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), C20.17D4⋊15C2, C4.67(D4⋊2D5), C20.48D4⋊31C2, (C2×C10).144C24, (C2×C20).501C23, (C22×C4).367D10, C23.11(C22×D5), Dic5.72(C4○D4), (D4×C10).118C22, C23.11D10⋊4C2, C23.D10⋊14C2, C22.5(D4⋊2D5), 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
Generators and relations for C4⋊C4.178D10
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 >
Subgroups: 670 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C2×C42, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, Dic10, C2×Dic5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C23.36C23, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×Dic5, C22×Dic5, C22×C20, D4×C10, D4×C10, C23.11D10, C23.D10, Dic5⋊3Q8, C4.Dic10, C2×C4×Dic5, C20.48D4, D4×Dic5, D4×Dic5, C23.18D10, C20.17D4, C5×C4⋊D4, C4⋊C4.178D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, C22×D5, C23.36C23, D4⋊2D5, C23×D5, C2×D4⋊2D5, D5×C4○D4, C4⋊C4.178D10
►On 160 points
Generators in S160
(1 27 33 46)(2 28 34 47)(3 29 35 48)(4 30 36 49)(5 21 37 50)(6 22 38 41)(7 23 39 42)(8 24 40 43)(9 25 31 44)(10 26 32 45)(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)(81 127 139 108)(82 128 140 109)(83 129 131 110)(84 130 132 101)(85 121 133 102)(86 122 134 103)(87 123 135 104)(88 124 136 105)(89 125 137 106)(90 126 138 107)
(1 116 111 6)(2 7 112 117)(3 118 113 8)(4 9 114 119)(5 120 115 10)(11 59 28 42)(12 43 29 60)(13 51 30 44)(14 45 21 52)(15 53 22 46)(16 47 23 54)(17 55 24 48)(18 49 25 56)(19 57 26 50)(20 41 27 58)(31 76 71 36)(32 37 72 77)(33 78 73 38)(34 39 74 79)(35 80 75 40)(61 66 108 103)(62 104 109 67)(63 68 110 105)(64 106 101 69)(65 70 102 107)(81 134 96 143)(82 144 97 135)(83 136 98 145)(84 146 99 137)(85 138 100 147)(86 148 91 139)(87 140 92 149)(88 150 93 131)(89 132 94 141)(90 142 95 133)(121 126 152 157)(122 158 153 127)(123 128 154 159)(124 160 155 129)(125 130 156 151)
(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 73 106)(2 155 74 105)(3 154 75 104)(4 153 76 103)(5 152 77 102)(6 151 78 101)(7 160 79 110)(8 159 80 109)(9 158 71 108)(10 157 72 107)(11 136 47 93)(12 135 48 92)(13 134 49 91)(14 133 50 100)(15 132 41 99)(16 131 42 98)(17 140 43 97)(18 139 44 96)(19 138 45 95)(20 137 46 94)(21 142 57 85)(22 141 58 84)(23 150 59 83)(24 149 60 82)(25 148 51 81)(26 147 52 90)(27 146 53 89)(28 145 54 88)(29 144 55 87)(30 143 56 86)(31 61 119 127)(32 70 120 126)(33 69 111 125)(34 68 112 124)(35 67 113 123)(36 66 114 122)(37 65 115 121)(38 64 116 130)(39 63 117 129)(40 62 118 128)
G:=sub<Sym(160)| (1,27,33,46)(2,28,34,47)(3,29,35,48)(4,30,36,49)(5,21,37,50)(6,22,38,41)(7,23,39,42)(8,24,40,43)(9,25,31,44)(10,26,32,45)(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)(81,127,139,108)(82,128,140,109)(83,129,131,110)(84,130,132,101)(85,121,133,102)(86,122,134,103)(87,123,135,104)(88,124,136,105)(89,125,137,106)(90,126,138,107), (1,116,111,6)(2,7,112,117)(3,118,113,8)(4,9,114,119)(5,120,115,10)(11,59,28,42)(12,43,29,60)(13,51,30,44)(14,45,21,52)(15,53,22,46)(16,47,23,54)(17,55,24,48)(18,49,25,56)(19,57,26,50)(20,41,27,58)(31,76,71,36)(32,37,72,77)(33,78,73,38)(34,39,74,79)(35,80,75,40)(61,66,108,103)(62,104,109,67)(63,68,110,105)(64,106,101,69)(65,70,102,107)(81,134,96,143)(82,144,97,135)(83,136,98,145)(84,146,99,137)(85,138,100,147)(86,148,91,139)(87,140,92,149)(88,150,93,131)(89,132,94,141)(90,142,95,133)(121,126,152,157)(122,158,153,127)(123,128,154,159)(124,160,155,129)(125,130,156,151), (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), (1,156,73,106)(2,155,74,105)(3,154,75,104)(4,153,76,103)(5,152,77,102)(6,151,78,101)(7,160,79,110)(8,159,80,109)(9,158,71,108)(10,157,72,107)(11,136,47,93)(12,135,48,92)(13,134,49,91)(14,133,50,100)(15,132,41,99)(16,131,42,98)(17,140,43,97)(18,139,44,96)(19,138,45,95)(20,137,46,94)(21,142,57,85)(22,141,58,84)(23,150,59,83)(24,149,60,82)(25,148,51,81)(26,147,52,90)(27,146,53,89)(28,145,54,88)(29,144,55,87)(30,143,56,86)(31,61,119,127)(32,70,120,126)(33,69,111,125)(34,68,112,124)(35,67,113,123)(36,66,114,122)(37,65,115,121)(38,64,116,130)(39,63,117,129)(40,62,118,128)>;
