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G = D103(C4⋊C4)  order 320 = 26·5

2nd semidirect product of D10 and C4⋊C4 acting via C4⋊C4/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2.6(C4×D20), D103(C4⋊C4), C10.33(C4×D4), D10⋊C45C4, C10.3C22≀C2, (C2×C4).112D20, (C2×C20).234D4, C22.62(D4×D5), C22.17(Q8×D5), C2.C428D5, (C22×C4).17D10, C22.25(C2×D20), (C22×D5).13Q8, C2.2(C22⋊D20), C52(C23.8Q8), C2.4(D10⋊Q8), C2.2(D102Q8), (C2×Dic5).187D4, (C22×D5).104D4, C10.25(C22⋊Q8), C10.10C422C2, C2.8(Dic54D4), C22.36(C4○D20), (C23×D5).97C22, C23.258(C22×D5), C2.4(D10.12D4), C22.37(D42D5), (C22×C20).333C22, (C22×C10).293C23, C10.9(C22.D4), (C22×Dic5).16C22, (C2×C4)⋊3(C4×D5), C2.8(D5×C4⋊C4), (C2×C20)⋊18(C2×C4), C10.29(C2×C4⋊C4), (C2×C4⋊Dic5)⋊1C2, C22.91(C2×C4×D5), (C2×Dic5)⋊5(C2×C4), (C2×C10).68(C2×Q8), (D5×C22×C4).15C2, (C2×C10).202(C2×D4), (C2×D10⋊C4).5C2, (C2×C10.D4)⋊31C2, (C22×D5).67(C2×C4), (C2×C10).132(C4○D4), (C2×C10).153(C22×C4), (C5×C2.C42)⋊15C2, SmallGroup(320,295)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D103(C4⋊C4)
C1C5C10C2×C10C22×C10C23×D5C2×D10⋊C4 — D103(C4⋊C4)
C5C2×C10 — D103(C4⋊C4)
C1C23C2.C42

Generators and relations for D103(C4⋊C4)
 G = < a,b,c,d | a10=b2=c4=d4=1, bab=cac-1=a-1, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 910 in 234 conjugacy classes, 77 normal (51 characteristic)
C1, C2 [×7], C2 [×4], C4 [×10], C22 [×7], C22 [×16], C5, C2×C4 [×4], C2×C4 [×26], C23, C23 [×10], D5 [×4], C10 [×7], C22⋊C4 [×6], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×9], C24, Dic5 [×5], C20 [×5], D10 [×4], D10 [×12], C2×C10 [×7], C2.C42, C2.C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C4×D5 [×8], C2×Dic5 [×4], C2×Dic5 [×7], C2×C20 [×4], C2×C20 [×7], C22×D5 [×6], C22×D5 [×4], C22×C10, C23.8Q8, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×4], D10⋊C4 [×2], C2×C4×D5 [×6], C22×Dic5 [×3], C22×C20 [×3], C23×D5, C10.10C42, C5×C2.C42, C2×C10.D4, C2×C4⋊Dic5, C2×D10⋊C4 [×2], D5×C22×C4, D103(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], D10 [×3], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C4×D5 [×2], D20 [×2], C22×D5, C23.8Q8, C2×C4×D5, C2×D20, C4○D20, D4×D5 [×2], D42D5, Q8×D5, C4×D20, Dic54D4, C22⋊D20, D10.12D4, D5×C4⋊C4, D10⋊Q8, D102Q8, D103(C4⋊C4)

Smallest permutation representation of D103(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 35)(2 34)(3 33)(4 32)(5 31)(6 40)(7 39)(8 38)(9 37)(10 36)(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 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 90)(70 89)(91 115)(92 114)(93 113)(94 112)(95 111)(96 120)(97 119)(98 118)(99 117)(100 116)(101 123)(102 122)(103 121)(104 130)(105 129)(106 128)(107 127)(108 126)(109 125)(110 124)(131 155)(132 154)(133 153)(134 152)(135 151)(136 160)(137 159)(138 158)(139 157)(140 156)
(1 105 30 91)(2 104 21 100)(3 103 22 99)(4 102 23 98)(5 101 24 97)(6 110 25 96)(7 109 26 95)(8 108 27 94)(9 107 28 93)(10 106 29 92)(11 90 151 76)(12 89 152 75)(13 88 153 74)(14 87 154 73)(15 86 155 72)(16 85 156 71)(17 84 157 80)(18 83 158 79)(19 82 159 78)(20 81 160 77)(31 130 45 116)(32 129 46 115)(33 128 47 114)(34 127 48 113)(35 126 49 112)(36 125 50 111)(37 124 41 120)(38 123 42 119)(39 122 43 118)(40 121 44 117)(51 150 65 136)(52 149 66 135)(53 148 67 134)(54 147 68 133)(55 146 69 132)(56 145 70 131)(57 144 61 140)(58 143 62 139)(59 142 63 138)(60 141 64 137)
(1 76 36 56)(2 77 37 57)(3 78 38 58)(4 79 39 59)(5 80 40 60)(6 71 31 51)(7 72 32 52)(8 73 33 53)(9 74 34 54)(10 75 35 55)(11 111 145 91)(12 112 146 92)(13 113 147 93)(14 114 148 94)(15 115 149 95)(16 116 150 96)(17 117 141 97)(18 118 142 98)(19 119 143 99)(20 120 144 100)(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)(101 157 121 137)(102 158 122 138)(103 159 123 139)(104 160 124 140)(105 151 125 131)(106 152 126 132)(107 153 127 133)(108 154 128 134)(109 155 129 135)(110 156 130 136)

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

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

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

68 conjugacy classes

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

68 irreducible representations

dim111111112222222222444
type++++++++++-++++--
imageC1C2C2C2C2C2C2C4D4D4D4Q8D5C4○D4D10C4×D5D20C4○D20D4×D5D42D5Q8×D5
kernelD103(C4⋊C4)C10.10C42C5×C2.C42C2×C10.D4C2×C4⋊Dic5C2×D10⋊C4D5×C22×C4D10⋊C4C2×Dic5C2×C20C22×D5C22×D5C2.C42C2×C10C22×C4C2×C4C2×C4C22C22C22C22
# reps111112182222246888422

Matrix representation of D103(C4⋊C4) in GL6(𝔽41)

34340000
600000
001000
000100
0000400
0000040
,
34330000
670000
0040000
0004000
000010
00002240
,
770000
40340000
00403900
001100
00001436
00002327
,
3200000
0320000
0032000
009900
0000320
000079

G:=sub<GL(6,GF(41))| [34,6,0,0,0,0,34,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,6,0,0,0,0,33,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,22,0,0,0,0,0,40],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,14,23,0,0,0,0,36,27],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,9,0,0,0,0,0,9,0,0,0,0,0,0,32,7,0,0,0,0,0,9] >;

D103(C4⋊C4) in GAP, Magma, Sage, TeX

D_{10}\rtimes_3(C_4\rtimes C_4)
% in TeX

G:=Group("D10:3(C4:C4)");
// GroupNames label

G:=SmallGroup(320,295);
// by ID

G=gap.SmallGroup(320,295);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^4=d^4=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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