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G = C4.Dic10order 160 = 25·5

3rd non-split extension by C4 of Dic10 acting via Dic10/Dic5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.3Q8, C4.3Dic10, C4⋊C4.6D5, C10.6(C2×Q8), (C2×C4).43D10, C4⋊Dic5.7C2, C53(C42.C2), (C4×Dic5).2C2, C2.8(C2×Dic10), C10.25(C4○D4), (C2×C20).22C22, (C2×C10).31C23, C2.4(Q82D5), C10.D4.3C2, C2.12(D42D5), C22.48(C22×D5), (C2×Dic5).10C22, (C5×C4⋊C4).7C2, SmallGroup(160,111)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4.Dic10
C1C5C10C2×C10C2×Dic5C4×Dic5 — C4.Dic10
C5C2×C10 — C4.Dic10
C1C22C4⋊C4

Generators and relations for C4.Dic10
 G = < a,b,c | a4=b20=1, c2=b10, bab-1=a-1, ac=ca, cbc-1=a2b-1 >

Subgroups: 144 in 56 conjugacy classes, 33 normal (19 characteristic)
C1, C2 [×3], C4 [×2], C4 [×6], C22, C5, C2×C4, C2×C4 [×2], C2×C4 [×4], C10 [×3], C42, C4⋊C4, C4⋊C4 [×5], Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C42.C2, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic5, C10.D4 [×2], C4⋊Dic5, C4⋊Dic5 [×2], C5×C4⋊C4, C4.Dic10
Quotients: C1, C2 [×7], C22 [×7], Q8 [×2], C23, D5, C2×Q8, C4○D4 [×2], D10 [×3], C42.C2, Dic10 [×2], C22×D5, C2×Dic10, D42D5, Q82D5, C4.Dic10

Smallest permutation representation of C4.Dic10
Regular action on 160 points
Generators in S160
(1 154 106 65)(2 66 107 155)(3 156 108 67)(4 68 109 157)(5 158 110 69)(6 70 111 159)(7 160 112 71)(8 72 113 141)(9 142 114 73)(10 74 115 143)(11 144 116 75)(12 76 117 145)(13 146 118 77)(14 78 119 147)(15 148 120 79)(16 80 101 149)(17 150 102 61)(18 62 103 151)(19 152 104 63)(20 64 105 153)(21 95 122 47)(22 48 123 96)(23 97 124 49)(24 50 125 98)(25 99 126 51)(26 52 127 100)(27 81 128 53)(28 54 129 82)(29 83 130 55)(30 56 131 84)(31 85 132 57)(32 58 133 86)(33 87 134 59)(34 60 135 88)(35 89 136 41)(36 42 137 90)(37 91 138 43)(38 44 139 92)(39 93 140 45)(40 46 121 94)
(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 43 11 53)(2 90 12 100)(3 41 13 51)(4 88 14 98)(5 59 15 49)(6 86 16 96)(7 57 17 47)(8 84 18 94)(9 55 19 45)(10 82 20 92)(21 160 31 150)(22 70 32 80)(23 158 33 148)(24 68 34 78)(25 156 35 146)(26 66 36 76)(27 154 37 144)(28 64 38 74)(29 152 39 142)(30 62 40 72)(42 117 52 107)(44 115 54 105)(46 113 56 103)(48 111 58 101)(50 109 60 119)(61 122 71 132)(63 140 73 130)(65 138 75 128)(67 136 77 126)(69 134 79 124)(81 106 91 116)(83 104 93 114)(85 102 95 112)(87 120 97 110)(89 118 99 108)(121 141 131 151)(123 159 133 149)(125 157 135 147)(127 155 137 145)(129 153 139 143)

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

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

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

C4.Dic10 is a maximal subgroup of
C10.C4≀C2  D4.Dic10  C4⋊C4.D10  D4.2Dic10  (C8×Dic5)⋊C2  Q8.Dic10  C408C4.C2  Q8.2Dic10  Q8⋊Dic5⋊C2  C403Q8  Dic10.Q8  C8.8Dic10  D20.Q8  C404Q8  Dic10.2Q8  C8.6Dic10  D20.2Q8  C10.12- 1+4  C10.52- 1+4  C10.112+ 1+4  C42.88D10  C42.90D10  C42.94D10  C42.95D10  D45Dic10  D46Dic10  C42.229D10  C42.116D10  Q85Dic10  Q86Dic10  C42.131D10  C42.134D10  C4⋊C4.178D10  C10.432+ 1+4  C10.1152+ 1+4  C10.472+ 1+4  C22⋊Q825D5  C10.152- 1+4  C10.212- 1+4  C10.772- 1+4  C10.572+ 1+4  C10.582+ 1+4  C10.802- 1+4  C10.632+ 1+4  C10.852- 1+4  C10.692+ 1+4  Dic107Q8  C42.147D10  D5×C42.C2  C42.148D10  D207Q8  C42.152D10  C42.154D10  C42.155D10  C42.159D10  C42.161D10  C42.162D10  C42.165D10  Dic108Q8  C42.241D10  C42.174D10  D209Q8  Dic3.Dic10  Dic3.2Dic10  C12.Dic10  C20.Dic6  C4.Dic30
C4.Dic10 is a maximal quotient of
C4⋊Dic515C4  C10.52(C4×D4)  (C2×C4).Dic10  C10.(C4⋊Q8)  C204(C4⋊C4)  C20.48(C4⋊C4)  (C2×C20).54D4  C206(C4⋊C4)  (C2×C20).55D4  Dic3.Dic10  Dic3.2Dic10  C12.Dic10  C20.Dic6  C4.Dic30

34 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J5A5B10A···10F20A···20L
order122244444444445510···1020···20
size11112244101010102020222···24···4

34 irreducible representations

dim111112222244
type+++++-++--+
imageC1C2C2C2C2Q8D5C4○D4D10Dic10D42D5Q82D5
kernelC4.Dic10C4×Dic5C10.D4C4⋊Dic5C5×C4⋊C4C20C4⋊C4C10C2×C4C4C2C2
# reps112312246822

Matrix representation of C4.Dic10 in GL4(𝔽41) generated by

1000
0100
00203
00321
,
11200
252700
0001
00400
,
182700
322300
00320
00032
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,20,3,0,0,3,21],[11,25,0,0,2,27,0,0,0,0,0,40,0,0,1,0],[18,32,0,0,27,23,0,0,0,0,32,0,0,0,0,32] >;

C4.Dic10 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_{10}
% in TeX

G:=Group("C4.Dic10");
// GroupNames label

G:=SmallGroup(160,111);
// by ID

G=gap.SmallGroup(160,111);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,506,188,50,4613]);
// Polycyclic

G:=Group<a,b,c|a^4=b^20=1,c^2=b^10,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1>;
// generators/relations

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