G:=Group( (1,27,33,46)(2,28,34,47)(3,29,35,48)(4,30,36,49)(5,21,37,50)(6,22,38,41)(7,23,39,42)(8,24,40,43)(9,25,31,44)(10,26,32,45)(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)(81,127,139,108)(82,128,140,109)(83,129,131,110)(84,130,132,101)(85,121,133,102)(86,122,134,103)(87,123,135,104)(88,124,136,105)(89,125,137,106)(90,126,138,107), (1,116,111,6)(2,7,112,117)(3,118,113,8)(4,9,114,119)(5,120,115,10)(11,59,28,42)(12,43,29,60)(13,51,30,44)(14,45,21,52)(15,53,22,46)(16,47,23,54)(17,55,24,48)(18,49,25,56)(19,57,26,50)(20,41,27,58)(31,76,71,36)(32,37,72,77)(33,78,73,38)(34,39,74,79)(35,80,75,40)(61,66,108,103)(62,104,109,67)(63,68,110,105)(64,106,101,69)(65,70,102,107)(81,134,96,143)(82,144,97,135)(83,136,98,145)(84,146,99,137)(85,138,100,147)(86,148,91,139)(87,140,92,149)(88,150,93,131)(89,132,94,141)(90,142,95,133)(121,126,152,157)(122,158,153,127)(123,128,154,159)(124,160,155,129)(125,130,156,151), (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), (1,156,73,106)(2,155,74,105)(3,154,75,104)(4,153,76,103)(5,152,77,102)(6,151,78,101)(7,160,79,110)(8,159,80,109)(9,158,71,108)(10,157,72,107)(11,136,47,93)(12,135,48,92)(13,134,49,91)(14,133,50,100)(15,132,41,99)(16,131,42,98)(17,140,43,97)(18,139,44,96)(19,138,45,95)(20,137,46,94)(21,142,57,85)(22,141,58,84)(23,150,59,83)(24,149,60,82)(25,148,51,81)(26,147,52,90)(27,146,53,89)(28,145,54,88)(29,144,55,87)(30,143,56,86)(31,61,119,127)(32,70,120,126)(33,69,111,125)(34,68,112,124)(35,67,113,123)(36,66,114,122)(37,65,115,121)(38,64,116,130)(39,63,117,129)(40,62,118,128) );
G=PermutationGroup([[(1,27,33,46),(2,28,34,47),(3,29,35,48),(4,30,36,49),(5,21,37,50),(6,22,38,41),(7,23,39,42),(8,24,40,43),(9,25,31,44),(10,26,32,45),(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),(81,127,139,108),(82,128,140,109),(83,129,131,110),(84,130,132,101),(85,121,133,102),(86,122,134,103),(87,123,135,104),(88,124,136,105),(89,125,137,106),(90,126,138,107)], [(1,116,111,6),(2,7,112,117),(3,118,113,8),(4,9,114,119),(5,120,115,10),(11,59,28,42),(12,43,29,60),(13,51,30,44),(14,45,21,52),(15,53,22,46),(16,47,23,54),(17,55,24,48),(18,49,25,56),(19,57,26,50),(20,41,27,58),(31,76,71,36),(32,37,72,77),(33,78,73,38),(34,39,74,79),(35,80,75,40),(61,66,108,103),(62,104,109,67),(63,68,110,105),(64,106,101,69),(65,70,102,107),(81,134,96,143),(82,144,97,135),(83,136,98,145),(84,146,99,137),(85,138,100,147),(86,148,91,139),(87,140,92,149),(88,150,93,131),(89,132,94,141),(90,142,95,133),(121,126,152,157),(122,158,153,127),(123,128,154,159),(124,160,155,129),(125,130,156,151)], [(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)], [(1,156,73,106),(2,155,74,105),(3,154,75,104),(4,153,76,103),(5,152,77,102),(6,151,78,101),(7,160,79,110),(8,159,80,109),(9,158,71,108),(10,157,72,107),(11,136,47,93),(12,135,48,92),(13,134,49,91),(14,133,50,100),(15,132,41,99),(16,131,42,98),(17,140,43,97),(18,139,44,96),(19,138,45,95),(20,137,46,94),(21,142,57,85),(22,141,58,84),(23,150,59,83),(24,149,60,82),(25,148,51,81),(26,147,52,90),(27,146,53,89),(28,145,54,88),(29,144,55,87),(30,143,56,86),(31,61,119,127),(32,70,120,126),(33,69,111,125),(34,68,112,124),(35,67,113,123),(36,66,114,122),(37,65,115,121),(38,64,116,130),(39,63,117,129),(40,62,118,128)]])
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 |
Matrix representation of C4⋊C4.178D10 ►in 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] >;
C4⋊C4.178D10 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