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G = D104(C4⋊C4)  order 320 = 26·5

3rd semidirect product of D10 and C4⋊C4 acting via C4⋊C4/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D104(C4⋊C4), C204(C22⋊C4), (C2×C4).143D20, (C2×C20).141D4, C43(D10⋊C4), C22.26(Q8×D5), C2.4(C202D4), C2.4(C4⋊D20), (C22×D5).88D4, C22.110(D4×D5), C22.47(C2×D20), (C22×D5).15Q8, C10.53(C4⋊D4), C54(C23.7Q8), C2.5(D102Q8), C2.3(D103Q8), (C22×C4).334D10, C10.72(C22⋊Q8), C23.295(C22×D5), C10.10C4219C2, C10.56(C42⋊C2), C22.59(D42D5), (C22×C10).351C23, (C22×C20).143C22, C22.27(Q82D5), (C23×D5).101C22, (C22×Dic5).58C22, (C2×C4×D5)⋊7C4, (C2×C4⋊C4)⋊5D5, (C10×C4⋊C4)⋊5C2, C2.21(D5×C4⋊C4), C10.43(C2×C4⋊C4), (D5×C22×C4).1C2, (C2×C4⋊Dic5)⋊30C2, (C2×C4).154(C4×D5), (C2×C10).83(C2×Q8), C22.136(C2×C4×D5), (C2×C20).258(C2×C4), (C2×C10).451(C2×D4), C10.83(C2×C22⋊C4), C22.66(C2×C5⋊D4), C2.13(C4⋊C47D5), C2.15(C2×D10⋊C4), (C2×C4).184(C5⋊D4), (C2×D10⋊C4).13C2, (C2×C10).156(C4○D4), (C2×C10).220(C22×C4), (C2×Dic5).151(C2×C4), (C22×D5).105(C2×C4), SmallGroup(320,614)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D104(C4⋊C4)
C1C5C10C2×C10C22×C10C23×D5D5×C22×C4 — D104(C4⋊C4)
C5C2×C10 — D104(C4⋊C4)
C1C23C2×C4⋊C4

Generators and relations for D104(C4⋊C4)
 G = < a,b,c,d | a10=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 894 in 234 conjugacy classes, 87 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C23×C4, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23.7Q8, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C23×D5, C10.10C42, C2×C4⋊Dic5, C2×D10⋊C4, C10×C4⋊C4, D5×C22×C4, D104(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D10, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C4×D5, D20, C5⋊D4, C22×D5, C23.7Q8, D10⋊C4, C2×C4×D5, C2×D20, D4×D5, D42D5, Q8×D5, Q82D5, C2×C5⋊D4, D5×C4⋊C4, C4⋊C47D5, C4⋊D20, D102Q8, C2×D10⋊C4, C202D4, D103Q8, D104(C4⋊C4)

Smallest permutation representation of D104(C4⋊C4)
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 33)(2 32)(3 31)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 149)(12 148)(13 147)(14 146)(15 145)(16 144)(17 143)(18 142)(19 141)(20 150)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 42)(28 41)(29 50)(30 49)(51 71)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(61 83)(62 82)(63 81)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 84)(91 116)(92 115)(93 114)(94 113)(95 112)(96 111)(97 120)(98 119)(99 118)(100 117)(101 128)(102 127)(103 126)(104 125)(105 124)(106 123)(107 122)(108 121)(109 130)(110 129)(131 151)(132 160)(133 159)(134 158)(135 157)(136 156)(137 155)(138 154)(139 153)(140 152)
(1 130 30 114)(2 121 21 115)(3 122 22 116)(4 123 23 117)(5 124 24 118)(6 125 25 119)(7 126 26 120)(8 127 27 111)(9 128 28 112)(10 129 29 113)(11 65 159 59)(12 66 160 60)(13 67 151 51)(14 68 152 52)(15 69 153 53)(16 70 154 54)(17 61 155 55)(18 62 156 56)(19 63 157 57)(20 64 158 58)(31 107 47 91)(32 108 48 92)(33 109 49 93)(34 110 50 94)(35 101 41 95)(36 102 42 96)(37 103 43 97)(38 104 44 98)(39 105 45 99)(40 106 46 100)(71 147 87 131)(72 148 88 132)(73 149 89 133)(74 150 90 134)(75 141 81 135)(76 142 82 136)(77 143 83 137)(78 144 84 138)(79 145 85 139)(80 146 86 140)
(1 74 34 54)(2 75 35 55)(3 76 36 56)(4 77 37 57)(5 78 38 58)(6 79 39 59)(7 80 40 60)(8 71 31 51)(9 72 32 52)(10 73 33 53)(11 119 145 99)(12 120 146 100)(13 111 147 91)(14 112 148 92)(15 113 149 93)(16 114 150 94)(17 115 141 95)(18 116 142 96)(19 117 143 97)(20 118 144 98)(21 81 41 61)(22 82 42 62)(23 83 43 63)(24 84 44 64)(25 85 45 65)(26 86 46 66)(27 87 47 67)(28 88 48 68)(29 89 49 69)(30 90 50 70)(101 155 121 135)(102 156 122 136)(103 157 123 137)(104 158 124 138)(105 159 125 139)(106 160 126 140)(107 151 127 131)(108 152 128 132)(109 153 129 133)(110 154 130 134)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B10A···10N20A···20X
order12···2222244444444444444445510···1020···20
size11···110101010222244441010101020202020222···24···4

68 irreducible representations

dim11111112222222224444
type++++++++-++++--+
imageC1C2C2C2C2C2C4D4D4Q8D5C4○D4D10C4×D5D20C5⋊D4D4×D5D42D5Q8×D5Q82D5
kernelD104(C4⋊C4)C10.10C42C2×C4⋊Dic5C2×D10⋊C4C10×C4⋊C4D5×C22×C4C2×C4×D5C2×C20C22×D5C22×D5C2×C4⋊C4C2×C10C22×C4C2×C4C2×C4C2×C4C22C22C22C22
# reps12121184222468882222

Matrix representation of D104(C4⋊C4) in GL6(𝔽41)

34340000
710000
007700
00344000
0000400
0000040
,
770000
40340000
00343400
001700
000010
0000940
,
100000
010000
001000
000100
0000320
000019
,
30320000
9110000
0032000
0003200
0000219
0000120

G:=sub<GL(6,GF(41))| [34,7,0,0,0,0,34,1,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,34,1,0,0,0,0,34,7,0,0,0,0,0,0,1,9,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,1,0,0,0,0,0,9],[30,9,0,0,0,0,32,11,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,21,1,0,0,0,0,9,20] >;

D104(C4⋊C4) in GAP, Magma, Sage, TeX

D_{10}\rtimes_4(C_4\rtimes C_4)
% in TeX

G:=Group("D10:4(C4:C4)");
// GroupNames label

G:=SmallGroup(320,614);
// by ID

G=gap.SmallGroup(320,614);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,422,387,58,12550]);
// Polycyclic

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

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