direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×M4(2)⋊C4, M4(2)⋊1C20, C8⋊1(C2×C20), C40⋊29(C2×C4), C4.Q8⋊2C10, C4.4(Q8×C10), C2.D8⋊10C10, C20.95(C4⋊C4), C20.93(C2×Q8), (C2×C20).43Q8, (C2×C20).522D4, C23.40(C5×D4), (C5×M4(2))⋊10C4, C4.27(C22×C20), C22.50(D4×C10), C20.244(C22×C4), (C2×C20).901C23, (C2×C40).269C22, C42⋊C2.7C10, (C22×C10).162D4, (C10×M4(2)).7C2, (C2×M4(2)).1C10, C10.128(C8⋊C22), C10.128(C8.C22), (C22×C20).414C22, C4.15(C5×C4⋊C4), C10.93(C2×C4⋊C4), C2.14(C10×C4⋊C4), (C2×C4).6(C5×Q8), (C5×C4.Q8)⋊11C2, (C5×C2.D8)⋊25C2, C2.3(C5×C8⋊C22), (C2×C4⋊C4).15C10, (C10×C4⋊C4).44C2, C4⋊C4.44(C2×C10), (C2×C4).25(C2×C20), (C2×C8).16(C2×C10), (C2×C4).125(C5×D4), C22.10(C5×C4⋊C4), (C2×C10).55(C4⋊C4), C2.3(C5×C8.C22), (C2×C20).371(C2×C4), (C2×C10).626(C2×D4), (C5×C4⋊C4).365C22, (C2×C4).76(C22×C10), (C22×C4).33(C2×C10), (C5×C42⋊C2).21C2, SmallGroup(320,929)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×M4(2)⋊C4
G = < a,b,c,d | a5=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, cd=dc >
Subgroups: 178 in 118 conjugacy classes, 82 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4, C20 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C2×M4(2), C40 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×6], C22×C10, M4(2)⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C5×C4⋊C4, C2×C40 [×2], C5×M4(2) [×4], C22×C20, C22×C20, C5×C4.Q8 [×2], C5×C2.D8 [×2], C10×C4⋊C4, C5×C42⋊C2, C10×M4(2), C5×M4(2)⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], Q8 [×2], C23, C10 [×7], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C20 [×4], C2×C10 [×7], C2×C4⋊C4, C8⋊C22, C8.C22, C2×C20 [×6], C5×D4 [×2], C5×Q8 [×2], C22×C10, M4(2)⋊C4, C5×C4⋊C4 [×4], C22×C20, D4×C10, Q8×C10, C10×C4⋊C4, C5×C8⋊C22, C5×C8.C22, C5×M4(2)⋊C4
►On 160 points
Generators in S160
(1 85 75 33 67)(2 86 76 34 68)(3 87 77 35 69)(4 88 78 36 70)(5 81 79 37 71)(6 82 80 38 72)(7 83 73 39 65)(8 84 74 40 66)(9 27 57 54 19)(10 28 58 55 20)(11 29 59 56 21)(12 30 60 49 22)(13 31 61 50 23)(14 32 62 51 24)(15 25 63 52 17)(16 26 64 53 18)(41 154 116 146 108)(42 155 117 147 109)(43 156 118 148 110)(44 157 119 149 111)(45 158 120 150 112)(46 159 113 151 105)(47 160 114 152 106)(48 153 115 145 107)(89 127 143 97 135)(90 128 144 98 136)(91 121 137 99 129)(92 122 138 100 130)(93 123 139 101 131)(94 124 140 102 132)(95 125 141 103 133)(96 126 142 104 134)
(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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)(34 38)(36 40)(41 45)(43 47)(49 53)(51 55)(58 62)(60 64)(66 70)(68 72)(74 78)(76 80)(82 86)(84 88)(89 93)(91 95)(97 101)(99 103)(106 110)(108 112)(114 118)(116 120)(121 125)(123 127)(129 133)(131 135)(137 141)(139 143)(146 150)(148 152)(154 158)(156 160)
(1 126 13 111)(2 125 14 110)(3 124 15 109)(4 123 16 108)(5 122 9 107)(6 121 10 106)(7 128 11 105)(8 127 12 112)(17 147 69 94)(18 146 70 93)(19 145 71 92)(20 152 72 91)(21 151 65 90)(22 150 66 89)(23 149 67 96)(24 148 68 95)(25 42 87 140)(26 41 88 139)(27 48 81 138)(28 47 82 137)(29 46 83 144)(30 45 84 143)(31 44 85 142)(32 43 86 141)(33 134 50 119)(34 133 51 118)(35 132 52 117)(36 131 53 116)(37 130 54 115)(38 129 55 114)(39 136 56 113)(40 135 49 120)(57 153 79 100)(58 160 80 99)(59 159 73 98)(60 158 74 97)(61 157 75 104)(62 156 76 103)(63 155 77 102)(64 154 78 101)
