metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.228D10, (C2×C20).284D4, D20⋊6C4⋊26C2, C4.87(C4○D20), C20.55D4⋊5C2, C20⋊7D4.10C2, C20.Q8⋊25C2, C10.67(C2×SD16), (C2×C10).41SD16, (C22×C4).95D10, C22.8(Q8⋊D5), C20.175(C4○D4), C10.84(C8⋊C22), (C2×C20).321C23, (C2×D20).95C22, (C22×C10).186D4, C5⋊4(C23.46D4), C23.78(C5⋊D4), C2.6(D4.D10), C4⋊Dic5.131C22, (C22×C20).136C22, C2.9(C23.23D10), C10.59(C22.D4), (C2×C4⋊C4)⋊4D5, (C10×C4⋊C4)⋊4C2, C2.5(C2×Q8⋊D5), (C2×C10).441(C2×D4), (C2×C4).32(C5⋊D4), (C5×C4⋊C4).259C22, (C2×C5⋊2C8).82C22, (C2×C4).421(C22×D5), C22.131(C2×C5⋊D4), SmallGroup(320,595)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.228D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2b2c-1 >
Subgroups: 462 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C4.Q8, C2×C4⋊C4, C4⋊D4, C5⋊2C8, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C23.46D4, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×D20, C2×C5⋊D4, C22×C20, C22×C20, C20.Q8, D20⋊6C4, C20.55D4, C20⋊7D4, C10×C4⋊C4, C4⋊C4.228D10
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C22.D4, C2×SD16, C8⋊C22, C5⋊D4, C22×D5, C23.46D4, Q8⋊D5, C4○D20, C2×C5⋊D4, C23.23D10, D4.D10, C2×Q8⋊D5, C4⋊C4.228D10
►On 160 points
Generators in S160
(1 105 123 70)(2 106 124 61)(3 107 125 62)(4 108 126 63)(5 109 127 64)(6 110 128 65)(7 101 129 66)(8 102 130 67)(9 103 121 68)(10 104 122 69)(11 31 85 142)(12 32 86 143)(13 33 87 144)(14 34 88 145)(15 35 89 146)(16 36 90 147)(17 37 81 148)(18 38 82 149)(19 39 83 150)(20 40 84 141)(21 59 132 94)(22 60 133 95)(23 51 134 96)(24 52 135 97)(25 53 136 98)(26 54 137 99)(27 55 138 100)(28 56 139 91)(29 57 140 92)(30 58 131 93)(41 114 152 79)(42 115 153 80)(43 116 154 71)(44 117 155 72)(45 118 156 73)(46 119 157 74)(47 120 158 75)(48 111 159 76)(49 112 160 77)(50 113 151 78)
(1 75 60 37)(2 76 51 38)(3 77 52 39)(4 78 53 40)(5 79 54 31)(6 80 55 32)(7 71 56 33)(8 72 57 34)(9 73 58 35)(10 74 59 36)(11 109 152 137)(12 110 153 138)(13 101 154 139)(14 102 155 140)(15 103 156 131)(16 104 157 132)(17 105 158 133)(18 106 159 134)(19 107 160 135)(20 108 151 136)(21 90 69 46)(22 81 70 47)(23 82 61 48)(24 83 62 49)(25 84 63 50)(26 85 64 41)(27 86 65 42)(28 87 66 43)(29 88 67 44)(30 89 68 45)(91 144 129 116)(92 145 130 117)(93 146 121 118)(94 147 122 119)(95 148 123 120)(96 149 124 111)(97 150 125 112)(98 141 126 113)(99 142 127 114)(100 143 128 115)
(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 10 95 94)(2 93 96 9)(3 8 97 92)(4 91 98 7)(5 6 99 100)(11 32 41 115)(12 114 42 31)(13 40 43 113)(14 112 44 39)(15 38 45 111)(16 120 46 37)(17 36 47 119)(18 118 48 35)(19 34 49 117)(20 116 50 33)(21 70 104 133)(22 132 105 69)(23 68 106 131)(24 140 107 67)(25 66 108 139)(26 138 109 65)(27 64 110 137)(28 136 101 63)(29 62 102 135)(30 134 103 61)(51 121 124 58)(52 57 125 130)(53 129 126 56)(54 55 127 128)(59 123 122 60)(71 151 144 84)(72 83 145 160)(73 159 146 82)(74 81 147 158)(75 157 148 90)(76 89 149 156)(77 155 150 88)(78 87 141 154)(79 153 142 86)(80 85 143 152)
