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## G = C10.442+ 1+4order 320 = 26·5

### 44th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.442+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.12D4 — C10.442+ 1+4
 Lower central C5 — C2×C10 — C10.442+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C10.442+ 1+4
G = < a,b,c,d,e | a10=b4=1, c2=e2=a5, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=a5b-1, bd=db, ebe-1=a5b, cd=dc, ce=ec, ede-1=a5b2d >

Subgroups: 910 in 240 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C41D4, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22.34C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C23.11D10, Dic54D4, D10.12D4, D10⋊D4, Dic5.Q8, D10.13D4, C4×C5⋊D4, C23.23D10, C23.18D10, C202D4, Dic5⋊D4, C20⋊D4, C5×C4⋊D4, C10.442+ 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.34C24, C23×D5, D46D10, D5×C4○D4, C10.442+ 1+4

Smallest permutation representation of C10.442+ 1+4
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)
(1 138 18 123)(2 139 19 124)(3 140 20 125)(4 131 11 126)(5 132 12 127)(6 133 13 128)(7 134 14 129)(8 135 15 130)(9 136 16 121)(10 137 17 122)(21 156 36 141)(22 157 37 142)(23 158 38 143)(24 159 39 144)(25 160 40 145)(26 151 31 146)(27 152 32 147)(28 153 33 148)(29 154 34 149)(30 155 35 150)(41 96 56 81)(42 97 57 82)(43 98 58 83)(44 99 59 84)(45 100 60 85)(46 91 51 86)(47 92 52 87)(48 93 53 88)(49 94 54 89)(50 95 55 90)(61 116 76 101)(62 117 77 102)(63 118 78 103)(64 119 79 104)(65 120 80 105)(66 111 71 106)(67 112 72 107)(68 113 73 108)(69 114 74 109)(70 115 75 110)
(1 88 6 83)(2 89 7 84)(3 90 8 85)(4 81 9 86)(5 82 10 87)(11 96 16 91)(12 97 17 92)(13 98 18 93)(14 99 19 94)(15 100 20 95)(21 106 26 101)(22 107 27 102)(23 108 28 103)(24 109 29 104)(25 110 30 105)(31 116 36 111)(32 117 37 112)(33 118 38 113)(34 119 39 114)(35 120 40 115)(41 126 46 121)(42 127 47 122)(43 128 48 123)(44 129 49 124)(45 130 50 125)(51 136 56 131)(52 137 57 132)(53 138 58 133)(54 139 59 134)(55 140 60 135)(61 146 66 141)(62 147 67 142)(63 148 68 143)(64 149 69 144)(65 150 70 145)(71 156 76 151)(72 157 77 152)(73 158 78 153)(74 159 79 154)(75 160 80 155)
(1 38 18 23)(2 37 19 22)(3 36 20 21)(4 35 11 30)(5 34 12 29)(6 33 13 28)(7 32 14 27)(8 31 15 26)(9 40 16 25)(10 39 17 24)(41 80 56 65)(42 79 57 64)(43 78 58 63)(44 77 59 62)(45 76 60 61)(46 75 51 70)(47 74 52 69)(48 73 53 68)(49 72 54 67)(50 71 55 66)(81 120 96 105)(82 119 97 104)(83 118 98 103)(84 117 99 102)(85 116 100 101)(86 115 91 110)(87 114 92 109)(88 113 93 108)(89 112 94 107)(90 111 95 106)(121 160 136 145)(122 159 137 144)(123 158 138 143)(124 157 139 142)(125 156 140 141)(126 155 131 150)(127 154 132 149)(128 153 133 148)(129 152 134 147)(130 151 135 146)
(1 28 6 23)(2 29 7 24)(3 30 8 25)(4 21 9 26)(5 22 10 27)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 61 46 66)(42 62 47 67)(43 63 48 68)(44 64 49 69)(45 65 50 70)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 60 80)(81 106 86 101)(82 107 87 102)(83 108 88 103)(84 109 89 104)(85 110 90 105)(91 116 96 111)(92 117 97 112)(93 118 98 113)(94 119 99 114)(95 120 100 115)(121 141 126 146)(122 142 127 147)(123 143 128 148)(124 144 129 149)(125 145 130 150)(131 151 136 156)(132 152 137 157)(133 153 138 158)(134 154 139 159)(135 155 140 160)```

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 D4⋊6D10 D5×C4○D4 kernel C10.442+ 1+4 C23.11D10 Dic5⋊4D4 D10.12D4 D10⋊D4 Dic5.Q8 D10.13D4 C4×C5⋊D4 C23.23D10 C23.18D10 C20⋊2D4 Dic5⋊D4 C20⋊D4 C5×C4⋊D4 C4⋊D4 Dic5 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 1 3 1 1 2 4 4 2 2 6 2 8 4

Matrix representation of C10.442+ 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 6 6 0 0 0 0 35 1 0 0 0 0 0 0 6 6 0 0 0 0 35 1
,
 9 36 0 0 0 0 0 32 0 0 0 0 0 0 39 28 17 0 0 0 13 2 0 17 0 0 17 0 2 13 0 0 0 17 28 39
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 18 35 0 0 0 0 6 23 0 0 23 6 0 0 0 0 35 18 0 0
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 1 0 0 0 0 0 6 40 0 0 1 0 0 0 0 0 6 40 0 0
,
 32 0 0 0 0 0 25 9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 40 0 0 0 0 0 0 40 0 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,6,1,0,0,0,0,0,0,6,35,0,0,0,0,6,1],[9,0,0,0,0,0,36,32,0,0,0,0,0,0,39,13,17,0,0,0,28,2,0,17,0,0,17,0,2,28,0,0,0,17,13,39],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,23,35,0,0,0,0,6,18,0,0,18,6,0,0,0,0,35,23,0,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,1,6,0,0,0,0,0,40,0,0],[32,25,0,0,0,0,0,9,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C10.442+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{44}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.44ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1287);`
`// by ID`

`G=gap.SmallGroup(320,1287);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,675,297,136,12550]);`
`// Polycyclic`

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

׿
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