G:=sub<Sym(160)| (1,85,75,33,67)(2,86,76,34,68)(3,87,77,35,69)(4,88,78,36,70)(5,81,79,37,71)(6,82,80,38,72)(7,83,73,39,65)(8,84,74,40,66)(9,27,57,54,19)(10,28,58,55,20)(11,29,59,56,21)(12,30,60,49,22)(13,31,61,50,23)(14,32,62,51,24)(15,25,63,52,17)(16,26,64,53,18)(41,154,116,146,108)(42,155,117,147,109)(43,156,118,148,110)(44,157,119,149,111)(45,158,120,150,112)(46,159,113,151,105)(47,160,114,152,106)(48,153,115,145,107)(89,127,143,97,135)(90,128,144,98,136)(91,121,137,99,129)(92,122,138,100,130)(93,123,139,101,131)(94,124,140,102,132)(95,125,141,103,133)(96,126,142,104,134), (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(26,30)(28,32)(34,38)(36,40)(41,45)(43,47)(49,53)(51,55)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80)(82,86)(84,88)(89,93)(91,95)(97,101)(99,103)(106,110)(108,112)(114,118)(116,120)(121,125)(123,127)(129,133)(131,135)(137,141)(139,143)(146,150)(148,152)(154,158)(156,160), (1,126,13,111)(2,125,14,110)(3,124,15,109)(4,123,16,108)(5,122,9,107)(6,121,10,106)(7,128,11,105)(8,127,12,112)(17,147,69,94)(18,146,70,93)(19,145,71,92)(20,152,72,91)(21,151,65,90)(22,150,66,89)(23,149,67,96)(24,148,68,95)(25,42,87,140)(26,41,88,139)(27,48,81,138)(28,47,82,137)(29,46,83,144)(30,45,84,143)(31,44,85,142)(32,43,86,141)(33,134,50,119)(34,133,51,118)(35,132,52,117)(36,131,53,116)(37,130,54,115)(38,129,55,114)(39,136,56,113)(40,135,49,120)(57,153,79,100)(58,160,80,99)(59,159,73,98)(60,158,74,97)(61,157,75,104)(62,156,76,103)(63,155,77,102)(64,154,78,101)>;
G:=Group( (1,85,75,33,67)(2,86,76,34,68)(3,87,77,35,69)(4,88,78,36,70)(5,81,79,37,71)(6,82,80,38,72)(7,83,73,39,65)(8,84,74,40,66)(9,27,57,54,19)(10,28,58,55,20)(11,29,59,56,21)(12,30,60,49,22)(13,31,61,50,23)(14,32,62,51,24)(15,25,63,52,17)(16,26,64,53,18)(41,154,116,146,108)(42,155,117,147,109)(43,156,118,148,110)(44,157,119,149,111)(45,158,120,150,112)(46,159,113,151,105)(47,160,114,152,106)(48,153,115,145,107)(89,127,143,97,135)(90,128,144,98,136)(91,121,137,99,129)(92,122,138,100,130)(93,123,139,101,131)(94,124,140,102,132)(95,125,141,103,133)(96,126,142,104,134), (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(26,30)(28,32)(34,38)(36,40)(41,45)(43,47)(49,53)(51,55)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80)(82,86)(84,88)(89,93)(91,95)(97,101)(99,103)(106,110)(108,112)(114,118)(116,120)(121,125)(123,127)(129,133)(131,135)(137,141)(139,143)(146,150)(148,152)(154,158)(156,160), (1,126,13,111)(2,125,14,110)(3,124,15,109)(4,123,16,108)(5,122,9,107)(6,121,10,106)(7,128,11,105)(8,127,12,112)(17,147,69,94)(18,146,70,93)(19,145,71,92)(20,152,72,91)(21,151,65,90)(22,150,66,89)(23,149,67,96)(24,148,68,95)(25,42,87,140)(26,41,88,139)(27,48,81,138)(28,47,82,137)(29,46,83,144)(30,45,84,143)(31,44,85,142)(32,43,86,141)(33,134,50,119)(34,133,51,118)(35,132,52,117)(36,131,53,116)(37,130,54,115)(38,129,55,114)(39,136,56,113)(40,135,49,120)(57,153,79,100)(58,160,80,99)(59,159,73,98)(60,158,74,97)(61,157,75,104)(62,156,76,103)(63,155,77,102)(64,154,78,101) );
