C22×C20, C4×Dic10 [×2], C202Q8, C20.6Q8, C23.D10 [×4], Dic5.Q8 [×4], C20.48D4 [×2], C5×C42⋊C2, C42.89D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2- 1+4 [×2], C22×D5 [×7], C22.35C24, C4○D20 [×2], C23×D5, C2×C4○D20, D4.10D10 [×2], C42.89D10

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 4A ··· 4F 4G 4H 4I 4J ··· 4Q 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20AB order 1 2 2 2 2 4 ··· 4 4 4 4 4 ··· 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 2 ··· 2 4 4 4 20 ··· 20 2 2 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 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 D5 C4○D4 D10 D10 D10 D10 C4○D20 2- 1+4 D4.10D10 kernel C42.89D10 C4×Dic10 C20⋊2Q8 C20.6Q8 C23.D10 Dic5.Q8 C20.48D4 C5×C42⋊C2 C42⋊C2 C20 C42 C22⋊C4 C4⋊C4 C22×C4 C4 C10 C2 # reps 1 2 1 1 4 4 2 1 2 4 4 4 4 2 16 2 8

Matrix representation of C42.89D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 28 0 0 0 0 38 40 0 0 0 0 35 6 30 28 0 0 19 32 22 11
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 23 10 7 0 0 6 30 10 33 0 0 2 21 36 40 0 0 36 38 8 15
,
 16 0 0 0 0 0 0 23 0 0 0 0 0 0 15 16 0 0 0 0 10 8 0 0 0 0 35 34 27 25 0 0 28 10 5 32
,
 0 18 0 0 0 0 25 0 0 0 0 0 0 0 6 26 37 19 0 0 34 20 10 31 0 0 17 13 27 9 0 0 7 6 17 29

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,38,35,19,0,0,28,40,6,32,0,0,0,0,30,22,0,0,0,0,28,11],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,6,2,36,0,0,23,30,21,38,0,0,10,10,36,8,0,0,7,33,40,15],[16,0,0,0,0,0,0,23,0,0,0,0,0,0,15,10,35,28,0,0,16,8,34,10,0,0,0,0,27,5,0,0,0,0,25,32],[0,25,0,0,0,0,18,0,0,0,0,0,0,0,6,34,17,7,0,0,26,20,13,6,0,0,37,10,27,17,0,0,19,31,9,29] >;`

C42.89D10 in GAP, Magma, Sage, TeX

`C_4^2._{89}D_{10}`
`% in TeX`

`G:=Group("C4^2.89D10");`
`// GroupNames label`

`G:=SmallGroup(320,1190);`
`// by ID`

`G=gap.SmallGroup(320,1190);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,100,675,297,136,12550]);`
`// Polycyclic`

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

׿
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