Copied to
clipboard

## G = C10.172- 1+4order 320 = 26·5

### 17th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.172- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C10.172- 1+4
 Lower central C5 — C2×C10 — C10.172- 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C10.172- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=a5b2, e2=b2, ab=ba, cac=dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a5b, be=eb, dcd-1=a5c, ce=ec, ede-1=b2d >

Subgroups: 1198 in 294 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C22⋊Q8, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C22×C10, C22.31C24, C10.D4, C4⋊Dic5, D10⋊C4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C4○D20, Q82D5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, Q8×C10, D10⋊D4, D5×C4⋊C4, C4⋊D20, C4⋊D20, D10⋊Q8, C207D4, D103Q8, C5×C22⋊Q8, C2×C4○D20, C2×Q82D5, C10.172- 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, 2- 1+4, C22×D5, C22.31C24, D4×D5, C23×D5, C2×D4×D5, Q8.10D10, D48D10, C10.172- 1+4

Smallest permutation representation of C10.172- 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)
(1 113 13 103)(2 114 14 104)(3 115 15 105)(4 116 16 106)(5 117 17 107)(6 118 18 108)(7 119 19 109)(8 120 20 110)(9 111 11 101)(10 112 12 102)(21 91 31 81)(22 92 32 82)(23 93 33 83)(24 94 34 84)(25 95 35 85)(26 96 36 86)(27 97 37 87)(28 98 38 88)(29 99 39 89)(30 100 40 90)(41 156 51 146)(42 157 52 147)(43 158 53 148)(44 159 54 149)(45 160 55 150)(46 151 56 141)(47 152 57 142)(48 153 58 143)(49 154 59 144)(50 155 60 145)(61 136 71 126)(62 137 72 127)(63 138 73 128)(64 139 74 129)(65 140 75 130)(66 131 76 121)(67 132 77 122)(68 133 78 123)(69 134 79 124)(70 135 80 125)
(1 128)(2 127)(3 126)(4 125)(5 124)(6 123)(7 122)(8 121)(9 130)(10 129)(11 140)(12 139)(13 138)(14 137)(15 136)(16 135)(17 134)(18 133)(19 132)(20 131)(21 155)(22 154)(23 153)(24 152)(25 151)(26 160)(27 159)(28 158)(29 157)(30 156)(31 145)(32 144)(33 143)(34 142)(35 141)(36 150)(37 149)(38 148)(39 147)(40 146)(41 100)(42 99)(43 98)(44 97)(45 96)(46 95)(47 94)(48 93)(49 92)(50 91)(51 90)(52 89)(53 88)(54 87)(55 86)(56 85)(57 84)(58 83)(59 82)(60 81)(61 105)(62 104)(63 103)(64 102)(65 101)(66 110)(67 109)(68 108)(69 107)(70 106)(71 115)(72 114)(73 113)(74 112)(75 111)(76 120)(77 119)(78 118)(79 117)(80 116)
(1 33 18 28)(2 32 19 27)(3 31 20 26)(4 40 11 25)(5 39 12 24)(6 38 13 23)(7 37 14 22)(8 36 15 21)(9 35 16 30)(10 34 17 29)(41 75 56 70)(42 74 57 69)(43 73 58 68)(44 72 59 67)(45 71 60 66)(46 80 51 65)(47 79 52 64)(48 78 53 63)(49 77 54 62)(50 76 55 61)(81 105 96 120)(82 104 97 119)(83 103 98 118)(84 102 99 117)(85 101 100 116)(86 110 91 115)(87 109 92 114)(88 108 93 113)(89 107 94 112)(90 106 95 111)(121 145 136 160)(122 144 137 159)(123 143 138 158)(124 142 139 157)(125 141 140 156)(126 150 131 155)(127 149 132 154)(128 148 133 153)(129 147 134 152)(130 146 135 151)
(1 93 13 83)(2 94 14 84)(3 95 15 85)(4 96 16 86)(5 97 17 87)(6 98 18 88)(7 99 19 89)(8 100 20 90)(9 91 11 81)(10 92 12 82)(21 111 31 101)(22 112 32 102)(23 113 33 103)(24 114 34 104)(25 115 35 105)(26 116 36 106)(27 117 37 107)(28 118 38 108)(29 119 39 109)(30 120 40 110)(41 131 51 121)(42 132 52 122)(43 133 53 123)(44 134 54 124)(45 135 55 125)(46 136 56 126)(47 137 57 127)(48 138 58 128)(49 139 59 129)(50 140 60 130)(61 151 71 141)(62 152 72 142)(63 153 73 143)(64 154 74 144)(65 155 75 145)(66 156 76 146)(67 157 77 147)(68 158 78 148)(69 159 79 149)(70 160 80 150)```

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 D10 D10 D10 D10 2+ 1+4 2- 1+4 D4×D5 Q8.10D10 D4⋊8D10 kernel C10.172- 1+4 D10⋊D4 D5×C4⋊C4 C4⋊D20 D10⋊Q8 C20⋊7D4 D10⋊3Q8 C5×C22⋊Q8 C2×C4○D20 C2×Q8⋊2D5 C4×D5 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C10 C10 C4 C2 C2 # reps 1 4 1 3 2 1 1 1 1 1 4 2 4 6 2 2 1 1 4 4 4

Matrix representation of C10.172- 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 35 35 0 0 0 0 6 40 0 0 0 0 33 39 35 35 0 0 32 8 6 40
,
 1 39 0 0 0 0 0 40 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 2 13 9 0 0 0 11 4 0 9
,
 1 39 0 0 0 0 0 40 0 0 0 0 0 0 30 36 4 26 0 0 35 17 9 37 0 0 2 13 9 0 0 0 16 37 32 26
,
 40 2 0 0 0 0 40 1 0 0 0 0 0 0 18 35 0 0 0 0 20 23 0 0 0 0 32 39 18 35 0 0 35 9 20 23
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 2 13 0 0 0 0 28 39 0 0 0 0 1 23 39 28 0 0 18 16 13 2

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,33,32,0,0,35,40,39,8,0,0,0,0,35,6,0,0,0,0,35,40],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,32,0,2,11,0,0,0,32,13,4,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,30,35,2,16,0,0,36,17,13,37,0,0,4,9,9,32,0,0,26,37,0,26],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,18,20,32,35,0,0,35,23,39,9,0,0,0,0,18,20,0,0,0,0,35,23],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,28,1,18,0,0,13,39,23,16,0,0,0,0,39,13,0,0,0,0,28,2] >;`

C10.172- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{17}2_-^{1+4}`
`% in TeX`

`G:=Group("C10.17ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1301);`
`// by ID`

`G=gap.SmallGroup(320,1301);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,297,12550]);`
`// Polycyclic`

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

׿
×
𝔽