G=PermutationGroup([(1,85,75,33,67),(2,86,76,34,68),(3,87,77,35,69),(4,88,78,36,70),(5,81,79,37,71),(6,82,80,38,72),(7,83,73,39,65),(8,84,74,40,66),(9,27,57,54,19),(10,28,58,55,20),(11,29,59,56,21),(12,30,60,49,22),(13,31,61,50,23),(14,32,62,51,24),(15,25,63,52,17),(16,26,64,53,18),(41,154,116,146,108),(42,155,117,147,109),(43,156,118,148,110),(44,157,119,149,111),(45,158,120,150,112),(46,159,113,151,105),(47,160,114,152,106),(48,153,115,145,107),(89,127,143,97,135),(90,128,144,98,136),(91,121,137,99,129),(92,122,138,100,130),(93,123,139,101,131),(94,124,140,102,132),(95,125,141,103,133),(96,126,142,104,134)], [(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)], [(2,6),(4,8),(10,14),(12,16),(18,22),(20,24),(26,30),(28,32),(34,38),(36,40),(41,45),(43,47),(49,53),(51,55),(58,62),(60,64),(66,70),(68,72),(74,78),(76,80),(82,86),(84,88),(89,93),(91,95),(97,101),(99,103),(106,110),(108,112),(114,118),(116,120),(121,125),(123,127),(129,133),(131,135),(137,141),(139,143),(146,150),(148,152),(154,158),(156,160)], [(1,126,13,111),(2,125,14,110),(3,124,15,109),(4,123,16,108),(5,122,9,107),(6,121,10,106),(7,128,11,105),(8,127,12,112),(17,147,69,94),(18,146,70,93),(19,145,71,92),(20,152,72,91),(21,151,65,90),(22,150,66,89),(23,149,67,96),(24,148,68,95),(25,42,87,140),(26,41,88,139),(27,48,81,138),(28,47,82,137),(29,46,83,144),(30,45,84,143),(31,44,85,142),(32,43,86,141),(33,134,50,119),(34,133,51,118),(35,132,52,117),(36,131,53,116),(37,130,54,115),(38,129,55,114),(39,136,56,113),(40,135,49,120),(57,153,79,100),(58,160,80,99),(59,159,73,98),(60,158,74,97),(61,157,75,104),(62,156,76,103),(63,155,77,102),(64,154,78,101)])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20AV | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 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 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | D4 | Q8 | D4 | C5×D4 | C5×Q8 | C5×D4 | C8⋊C22 | C8.C22 | C5×C8⋊C22 | C5×C8.C22 |
| kernel | C5×M4(2)⋊C4 | C5×C4.Q8 | C5×C2.D8 | C10×C4⋊C4 | C5×C42⋊C2 | C10×M4(2) | C5×M4(2) | M4(2)⋊C4 | C4.Q8 | C2.D8 | C2×C4⋊C4 | C42⋊C2 | C2×M4(2) | M4(2) | C2×C20 | C2×C20 | C22×C10 | C2×C4 | C2×C4 | C23 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 8 | 8 | 4 | 4 | 4 | 32 | 1 | 2 | 1 | 4 | 8 | 4 | 1 | 1 | 4 | 4 |
Matrix representation of C5×M4(2)⋊C4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 40 | 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 18 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 24 | 0 | 0 |
| 0 | 0 | 24 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 9 |
| 0 | 0 | 0 | 0 | 9 | 24 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[40,1,0,0,0,0,39,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,18,32,0,0,0,0,0,0,9,24,0,0,0,0,24,32,0,0,0,0,0,0,17,9,0,0,0,0,9,24] >;
C5×M4(2)⋊C4 in GAP, Magma, Sage, TeX
C_5\times M_4(2)\rtimes C_4
% in TeX
G:=Group("C5xM4(2):C4"); // GroupNames label
G:=SmallGroup(320,929);
// by ID
G=gap.SmallGroup(320,929);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,288,1731,7004,172]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b^-1,c*d=d*c>;
// generators/relations