G:=sub<Sym(160)| (1,105,123,70)(2,106,124,61)(3,107,125,62)(4,108,126,63)(5,109,127,64)(6,110,128,65)(7,101,129,66)(8,102,130,67)(9,103,121,68)(10,104,122,69)(11,31,85,142)(12,32,86,143)(13,33,87,144)(14,34,88,145)(15,35,89,146)(16,36,90,147)(17,37,81,148)(18,38,82,149)(19,39,83,150)(20,40,84,141)(21,59,132,94)(22,60,133,95)(23,51,134,96)(24,52,135,97)(25,53,136,98)(26,54,137,99)(27,55,138,100)(28,56,139,91)(29,57,140,92)(30,58,131,93)(41,114,152,79)(42,115,153,80)(43,116,154,71)(44,117,155,72)(45,118,156,73)(46,119,157,74)(47,120,158,75)(48,111,159,76)(49,112,160,77)(50,113,151,78), (1,75,60,37)(2,76,51,38)(3,77,52,39)(4,78,53,40)(5,79,54,31)(6,80,55,32)(7,71,56,33)(8,72,57,34)(9,73,58,35)(10,74,59,36)(11,109,152,137)(12,110,153,138)(13,101,154,139)(14,102,155,140)(15,103,156,131)(16,104,157,132)(17,105,158,133)(18,106,159,134)(19,107,160,135)(20,108,151,136)(21,90,69,46)(22,81,70,47)(23,82,61,48)(24,83,62,49)(25,84,63,50)(26,85,64,41)(27,86,65,42)(28,87,66,43)(29,88,67,44)(30,89,68,45)(91,144,129,116)(92,145,130,117)(93,146,121,118)(94,147,122,119)(95,148,123,120)(96,149,124,111)(97,150,125,112)(98,141,126,113)(99,142,127,114)(100,143,128,115), (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), (1,10,95,94)(2,93,96,9)(3,8,97,92)(4,91,98,7)(5,6,99,100)(11,32,41,115)(12,114,42,31)(13,40,43,113)(14,112,44,39)(15,38,45,111)(16,120,46,37)(17,36,47,119)(18,118,48,35)(19,34,49,117)(20,116,50,33)(21,70,104,133)(22,132,105,69)(23,68,106,131)(24,140,107,67)(25,66,108,139)(26,138,109,65)(27,64,110,137)(28,136,101,63)(29,62,102,135)(30,134,103,61)(51,121,124,58)(52,57,125,130)(53,129,126,56)(54,55,127,128)(59,123,122,60)(71,151,144,84)(72,83,145,160)(73,159,146,82)(74,81,147,158)(75,157,148,90)(76,89,149,156)(77,155,150,88)(78,87,141,154)(79,153,142,86)(80,85,143,152)>;
G:=Group( (1,105,123,70)(2,106,124,61)(3,107,125,62)(4,108,126,63)(5,109,127,64)(6,110,128,65)(7,101,129,66)(8,102,130,67)(9,103,121,68)(10,104,122,69)(11,31,85,142)(12,32,86,143)(13,33,87,144)(14,34,88,145)(15,35,89,146)(16,36,90,147)(17,37,81,148)(18,38,82,149)(19,39,83,150)(20,40,84,141)(21,59,132,94)(22,60,133,95)(23,51,134,96)(24,52,135,97)(25,53,136,98)(26,54,137,99)(27,55,138,100)(28,56,139,91)(29,57,140,92)(30,58,131,93)(41,114,152,79)(42,115,153,80)(43,116,154,71)(44,117,155,72)(45,118,156,73)(46,119,157,74)(47,120,158,75)(48,111,159,76)(49,112,160,77)(50,113,151,78), (1,75,60,37)(2,76,51,38)(3,77,52,39)(4,78,53,40)(5,79,54,31)(6,80,55,32)(7,71,56,33)(8,72,57,34)(9,73,58,35)(10,74,59,36)(11,109,152,137)(12,110,153,138)(13,101,154,139)(14,102,155,140)(15,103,156,131)(16,104,157,132)(17,105,158,133)(18,106,159,134)(19,107,160,135)(20,108,151,136)(21,90,69,46)(22,81,70,47)(23,82,61,48)(24,83,62,49)(25,84,63,50)(26,85,64,41)(27,86,65,42)(28,87,66,43)(29,88,67,44)(30,89,68,45)(91,144,129,116)(92,145,130,117)(93,146,121,118)(94,147,122,119)(95,148,123,120)(96,149,124,111)(97,150,125,112)(98,141,126,113)(99,142,127,114)(100,143,128,115), (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), (1,10,95,94)(2,93,96,9)(3,8,97,92)(4,91,98,7)(5,6,99,100)(11,32,41,115)(12,114,42,31)(13,40,43,113)(14,112,44,39)(15,38,45,111)(16,120,46,37)(17,36,47,119)(18,118,48,35)(19,34,49,117)(20,116,50,33)(21,70,104,133)(22,132,105,69)(23,68,106,131)(24,140,107,67)(25,66,108,139)(26,138,109,65)(27,64,110,137)(28,136,101,63)(29,62,102,135)(30,134,103,61)(51,121,124,58)(52,57,125,130)(53,129,126,56)(54,55,127,128)(59,123,122,60)(71,151,144,84)(72,83,145,160)(73,159,146,82)(74,81,147,158)(75,157,148,90)(76,89,149,156)(77,155,150,88)(78,87,141,154)(79,153,142,86)(80,85,143,152) );
