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## G = C10.802- 1+4order 320 = 26·5

### 35th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.802- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C10.D4 — C10.802- 1+4
 Lower central C5 — C2×C10 — C10.802- 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C10.802- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=e2=a5b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=a5b-1, bd=db, be=eb, cd=dc, ece-1=a5c, ede-1=b2d >

Subgroups: 614 in 214 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×14], C22, C22 [×2], C22 [×5], C5, C2×C4 [×5], C2×C4 [×16], D4 [×2], Q8 [×2], C23 [×2], C10 [×3], C10 [×3], C42 [×5], C22⋊C4 [×3], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4, C22×C4 [×3], C2×D4, C2×Q8, Dic5 [×2], Dic5 [×7], C20 [×5], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4, C22.D4, C42.C2 [×3], C422C2 [×2], Dic10 [×2], C2×Dic5 [×8], C2×Dic5 [×6], C2×C20 [×5], C2×C20 [×2], C5×D4 [×2], C22×C10 [×2], C22.46C24, C4×Dic5 [×5], C10.D4 [×10], C4⋊Dic5 [×4], C23.D5 [×5], C5×C22⋊C4 [×3], C5×C4⋊C4 [×2], C2×Dic10, C22×Dic5 [×3], C22×C20, D4×C10, C23.11D10 [×2], Dic5.14D4 [×2], C23.D10 [×2], Dic53Q8, Dic5.Q8 [×2], C4.Dic10, C2×C10.D4, C23.21D10, D4×Dic5, C23.18D10, C5×C22.D4, C10.802- 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- 1+4, C22×D5 [×7], C22.46C24, D42D5 [×2], C23×D5, C2×D42D5, D5×C4○D4, D4.10D10, C10.802- 1+4

Smallest permutation representation of C10.802- 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)
(1 112 26 125)(2 111 27 124)(3 120 28 123)(4 119 29 122)(5 118 30 121)(6 117 21 130)(7 116 22 129)(8 115 23 128)(9 114 24 127)(10 113 25 126)(11 64 153 52)(12 63 154 51)(13 62 155 60)(14 61 156 59)(15 70 157 58)(16 69 158 57)(17 68 159 56)(18 67 160 55)(19 66 151 54)(20 65 152 53)(31 94 43 107)(32 93 44 106)(33 92 45 105)(34 91 46 104)(35 100 47 103)(36 99 48 102)(37 98 49 101)(38 97 50 110)(39 96 41 109)(40 95 42 108)(71 134 84 147)(72 133 85 146)(73 132 86 145)(74 131 87 144)(75 140 88 143)(76 139 89 142)(77 138 90 141)(78 137 81 150)(79 136 82 149)(80 135 83 148)
(11 16)(12 17)(13 18)(14 19)(15 20)(31 48)(32 49)(33 50)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 61)(60 62)(71 76)(72 77)(73 78)(74 79)(75 80)(81 86)(82 87)(83 88)(84 89)(85 90)(111 129)(112 130)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)(131 144)(132 145)(133 146)(134 147)(135 148)(136 149)(137 150)(138 141)(139 142)(140 143)(151 156)(152 157)(153 158)(154 159)(155 160)
(1 92 21 110)(2 91 22 109)(3 100 23 108)(4 99 24 107)(5 98 25 106)(6 97 26 105)(7 96 27 104)(8 95 28 103)(9 94 29 102)(10 93 30 101)(11 85 158 77)(12 84 159 76)(13 83 160 75)(14 82 151 74)(15 81 152 73)(16 90 153 72)(17 89 154 71)(18 88 155 80)(19 87 156 79)(20 86 157 78)(31 119 48 127)(32 118 49 126)(33 117 50 125)(34 116 41 124)(35 115 42 123)(36 114 43 122)(37 113 44 121)(38 112 45 130)(39 111 46 129)(40 120 47 128)(51 134 68 142)(52 133 69 141)(53 132 70 150)(54 131 61 149)(55 140 62 148)(56 139 63 147)(57 138 64 146)(58 137 65 145)(59 136 66 144)(60 135 67 143)
(1 81 21 73)(2 82 22 74)(3 83 23 75)(4 84 24 76)(5 85 25 77)(6 86 26 78)(7 87 27 79)(8 88 28 80)(9 89 29 71)(10 90 30 72)(11 101 158 93)(12 102 159 94)(13 103 160 95)(14 104 151 96)(15 105 152 97)(16 106 153 98)(17 107 154 99)(18 108 155 100)(19 109 156 91)(20 110 157 92)(31 51 48 68)(32 52 49 69)(33 53 50 70)(34 54 41 61)(35 55 42 62)(36 56 43 63)(37 57 44 64)(38 58 45 65)(39 59 46 66)(40 60 47 67)(111 149 129 131)(112 150 130 132)(113 141 121 133)(114 142 122 134)(115 143 123 135)(116 144 124 136)(117 145 125 137)(118 146 126 138)(119 147 127 139)(120 148 128 140)```

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

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

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

53 conjugacy classes

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

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D10 2- 1+4 D4⋊2D5 D5×C4○D4 D4.10D10 kernel C10.802- 1+4 C23.11D10 Dic5.14D4 C23.D10 Dic5⋊3Q8 Dic5.Q8 C4.Dic10 C2×C10.D4 C23.21D10 D4×Dic5 C23.18D10 C5×C22.D4 C22.D4 Dic5 C2×C10 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C22 C2 C2 # reps 1 2 2 2 1 2 1 1 1 1 1 1 2 4 4 6 4 2 2 1 4 4 4

Matrix representation of C10.802- 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 35 0 0 0 0 7 34 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 32 0 0 0 0 9 0 0 0 0 0 0 0 15 3 0 0 0 0 21 26 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 40 40
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 15 3 0 0 0 0 21 26 0 0 0 0 0 0 1 0 0 0 0 0 40 40
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 18 0 0 0 0 32 32

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,7,0,0,0,0,35,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,15,21,0,0,0,0,3,26,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,15,21,0,0,0,0,3,26,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,32,0,0,0,0,18,32] >;`

C10.802- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{80}2_-^{1+4}`
`% in TeX`

`G:=Group("C10.80ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1322);`
`// by ID`

`G=gap.SmallGroup(320,1322);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,100,346,297,12550]);`
`// Polycyclic`

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

׿
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