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G = D102(C4⋊C4)  order 320 = 26·5

1st semidirect product of D10 and C4⋊C4 acting via C4⋊C4/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D102(C4⋊C4), C22.61(D4×D5), C22.16(Q8×D5), C10.4(C4⋊D4), (C22×D5).76D4, C2.C422D5, (C22×C4).16D10, (C22×D5).12Q8, C2.3(D10⋊D4), C52(C23.7Q8), Dic53(C22⋊C4), C2.3(D10⋊Q8), (C2×Dic5).186D4, C2.9(C42⋊D5), C10.24(C22⋊Q8), C22.35(C4○D20), (C22×C20).13C22, (C23×D5).96C22, C23.257(C22×D5), C10.10C4236C2, C10.24(C42⋊C2), (C22×C10).292C23, (C22×Dic5).15C22, (C2×C4×D5)⋊14C4, C2.7(D5×C4⋊C4), C10.28(C2×C4⋊C4), C2.8(D5×C22⋊C4), C22.90(C2×C4×D5), (C2×C4).125(C4×D5), (C2×C10).67(C2×Q8), (D5×C22×C4).14C2, (C2×C20).316(C2×C4), (C2×C10.D4)⋊1C2, (C2×C10).201(C2×D4), C10.47(C2×C22⋊C4), (C2×D10⋊C4).4C2, (C2×C10).58(C4○D4), (C22×D5).93(C2×C4), (C5×C2.C42)⋊20C2, (C2×C10).152(C22×C4), (C2×Dic5).133(C2×C4), SmallGroup(320,294)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D102(C4⋊C4)
C1C5C10C2×C10C22×C10C23×D5D5×C22×C4 — D102(C4⋊C4)
C5C2×C10 — D102(C4⋊C4)
C1C23C2.C42

Generators and relations for D102(C4⋊C4)
 G = < a,b,c,d | a10=b2=c4=d4=1, bab=cac-1=a-1, ad=da, cbc-1=a8b, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 910 in 234 conjugacy classes, 79 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×10], C22 [×3], C22 [×4], C22 [×16], C5, C2×C4 [×2], C2×C4 [×28], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×11], C24, Dic5 [×4], Dic5 [×2], C20 [×4], D10 [×4], D10 [×12], C2×C10 [×3], C2×C10 [×4], C2.C42, C2.C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C4×D5 [×8], C2×Dic5 [×6], C2×Dic5 [×6], C2×C20 [×2], C2×C20 [×8], C22×D5 [×6], C22×D5 [×4], C22×C10, C23.7Q8, C10.D4 [×4], D10⋊C4 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, C10.10C42, C5×C2.C42, C2×C10.D4 [×2], C2×D10⋊C4 [×2], D5×C22×C4, D102(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], C22×D5, C23.7Q8, C2×C4×D5, C4○D20 [×2], D4×D5 [×3], Q8×D5, C42⋊D5, D5×C22⋊C4, D10⋊D4 [×2], D5×C4⋊C4, D10⋊Q8 [×2], D102(C4⋊C4)

Smallest permutation representation of D102(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 40)(9 39)(10 38)(11 132)(12 131)(13 140)(14 139)(15 138)(16 137)(17 136)(18 135)(19 134)(20 133)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 50)(28 49)(29 48)(30 47)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 90)(58 89)(59 88)(60 87)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 80)(68 79)(69 78)(70 77)(91 121)(92 130)(93 129)(94 128)(95 127)(96 126)(97 125)(98 124)(99 123)(100 122)(101 111)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 160)(148 159)(149 158)(150 157)
(1 110 28 98)(2 109 29 97)(3 108 30 96)(4 107 21 95)(5 106 22 94)(6 105 23 93)(7 104 24 92)(8 103 25 91)(9 102 26 100)(10 101 27 99)(11 73 153 85)(12 72 154 84)(13 71 155 83)(14 80 156 82)(15 79 157 81)(16 78 158 90)(17 77 159 89)(18 76 160 88)(19 75 151 87)(20 74 152 86)(31 120 43 122)(32 119 44 121)(33 118 45 130)(34 117 46 129)(35 116 47 128)(36 115 48 127)(37 114 49 126)(38 113 50 125)(39 112 41 124)(40 111 42 123)(51 135 63 147)(52 134 64 146)(53 133 65 145)(54 132 66 144)(55 131 67 143)(56 140 68 142)(57 139 69 141)(58 138 70 150)(59 137 61 149)(60 136 62 148)
(1 85 45 65)(2 86 46 66)(3 87 47 67)(4 88 48 68)(5 89 49 69)(6 90 50 70)(7 81 41 61)(8 82 42 62)(9 83 43 63)(10 84 44 64)(11 118 145 98)(12 119 146 99)(13 120 147 100)(14 111 148 91)(15 112 149 92)(16 113 150 93)(17 114 141 94)(18 115 142 95)(19 116 143 96)(20 117 144 97)(21 76 36 56)(22 77 37 57)(23 78 38 58)(24 79 39 59)(25 80 40 60)(26 71 31 51)(27 72 32 52)(28 73 33 53)(29 74 34 54)(30 75 35 55)(101 154 121 134)(102 155 122 135)(103 156 123 136)(104 157 124 137)(105 158 125 138)(106 159 126 139)(107 160 127 140)(108 151 128 131)(109 152 129 132)(110 153 130 133)

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

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

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

68 conjugacy classes

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

68 irreducible representations

dim11111112222222244
type++++++++-+++-
imageC1C2C2C2C2C2C4D4D4Q8D5C4○D4D10C4×D5C4○D20D4×D5Q8×D5
kernelD102(C4⋊C4)C10.10C42C5×C2.C42C2×C10.D4C2×D10⋊C4D5×C22×C4C2×C4×D5C2×Dic5C22×D5C22×D5C2.C42C2×C10C22×C4C2×C4C22C22C22
# reps111221842224681662

Matrix representation of D102(C4⋊C4) in GL6(𝔽41)

160000
3560000
00343400
007100
0000400
0000040
,
35400000
3560000
00343400
001700
000010
0000140
,
9130000
0320000
0032000
0022900
0000400
0000040
,
2280000
13390000
00174000
0012400
0000139
0000040

G:=sub<GL(6,GF(41))| [1,35,0,0,0,0,6,6,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,35,0,0,0,0,40,6,0,0,0,0,0,0,34,1,0,0,0,0,34,7,0,0,0,0,0,0,1,1,0,0,0,0,0,40],[9,0,0,0,0,0,13,32,0,0,0,0,0,0,32,22,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[2,13,0,0,0,0,28,39,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,0,0,0,1,0,0,0,0,0,39,40] >;

D102(C4⋊C4) in GAP, Magma, Sage, TeX

D_{10}\rtimes_2(C_4\rtimes C_4)
% in TeX

G:=Group("D10:2(C4:C4)");
// GroupNames label

G:=SmallGroup(320,294);
// by ID

G=gap.SmallGroup(320,294);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,1094,387,58,12550]);
// 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^8*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

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