G=PermutationGroup([[(1,105,123,70),(2,106,124,61),(3,107,125,62),(4,108,126,63),(5,109,127,64),(6,110,128,65),(7,101,129,66),(8,102,130,67),(9,103,121,68),(10,104,122,69),(11,31,85,142),(12,32,86,143),(13,33,87,144),(14,34,88,145),(15,35,89,146),(16,36,90,147),(17,37,81,148),(18,38,82,149),(19,39,83,150),(20,40,84,141),(21,59,132,94),(22,60,133,95),(23,51,134,96),(24,52,135,97),(25,53,136,98),(26,54,137,99),(27,55,138,100),(28,56,139,91),(29,57,140,92),(30,58,131,93),(41,114,152,79),(42,115,153,80),(43,116,154,71),(44,117,155,72),(45,118,156,73),(46,119,157,74),(47,120,158,75),(48,111,159,76),(49,112,160,77),(50,113,151,78)], [(1,75,60,37),(2,76,51,38),(3,77,52,39),(4,78,53,40),(5,79,54,31),(6,80,55,32),(7,71,56,33),(8,72,57,34),(9,73,58,35),(10,74,59,36),(11,109,152,137),(12,110,153,138),(13,101,154,139),(14,102,155,140),(15,103,156,131),(16,104,157,132),(17,105,158,133),(18,106,159,134),(19,107,160,135),(20,108,151,136),(21,90,69,46),(22,81,70,47),(23,82,61,48),(24,83,62,49),(25,84,63,50),(26,85,64,41),(27,86,65,42),(28,87,66,43),(29,88,67,44),(30,89,68,45),(91,144,129,116),(92,145,130,117),(93,146,121,118),(94,147,122,119),(95,148,123,120),(96,149,124,111),(97,150,125,112),(98,141,126,113),(99,142,127,114),(100,143,128,115)], [(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)], [(1,10,95,94),(2,93,96,9),(3,8,97,92),(4,91,98,7),(5,6,99,100),(11,32,41,115),(12,114,42,31),(13,40,43,113),(14,112,44,39),(15,38,45,111),(16,120,46,37),(17,36,47,119),(18,118,48,35),(19,34,49,117),(20,116,50,33),(21,70,104,133),(22,132,105,69),(23,68,106,131),(24,140,107,67),(25,66,108,139),(26,138,109,65),(27,64,110,137),(28,136,101,63),(29,62,102,135),(30,134,103,61),(51,121,124,58),(52,57,125,130),(53,129,126,56),(54,55,127,128),(59,123,122,60),(71,151,144,84),(72,83,145,160),(73,159,146,82),(74,81,147,158),(75,157,148,90),(76,89,149,156),(77,155,150,88),(78,87,141,154),(79,153,142,86),(80,85,143,152)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 40 | 2 | 2 | 4 | ··· | 4 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | SD16 | D10 | D10 | C5⋊D4 | C5⋊D4 | C4○D20 | C8⋊C22 | Q8⋊D5 | D4.D10 |
| kernel | C4⋊C4.228D10 | C20.Q8 | D20⋊6C4 | C20.55D4 | C20⋊7D4 | C10×C4⋊C4 | C2×C20 | C22×C10 | C2×C4⋊C4 | C20 | C2×C10 | C4⋊C4 | C22×C4 | C2×C4 | C23 | C4 | C10 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of C4⋊C4.228D10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 26 | 26 |
| 0 | 0 | 26 | 15 |
| 35 | 26 | 0 | 0 |
| 38 | 20 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 21 | 26 | 0 | 0 |
| 24 | 20 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0],[32,0,0,0,0,32,0,0,0,0,26,26,0,0,26,15],[35,38,0,0,26,20,0,0,0,0,40,0,0,0,0,40],[21,24,0,0,26,20,0,0,0,0,40,0,0,0,0,1] >;
C4⋊C4.228D10 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{228}D_{10} % in TeX
G:=Group("C4:C4.228D10"); // GroupNames label
G:=SmallGroup(320,595);
// by ID
G=gap.SmallGroup(320,595);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,268,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2*b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*b^2*c^-1>;
// generators/relations