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## G = C10.772- 1+4order 320 = 26·5

### 32nd 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 — C10.772- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — C4×C5⋊D4 — C10.772- 1+4
 Lower central C5 — C2×C10 — C10.772- 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C10.772- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5b2, bab-1=cac=dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a5b, be=eb, dcd-1=a5c, ce=ec, ede-1=a5b2d >

Subgroups: 646 in 214 conjugacy classes, 99 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×Q8, C22×D5, C22×C10, C22.46C24, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, Q8×C10, C23.11D10, C22.D20, C4.Dic10, C4.Dic10, C4⋊C47D5, C4⋊C4⋊D5, C2×C4⋊Dic5, C4×C5⋊D4, Q8×Dic5, D103Q8, C5×C22⋊Q8, C10.772- 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.46C24, D42D5, Q82D5, C23×D5, C2×D42D5, C2×Q82D5, D4.10D10, C10.772- 1+4

Smallest permutation representation of C10.772- 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)
(1 110 23 93)(2 109 24 92)(3 108 25 91)(4 107 26 100)(5 106 27 99)(6 105 28 98)(7 104 29 97)(8 103 30 96)(9 102 21 95)(10 101 22 94)(11 85 158 78)(12 84 159 77)(13 83 160 76)(14 82 151 75)(15 81 152 74)(16 90 153 73)(17 89 154 72)(18 88 155 71)(19 87 156 80)(20 86 157 79)(31 122 48 115)(32 121 49 114)(33 130 50 113)(34 129 41 112)(35 128 42 111)(36 127 43 120)(37 126 44 119)(38 125 45 118)(39 124 46 117)(40 123 47 116)(51 140 68 147)(52 139 69 146)(53 138 70 145)(54 137 61 144)(55 136 62 143)(56 135 63 142)(57 134 64 141)(58 133 65 150)(59 132 66 149)(60 131 67 148)
(2 10)(3 9)(4 8)(5 7)(11 16)(12 15)(13 14)(17 20)(18 19)(21 25)(22 24)(26 30)(27 29)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 50)(39 49)(40 48)(51 55)(52 54)(56 60)(57 59)(61 69)(62 68)(63 67)(64 66)(71 87)(72 86)(73 85)(74 84)(75 83)(76 82)(77 81)(78 90)(79 89)(80 88)(91 102)(92 101)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(111 120)(112 119)(113 118)(114 117)(115 116)(121 124)(122 123)(125 130)(126 129)(127 128)(131 142)(132 141)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 143)(151 160)(152 159)(153 158)(154 157)(155 156)
(1 113 23 130)(2 112 24 129)(3 111 25 128)(4 120 26 127)(5 119 27 126)(6 118 28 125)(7 117 29 124)(8 116 30 123)(9 115 21 122)(10 114 22 121)(11 58 158 65)(12 57 159 64)(13 56 160 63)(14 55 151 62)(15 54 152 61)(16 53 153 70)(17 52 154 69)(18 51 155 68)(19 60 156 67)(20 59 157 66)(31 100 48 107)(32 99 49 106)(33 98 50 105)(34 97 41 104)(35 96 42 103)(36 95 43 102)(37 94 44 101)(38 93 45 110)(39 92 46 109)(40 91 47 108)(71 142 88 135)(72 141 89 134)(73 150 90 133)(74 149 81 132)(75 148 82 131)(76 147 83 140)(77 146 84 139)(78 145 85 138)(79 144 86 137)(80 143 87 136)
(1 53 28 65)(2 54 29 66)(3 55 30 67)(4 56 21 68)(5 57 22 69)(6 58 23 70)(7 59 24 61)(8 60 25 62)(9 51 26 63)(10 52 27 64)(11 125 153 113)(12 126 154 114)(13 127 155 115)(14 128 156 116)(15 129 157 117)(16 130 158 118)(17 121 159 119)(18 122 160 120)(19 123 151 111)(20 124 152 112)(31 83 43 71)(32 84 44 72)(33 85 45 73)(34 86 46 74)(35 87 47 75)(36 88 48 76)(37 89 49 77)(38 90 50 78)(39 81 41 79)(40 82 42 80)(91 143 103 131)(92 144 104 132)(93 145 105 133)(94 146 106 134)(95 147 107 135)(96 148 108 136)(97 149 109 137)(98 150 110 138)(99 141 101 139)(100 142 102 140)```

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

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

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

53 conjugacy classes

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

53 irreducible representations

 dim 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 D5 C4○D4 C4○D4 D10 D10 D10 D10 2- 1+4 D4⋊2D5 Q8⋊2D5 D4.10D10 kernel C10.772- 1+4 C23.11D10 C22.D20 C4.Dic10 C4⋊C4⋊7D5 C4⋊C4⋊D5 C2×C4⋊Dic5 C4×C5⋊D4 Q8×Dic5 D10⋊3Q8 C5×C22⋊Q8 C22⋊Q8 C20 C2×C10 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C10 C4 C22 C2 # reps 1 2 2 3 1 2 1 1 1 1 1 2 4 4 4 6 2 2 1 4 4 4

Matrix representation of C10.772- 1+4 in GL6(𝔽41)

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

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

C10.772- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{77}2_-^{1+4}`
`% in TeX`

`G:=Group("C10.77ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1314);`
`// by ID`

`G=gap.SmallGroup(320,1314);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,100,675,570,185,192,12550]);`
`// Polycyclic`

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

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