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G = D105(C4⋊C4)  order 320 = 26·5

4th semidirect product of D10 and C4⋊C4 acting via C4⋊C4/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D105(C4⋊C4), C10.98(C4×D4), D10⋊C49C4, (C2×C20).251D4, C10.41C22≀C2, C22.27(Q8×D5), C22.112(D4×D5), (C22×C4).42D10, (C22×D5).16Q8, C2.4(C23⋊D10), C55(C23.8Q8), C2.7(D10⋊Q8), C2.4(D103Q8), (C2×Dic5).233D4, (C22×D5).126D4, C10.49(C22⋊Q8), C2.19(D208C4), C22.58(C4○D20), C23.297(C22×D5), C10.10C4240C2, C2.6(D10.13D4), (C22×C20).347C22, (C22×C10).353C23, C22.29(Q82D5), (C23×D5).102C22, C10.51(C22.D4), (C22×Dic5).59C22, (C2×C4⋊C4)⋊7D5, (C2×C4)⋊4(C4×D5), (C10×C4⋊C4)⋊24C2, C2.22(D5×C4⋊C4), (C2×C20)⋊34(C2×C4), C10.44(C2×C4⋊C4), C2.13(C4×C5⋊D4), (C2×Dic5)⋊9(C2×C4), (C2×C10).84(C2×Q8), (D5×C22×C4).19C2, C22.138(C2×C4×D5), (C2×C10).334(C2×D4), C22.68(C2×C5⋊D4), (C2×C10.D4)⋊13C2, (C2×C4).169(C5⋊D4), (C22×D5).79(C2×C4), (C2×D10⋊C4).14C2, (C2×C10).190(C4○D4), (C2×C10).222(C22×C4), SmallGroup(320,616)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D105(C4⋊C4)
C1C5C10C2×C10C22×C10C23×D5C2×D10⋊C4 — D105(C4⋊C4)
C5C2×C10 — D105(C4⋊C4)
C1C23C2×C4⋊C4

Generators and relations for D105(C4⋊C4)
 G = < a,b,c,d | a10=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >

Subgroups: 894 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 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4, 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], D10⋊C4 [×4], D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C4×D5 [×6], C22×Dic5 [×3], C22×C20 [×3], C23×D5, C10.10C42 [×2], C2×C10.D4, C2×D10⋊C4 [×2], C10×C4⋊C4, D5×C22×C4, D105(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], C5⋊D4 [×2], C22×D5, C23.8Q8, C2×C4×D5, C4○D20, D4×D5 [×2], Q8×D5, Q82D5, C2×C5⋊D4, D5×C4⋊C4, D208C4, D10.13D4, D10⋊Q8, C4×C5⋊D4, C23⋊D10, D103Q8, D105(C4⋊C4)

Smallest permutation representation of D105(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 40)(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 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 80)(60 79)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 90)(70 89)(91 113)(92 112)(93 111)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)(101 123)(102 122)(103 121)(104 130)(105 129)(106 128)(107 127)(108 126)(109 125)(110 124)(131 153)(132 152)(133 151)(134 160)(135 159)(136 158)(137 157)(138 156)(139 155)(140 154)
(1 130 30 120)(2 121 21 111)(3 122 22 112)(4 123 23 113)(5 124 24 114)(6 125 25 115)(7 126 26 116)(8 127 27 117)(9 128 28 118)(10 129 29 119)(11 65 155 55)(12 66 156 56)(13 67 157 57)(14 68 158 58)(15 69 159 59)(16 70 160 60)(17 61 151 51)(18 62 152 52)(19 63 153 53)(20 64 154 54)(31 101 41 91)(32 102 42 92)(33 103 43 93)(34 104 44 94)(35 105 45 95)(36 106 46 96)(37 107 47 97)(38 108 48 98)(39 109 49 99)(40 110 50 100)(71 141 81 131)(72 142 82 132)(73 143 83 133)(74 144 84 134)(75 145 85 135)(76 146 86 136)(77 147 87 137)(78 148 88 138)(79 149 89 139)(80 150 90 140)
(1 80 40 60)(2 71 31 51)(3 72 32 52)(4 73 33 53)(5 74 34 54)(6 75 35 55)(7 76 36 56)(8 77 37 57)(9 78 38 58)(10 79 39 59)(11 115 145 95)(12 116 146 96)(13 117 147 97)(14 118 148 98)(15 119 149 99)(16 120 150 100)(17 111 141 91)(18 112 142 92)(19 113 143 93)(20 114 144 94)(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)(101 151 121 131)(102 152 122 132)(103 153 123 133)(104 154 124 134)(105 155 125 135)(106 156 126 136)(107 157 127 137)(108 158 128 138)(109 159 129 139)(110 160 130 140)

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

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

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

68 conjugacy classes

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

68 irreducible representations

dim11111112222222222444
type+++++++++-+++-+
imageC1C2C2C2C2C2C4D4D4D4Q8D5C4○D4D10C4×D5C5⋊D4C4○D20D4×D5Q8×D5Q82D5
kernelD105(C4⋊C4)C10.10C42C2×C10.D4C2×D10⋊C4C10×C4⋊C4D5×C22×C4D10⋊C4C2×Dic5C2×C20C22×D5C22×D5C2×C4⋊C4C2×C10C22×C4C2×C4C2×C4C22C22C22C22
# reps12121182222246888422

Matrix representation of D105(C4⋊C4) in GL6(𝔽41)

34340000
710000
0040700
0034700
000010
000001
,
34340000
170000
001000
0074000
0000400
0000040
,
17400000
1240000
0040000
0004000
00003313
0000368
,
4000000
0400000
0032000
0003200
00003237
000009

G:=sub<GL(6,GF(41))| [34,7,0,0,0,0,34,1,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[34,1,0,0,0,0,34,7,0,0,0,0,0,0,1,7,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[17,1,0,0,0,0,40,24,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,33,36,0,0,0,0,13,8],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,37,9] >;

D105(C4⋊C4) in GAP, Magma, Sage, TeX

D_{10}\rtimes_5(C_4\rtimes C_4)
% in TeX

G:=Group("D10:5(C4:C4)");
// GroupNames label

G:=SmallGroup(320,616);
// by ID

G=gap.SmallGroup(320,616);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,387,58,12550]);
// 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,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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