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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 [×7], C2 [×4], C4 [×4], C4 [×6], C22 [×7], C22 [×16], C5, C2×C4 [×6], C2×C4 [×24], C23, C23 [×10], D5 [×4], C10 [×7], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×11], C24, Dic5 [×4], C20 [×4], C20 [×2], D10 [×4], D10 [×12], C2×C10 [×7], C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4, C23×C4, C4×D5 [×8], C2×Dic5 [×2], C2×Dic5 [×8], C2×C20 [×6], C2×C20 [×6], C22×D5 [×6], C22×D5 [×4], C22×C10, C23.7Q8, C4⋊Dic5 [×2], D10⋊C4 [×4], C5×C4⋊C4 [×2], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, C10.10C42 [×2], C2×C4⋊Dic5, C2×D10⋊C4 [×2], C10×C4⋊C4, D5×C22×C4, D104(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, D5, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], D10 [×3], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.7Q8, D10⋊C4 [×4], 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 31)(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(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 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 80)(59 79)(60 78)(61 83)(62 82)(63 81)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 84)(91 112)(92 111)(93 120)(94 119)(95 118)(96 117)(97 116)(98 115)(99 114)(100 113)(101 128)(102 127)(103 126)(104 125)(105 124)(106 123)(107 122)(108 121)(109 130)(110 129)(131 157)(132 156)(133 155)(134 154)(135 153)(136 152)(137 151)(138 160)(139 159)(140 158)
(1 130 30 112)(2 121 21 113)(3 122 22 114)(4 123 23 115)(5 124 24 116)(6 125 25 117)(7 126 26 118)(8 127 27 119)(9 128 28 120)(10 129 29 111)(11 65 157 57)(12 66 158 58)(13 67 159 59)(14 68 160 60)(15 69 151 51)(16 70 152 52)(17 61 153 53)(18 62 154 54)(19 63 155 55)(20 64 156 56)(31 109 49 91)(32 110 50 92)(33 101 41 93)(34 102 42 94)(35 103 43 95)(36 104 44 96)(37 105 45 97)(38 106 46 98)(39 107 47 99)(40 108 48 100)(71 149 89 131)(72 150 90 132)(73 141 81 133)(74 142 82 134)(75 143 83 135)(76 144 84 136)(77 145 85 137)(78 146 86 138)(79 147 87 139)(80 148 88 140)
(1 72 32 52)(2 73 33 53)(3 74 34 54)(4 75 35 55)(5 76 36 56)(6 77 37 57)(7 78 38 58)(8 79 39 59)(9 80 40 60)(10 71 31 51)(11 117 145 97)(12 118 146 98)(13 119 147 99)(14 120 148 100)(15 111 149 91)(16 112 150 92)(17 113 141 93)(18 114 142 94)(19 115 143 95)(20 116 144 96)(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)(101 153 121 133)(102 154 122 134)(103 155 123 135)(104 156 124 136)(105 157 125 137)(106 158 126 138)(107 159 127 139)(108 160 128 140)(109 151 129 131)(110 152 130 132)

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

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

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

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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