metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.178D10, (D4×Dic5)⋊16C2, C4⋊D4.10D5, (C2×D4).152D10, C22⋊C4.47D10, C4.Dic10⋊18C2, Dic5⋊3Q8⋊21C2, C20.201(C4○D4), C20.48D4⋊31C2, C4.67(D4⋊2D5), C20.17D4⋊15C2, (C2×C10).144C24, (C2×C20).501C23, (C22×C4).367D10, C23.11(C22×D5), Dic5.72(C4○D4), (D4×C10).118C22, C22.5(D4⋊2D5), C23.D10⋊14C2, C23.11D10⋊4C2, C23.18D10⋊7C2, C4⋊Dic5.205C22, (C22×C10).15C23, (C4×Dic5).99C22, C22.165(C23×D5), C23.D5.21C22, (C22×C20).238C22, C5⋊6(C23.36C23), (C2×Dic5).236C23, C10.D4.15C22, (C2×Dic10).158C22, (C22×Dic5).105C22, (C2×C4×Dic5)⋊8C2, C2.35(D5×C4○D4), (C5×C4⋊D4).7C2, C10.149(C2×C4○D4), C2.32(C2×D4⋊2D5), (C2×C10).20(C4○D4), (C5×C4⋊C4).140C22, (C2×C4).292(C22×D5), (C5×C22⋊C4).9C22, SmallGroup(320,1272)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 670 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×8], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C23 [×2], C10 [×3], C10 [×4], C42 [×6], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, Dic5 [×2], Dic5 [×7], C20 [×2], C20 [×3], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], Dic10 [×2], C2×Dic5 [×4], C2×Dic5 [×4], C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×C10, C22×C10 [×2], C23.36C23, C4×Dic5 [×4], C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×2], C23.D5 [×8], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, D4×C10, D4×C10 [×2], C23.11D10 [×2], C23.D10 [×2], Dic5⋊3Q8, C4.Dic10, C2×C4×Dic5, C20.48D4, D4×Dic5, D4×Dic5 [×2], C23.18D10 [×2], C20.17D4, C5×C4⋊D4, C4⋊C4.178D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, D4⋊2D5 [×4], C23×D5, C2×D4⋊2D5 [×2], D5×C4○D4, C4⋊C4.178D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 27 32 50)(2 28 33 41)(3 29 34 42)(4 30 35 43)(5 21 36 44)(6 22 37 45)(7 23 38 46)(8 24 39 47)(9 25 40 48)(10 26 31 49)(11 74 54 112)(12 75 55 113)(13 76 56 114)(14 77 57 115)(15 78 58 116)(16 79 59 117)(17 80 60 118)(18 71 51 119)(19 72 52 120)(20 73 53 111)(61 96 158 148)(62 97 159 149)(63 98 160 150)(64 99 151 141)(65 100 152 142)(66 91 153 143)(67 92 154 144)(68 93 155 145)(69 94 156 146)(70 95 157 147)(81 123 133 106)(82 124 134 107)(83 125 135 108)(84 126 136 109)(85 127 137 110)(86 128 138 101)(87 129 139 102)(88 130 140 103)(89 121 131 104)(90 122 132 105)
(1 116 111 6)(2 7 112 117)(3 118 113 8)(4 9 114 119)(5 120 115 10)(11 59 28 46)(12 47 29 60)(13 51 30 48)(14 49 21 52)(15 53 22 50)(16 41 23 54)(17 55 24 42)(18 43 25 56)(19 57 26 44)(20 45 27 58)(31 36 72 77)(32 78 73 37)(33 38 74 79)(34 80 75 39)(35 40 76 71)(61 66 104 109)(62 110 105 67)(63 68 106 101)(64 102 107 69)(65 70 108 103)(81 138 98 145)(82 146 99 139)(83 140 100 147)(84 148 91 131)(85 132 92 149)(86 150 93 133)(87 134 94 141)(88 142 95 135)(89 136 96 143)(90 144 97 137)(121 126 158 153)(122 154 159 127)(123 128 160 155)(124 156 151 129)(125 130 152 157)
(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)
(1 156 73 102)(2 155 74 101)(3 154 75 110)(4 153 76 109)(5 152 77 108)(6 151 78 107)(7 160 79 106)(8 159 80 105)(9 158 71 104)(10 157 72 103)(11 138 41 93)(12 137 42 92)(13 136 43 91)(14 135 44 100)(15 134 45 99)(16 133 46 98)(17 132 47 97)(18 131 48 96)(19 140 49 95)(20 139 50 94)(21 142 57 83)(22 141 58 82)(23 150 59 81)(24 149 60 90)(25 148 51 89)(26 147 52 88)(27 146 53 87)(28 145 54 86)(29 144 55 85)(30 143 56 84)(31 70 120 130)(32 69 111 129)(33 68 112 128)(34 67 113 127)(35 66 114 126)(36 65 115 125)(37 64 116 124)(38 63 117 123)(39 62 118 122)(40 61 119 121)
