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G = C42.96D10order 320 = 26·5

96th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.96D10, C10.522- 1+4, C4⋊C4.273D10, D10⋊Q85C2, C422D55C2, C42⋊D53C2, C20.6Q87C2, (C4×Dic10)⋊10C2, C42⋊C215D5, (C4×C20).26C22, (C2×C10).75C24, C22⋊C4.99D10, (C2×C20).150C23, Dic54D4.6C2, Dic5.Q85C2, (C22×C4).196D10, C4⋊Dic5.35C22, Dic5.33(C4○D4), Dic5.14D45C2, C23.D5.5C22, C22.19(C4○D20), (C2×Dic5).28C23, C22.D20.2C2, (C22×D5).23C23, C22.104(C23×D5), C23.160(C22×D5), D10⋊C4.63C22, C23.11D1026C2, (C22×C10).145C23, (C22×C20).436C22, C53(C22.46C24), (C4×Dic5).217C22, C23.23D10.4C2, C10.D4.99C22, C2.10(D4.10D10), (C2×Dic10).239C22, (C22×Dic5).89C22, C4⋊C4⋊D55C2, C2.14(D5×C4○D4), C2.34(C2×C4○D20), C10.31(C2×C4○D4), (C2×C4×D5).244C22, (C5×C42⋊C2)⋊17C2, (C2×C10).42(C4○D4), (C2×C10.D4)⋊46C2, (C5×C4⋊C4).311C22, (C2×C4).277(C22×D5), (C2×C5⋊D4).10C22, (C5×C22⋊C4).139C22, SmallGroup(320,1203)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.96D10
C1C5C10C2×C10C22×D5C2×C5⋊D4Dic54D4 — C42.96D10
C5C2×C10 — C42.96D10
C1C22C42⋊C2

Generators and relations for C42.96D10
 G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=b2c-1 >

Subgroups: 662 in 214 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×14], C22, C22 [×2], C22 [×5], C5, C2×C4 [×6], C2×C4 [×15], D4 [×2], Q8 [×2], C23, C23, D5, C10 [×3], C10 [×2], C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4, C22×C4 [×3], C2×D4, C2×Q8, Dic5 [×2], Dic5 [×6], C20 [×6], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4, C42⋊C2, C42⋊C2 [×2], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C422C2 [×2], Dic10 [×2], C4×D5 [×2], C2×Dic5 [×7], C2×Dic5 [×4], C5⋊D4 [×2], C2×C20 [×6], C2×C20 [×2], C22×D5, C22×C10, C22.46C24, C4×Dic5 [×3], C10.D4 [×11], C4⋊Dic5 [×3], D10⋊C4 [×5], C23.D5, C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5, C22×Dic5 [×2], C2×C5⋊D4, C22×C20, C4×Dic10, C20.6Q8, C42⋊D5, C422D5, C23.11D10, Dic5.14D4, Dic54D4, C22.D20, Dic5.Q8 [×2], D10⋊Q8, C4⋊C4⋊D5, C2×C10.D4, C23.23D10, C5×C42⋊C2, C42.96D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- 1+4, C22×D5 [×7], C22.46C24, C4○D20 [×2], C23×D5, C2×C4○D20, D5×C4○D4, D4.10D10, C42.96D10