G:=sub<Sym(160)| (1,27,32,50)(2,28,33,41)(3,29,34,42)(4,30,35,43)(5,21,36,44)(6,22,37,45)(7,23,38,46)(8,24,39,47)(9,25,40,48)(10,26,31,49)(11,74,54,112)(12,75,55,113)(13,76,56,114)(14,77,57,115)(15,78,58,116)(16,79,59,117)(17,80,60,118)(18,71,51,119)(19,72,52,120)(20,73,53,111)(61,96,158,148)(62,97,159,149)(63,98,160,150)(64,99,151,141)(65,100,152,142)(66,91,153,143)(67,92,154,144)(68,93,155,145)(69,94,156,146)(70,95,157,147)(81,123,133,106)(82,124,134,107)(83,125,135,108)(84,126,136,109)(85,127,137,110)(86,128,138,101)(87,129,139,102)(88,130,140,103)(89,121,131,104)(90,122,132,105), (1,116,111,6)(2,7,112,117)(3,118,113,8)(4,9,114,119)(5,120,115,10)(11,59,28,46)(12,47,29,60)(13,51,30,48)(14,49,21,52)(15,53,22,50)(16,41,23,54)(17,55,24,42)(18,43,25,56)(19,57,26,44)(20,45,27,58)(31,36,72,77)(32,78,73,37)(33,38,74,79)(34,80,75,39)(35,40,76,71)(61,66,104,109)(62,110,105,67)(63,68,106,101)(64,102,107,69)(65,70,108,103)(81,138,98,145)(82,146,99,139)(83,140,100,147)(84,148,91,131)(85,132,92,149)(86,150,93,133)(87,134,94,141)(88,142,95,135)(89,136,96,143)(90,144,97,137)(121,126,158,153)(122,154,159,127)(123,128,160,155)(124,156,151,129)(125,130,152,157), (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), (1,156,73,102)(2,155,74,101)(3,154,75,110)(4,153,76,109)(5,152,77,108)(6,151,78,107)(7,160,79,106)(8,159,80,105)(9,158,71,104)(10,157,72,103)(11,138,41,93)(12,137,42,92)(13,136,43,91)(14,135,44,100)(15,134,45,99)(16,133,46,98)(17,132,47,97)(18,131,48,96)(19,140,49,95)(20,139,50,94)(21,142,57,83)(22,141,58,82)(23,150,59,81)(24,149,60,90)(25,148,51,89)(26,147,52,88)(27,146,53,87)(28,145,54,86)(29,144,55,85)(30,143,56,84)(31,70,120,130)(32,69,111,129)(33,68,112,128)(34,67,113,127)(35,66,114,126)(36,65,115,125)(37,64,116,124)(38,63,117,123)(39,62,118,122)(40,61,119,121)>;
G:=Group( (1,27,32,50)(2,28,33,41)(3,29,34,42)(4,30,35,43)(5,21,36,44)(6,22,37,45)(7,23,38,46)(8,24,39,47)(9,25,40,48)(10,26,31,49)(11,74,54,112)(12,75,55,113)(13,76,56,114)(14,77,57,115)(15,78,58,116)(16,79,59,117)(17,80,60,118)(18,71,51,119)(19,72,52,120)(20,73,53,111)(61,96,158,148)(62,97,159,149)(63,98,160,150)(64,99,151,141)(65,100,152,142)(66,91,153,143)(67,92,154,144)(68,93,155,145)(69,94,156,146)(70,95,157,147)(81,123,133,106)(82,124,134,107)(83,125,135,108)(84,126,136,109)(85,127,137,110)(86,128,138,101)(87,129,139,102)(88,130,140,103)(89,121,131,104)(90,122,132,105), (1,116,111,6)(2,7,112,117)(3,118,113,8)(4,9,114,119)(5,120,115,10)(11,59,28,46)(12,47,29,60)(13,51,30,48)(14,49,21,52)(15,53,22,50)(16,41,23,54)(17,55,24,42)(18,43,25,56)(19,57,26,44)(20,45,27,58)(31,36,72,77)(32,78,73,37)(33,38,74,79)(34,80,75,39)(35,40,76,71)(61,66,104,109)(62,110,105,67)(63,68,106,101)(64,102,107,69)(65,70,108,103)(81,138,98,145)(82,146,99,139)(83,140,100,147)(84,148,91,131)(85,132,92,149)(86,150,93,133)(87,134,94,141)(88,142,95,135)(89,136,96,143)(90,144,97,137)(121,126,158,153)(122,154,159,127)(123,128,160,155)(124,156,151,129)(125,130,152,157), (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), (1,156,73,102)(2,155,74,101)(3,154,75,110)(4,153,76,109)(5,152,77,108)(6,151,78,107)(7,160,79,106)(8,159,80,105)(9,158,71,104)(10,157,72,103)(11,138,41,93)(12,137,42,92)(13,136,43,91)(14,135,44,100)(15,134,45,99)(16,133,46,98)(17,132,47,97)(18,131,48,96)(19,140,49,95)(20,139,50,94)(21,142,57,83)(22,141,58,82)(23,150,59,81)(24,149,60,90)(25,148,51,89)(26,147,52,88)(27,146,53,87)(28,145,54,86)(29,144,55,85)(30,143,56,84)(31,70,120,130)(32,69,111,129)(33,68,112,128)(34,67,113,127)(35,66,114,126)(36,65,115,125)(37,64,116,124)(38,63,117,123)(39,62,118,122)(40,61,119,121) );
G=PermutationGroup([(1,27,32,50),(2,28,33,41),(3,29,34,42),(4,30,35,43),(5,21,36,44),(6,22,37,45),(7,23,38,46),(8,24,39,47),(9,25,40,48),(10,26,31,49),(11,74,54,112),(12,75,55,113),(13,76,56,114),(14,77,57,115),(15,78,58,116),(16,79,59,117),(17,80,60,118),(18,71,51,119),(19,72,52,120),(20,73,53,111),(61,96,158,148),(62,97,159,149),(63,98,160,150),(64,99,151,141),(65,100,152,142),(66,91,153,143),(67,92,154,144),(68,93,155,145),(69,94,156,146),(70,95,157,147),(81,123,133,106),(82,124,134,107),(83,125,135,108),(84,126,136,109),(85,127,137,110),(86,128,138,101),(87,129,139,102),(88,130,140,103),(89,121,131,104),(90,122,132,105)], [(1,116,111,6),(2,7,112,117),(3,118,113,8),(4,9,114,119),(5,120,115,10),(11,59,28,46),(12,47,29,60),(13,51,30,48),(14,49,21,52),(15,53,22,50),(16,41,23,54),(17,55,24,42),(18,43,25,56),(19,57,26,44),(20,45,27,58),(31,36,72,77),(32,78,73,37),(33,38,74,79),(34,80,75,39),(35,40,76,71),(61,66,104,109),(62,110,105,67),(63,68,106,101),(64,102,107,69),(65,70,108,103),(81,138,98,145),(82,146,99,139),(83,140,100,147),(84,148,91,131),(85,132,92,149),(86,150,93,133),(87,134,94,141),(88,142,95,135),(89,136,96,143),(90,144,97,137),(121,126,158,153),(122,154,159,127),(123,128,160,155),(124,156,151,129),(125,130,152,157)], [(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)], [(1,156,73,102),(2,155,74,101),(3,154,75,110),(4,153,76,109),(5,152,77,108),(6,151,78,107),(7,160,79,106),(8,159,80,105),(9,158,71,104),(10,157,72,103),(11,138,41,93),(12,137,42,92),(13,136,43,91),(14,135,44,100),(15,134,45,99),(16,133,46,98),(17,132,47,97),(18,131,48,96),(19,140,49,95),(20,139,50,94),(21,142,57,83),(22,141,58,82),(23,150,59,81),(24,149,60,90),(25,148,51,89),(26,147,52,88),(27,146,53,87),(28,145,54,86),(29,144,55,85),(30,143,56,84),(31,70,120,130),(32,69,111,129),(33,68,112,128),(34,67,113,127),(35,66,114,126),(36,65,115,125),(37,64,116,124),(38,63,117,123),(39,62,118,122),(40,61,119,121)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 5 | 0 | 0 | 0 | 0 |
| 16 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 25 | 0 | 0 |
| 0 | 0 | 36 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 9 | 4 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 25 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 4 | 0 | 0 | 0 | 0 |
| 21 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 6 |
| 0 | 0 | 0 | 0 | 34 | 0 |
| 32 | 37 | 0 | 0 | 0 | 0 |
| 20 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 8 | 1 |
G:=sub<GL(6,GF(41))| [1,16,0,0,0,0,5,40,0,0,0,0,0,0,1,36,0,0,0,0,25,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,4,32,0,0,0,0,0,0,1,0,0,0,0,0,25,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,21,0,0,0,0,4,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,34,0,0,0,0,6,0],[32,20,0,0,0,0,37,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,8,0,0,0,0,0,1] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 4Q | 4R | 4S | 4T | 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 | 4 | 4 | 4 | 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 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D4⋊2D5 | D4⋊2D5 | D5×C4○D4 |
| kernel | C4⋊C4.178D10 | C23.11D10 | C23.D10 | Dic5⋊3Q8 | C4.Dic10 | C2×C4×Dic5 | C20.48D4 | D4×Dic5 | C23.18D10 | C20.17D4 | C5×C4⋊D4 | C4⋊D4 | Dic5 | C20 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 6 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4._{178}D_{10} % in TeX
G:=Group("C4:C4.178D10"); // GroupNames label
G:=SmallGroup(320,1272);
// by ID
G=gap.SmallGroup(320,1272);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,794,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2*b^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