Smallest permutation representation of C42.96D10
On 160 points
Generators in S160
(1 50 30 113)(2 71 21 144)(3 42 22 115)(4 73 23 146)(5 44 24 117)(6 75 25 148)(7 46 26 119)(8 77 27 150)(9 48 28 111)(10 79 29 142)(11 101 93 38)(12 132 94 69)(13 103 95 40)(14 134 96 61)(15 105 97 32)(16 136 98 63)(17 107 99 34)(18 138 100 65)(19 109 91 36)(20 140 92 67)(31 154 104 87)(33 156 106 89)(35 158 108 81)(37 160 110 83)(39 152 102 85)(41 51 114 124)(43 53 116 126)(45 55 118 128)(47 57 120 130)(49 59 112 122)(52 145 125 72)(54 147 127 74)(56 149 129 76)(58 141 121 78)(60 143 123 80)(62 155 135 88)(64 157 137 90)(66 159 139 82)(68 151 131 84)(70 153 133 86)
(1 133 123 103)(2 134 124 104)(3 135 125 105)(4 136 126 106)(5 137 127 107)(6 138 128 108)(7 139 129 109)(8 140 130 110)(9 131 121 101)(10 132 122 102)(11 111 151 141)(12 112 152 142)(13 113 153 143)(14 114 154 144)(15 115 155 145)(16 116 156 146)(17 117 157 147)(18 118 158 148)(19 119 159 149)(20 120 160 150)(21 61 51 31)(22 62 52 32)(23 63 53 33)(24 64 54 34)(25 65 55 35)(26 66 56 36)(27 67 57 37)(28 68 58 38)(29 69 59 39)(30 70 60 40)(41 87 71 96)(42 88 72 97)(43 89 73 98)(44 90 74 99)(45 81 75 100)(46 82 76 91)(47 83 77 92)(48 84 78 93)(49 85 79 94)(50 86 80 95)
(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 10 123 122)(2 121 124 9)(3 8 125 130)(4 129 126 7)(5 6 127 128)(11 96 151 87)(12 86 152 95)(13 94 153 85)(14 84 154 93)(15 92 155 83)(16 82 156 91)(17 100 157 81)(18 90 158 99)(19 98 159 89)(20 88 160 97)(21 58 51 28)(22 27 52 57)(23 56 53 26)(24 25 54 55)(29 60 59 30)(31 131 61 101)(32 110 62 140)(33 139 63 109)(34 108 64 138)(35 137 65 107)(36 106 66 136)(37 135 67 105)(38 104 68 134)(39 133 69 103)(40 102 70 132)(41 78 71 48)(42 47 72 77)(43 76 73 46)(44 45 74 75)(49 80 79 50)(111 114 141 144)(112 143 142 113)(115 120 145 150)(116 149 146 119)(117 118 147 148)

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F4A···4F4G4H4I4J4K4L4M4N···4R5A5B10A···10F10G10H10I10J20A···20H20I···20AB
order12222224···444444444···45510···101010101020···2020···20
size111122202···24441010101020···20222···244442···24···4

65 irreducible representations

dim11111111111111122222222444
type++++++++++++++++++++--
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2D5C4○D4C4○D4D10D10D10D10C4○D202- 1+4D5×C4○D4D4.10D10
kernelC42.96D10C4×Dic10C20.6Q8C42⋊D5C422D5C23.11D10Dic5.14D4Dic54D4C22.D20Dic5.Q8D10⋊Q8C4⋊C4⋊D5C2×C10.D4C23.23D10C5×C42⋊C2C42⋊C2Dic5C2×C10C42C22⋊C4C4⋊C4C22×C4C22C10C2C2
# reps111111111211111244444216144

Matrix representation of C42.96D10 in GL4(𝔽41) generated by

9000
0900
00400
00401
,
183500
62300
00320
00032
,
6600
35100
00139
00040
,
6600
13500
00402
00401
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,40,40,0,0,0,1],[18,6,0,0,35,23,0,0,0,0,32,0,0,0,0,32],[6,35,0,0,6,1,0,0,0,0,1,0,0,0,39,40],[6,1,0,0,6,35,0,0,0,0,40,40,0,0,2,1] >;

C42.96D10 in GAP, Magma, Sage, TeX

C_4^2._{96}D_{10}
% in TeX

G:=Group("C4^2.96D10");
// GroupNames label

G:=SmallGroup(320,1203);
// by ID

G=gap.SmallGroup(320,1203);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,100,675,136,12550]);
// Polycyclic

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

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