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G = C4⋊C4.D10order 320 = 26·5

5th non-split extension by C4⋊C4 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.5D10, C408C416C2, D4⋊C415D5, D206C43C2, (C2×D4).19D10, C20.3(C4○D4), (C2×C8).166D10, C20⋊D4.6C2, D4⋊Dic52C2, C4.Dic102C2, D205C416C2, C4.21(C4○D20), C2.8(D40⋊C2), (C2×Dic5).23D4, C22.166(D4×D5), C2.11(D8⋊D5), C4.47(D42D5), C10.53(C8⋊C22), (C2×C40).179C22, (C2×C20).204C23, (C2×D20).50C22, (D4×C10).25C22, C4⋊Dic5.63C22, C10.23(C4.4D4), (C4×Dic5).14C22, C52(C42.29C22), C2.13(Dic5.5D4), (C5×C4⋊C4).9C22, (C5×D4⋊C4)⋊17C2, (C2×C10).217(C2×D4), (C2×C52C8).10C22, (C2×C4).311(C22×D5), SmallGroup(320,391)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4⋊C4.D10
C1C5C10C20C2×C20C4×Dic5C20⋊D4 — C4⋊C4.D10
C5C10C2×C20 — C4⋊C4.D10
C1C22C2×C4D4⋊C4

Generators and relations for C4⋊C4.D10
 G = < a,b,c,d | a4=b4=c10=d2=1, bab-1=cac-1=dad=a-1, cbc-1=a-1b-1, dbd=ab, dcd=b2c-1 >

Subgroups: 542 in 110 conjugacy classes, 37 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×4], D4 [×8], C23 [×2], D5, C10 [×3], C10, C42, C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, C2×D4, C2×D4 [×3], Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C8⋊C4, D4⋊C4, D4⋊C4 [×3], C42.C2, C41D4, C52C8, C40, D20 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C22×D5, C22×C10, C42.29C22, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×D20, C2×C5⋊D4 [×2], D4×C10, D206C4, C408C4, D205C4, D4⋊Dic5, C5×D4⋊C4, C4.Dic10, C20⋊D4, C4⋊C4.D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C8⋊C22 [×2], C22×D5, C42.29C22, C4○D20, D4×D5, D42D5, Dic5.5D4, D8⋊D5, D40⋊C2, C4⋊C4.D10

Smallest permutation representation of C4⋊C4.D10
On 160 points
Generators in S160
(1 70 123 103)(2 104 124 61)(3 62 125 105)(4 106 126 63)(5 64 127 107)(6 108 128 65)(7 66 129 109)(8 110 130 67)(9 68 121 101)(10 102 122 69)(11 35 81 148)(12 149 82 36)(13 37 83 150)(14 141 84 38)(15 39 85 142)(16 143 86 40)(17 31 87 144)(18 145 88 32)(19 33 89 146)(20 147 90 34)(21 51 134 94)(22 95 135 52)(23 53 136 96)(24 97 137 54)(25 55 138 98)(26 99 139 56)(27 57 140 100)(28 91 131 58)(29 59 132 92)(30 93 133 60)(41 114 154 71)(42 72 155 115)(43 116 156 73)(44 74 157 117)(45 118 158 75)(46 76 159 119)(47 120 160 77)(48 78 151 111)(49 112 152 79)(50 80 153 113)
(1 113 93 40)(2 17 94 41)(3 115 95 32)(4 19 96 43)(5 117 97 34)(6 11 98 45)(7 119 99 36)(8 13 100 47)(9 111 91 38)(10 15 92 49)(12 109 46 26)(14 101 48 28)(16 103 50 30)(18 105 42 22)(20 107 44 24)(21 71 104 144)(23 73 106 146)(25 75 108 148)(27 77 110 150)(29 79 102 142)(31 134 114 61)(33 136 116 63)(35 138 118 65)(37 140 120 67)(39 132 112 69)(51 154 124 87)(52 145 125 72)(53 156 126 89)(54 147 127 74)(55 158 128 81)(56 149 129 76)(57 160 130 83)(58 141 121 78)(59 152 122 85)(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 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 5)(2 96)(4 94)(6 92)(7 9)(8 100)(10 98)(11 79)(12 141)(13 77)(14 149)(15 75)(16 147)(17 73)(18 145)(19 71)(20 143)(21 63)(22 135)(23 61)(24 133)(25 69)(26 131)(27 67)(28 139)(29 65)(30 137)(31 156)(32 88)(33 154)(34 86)(35 152)(36 84)(37 160)(38 82)(39 158)(40 90)(41 146)(42 72)(43 144)(44 80)(45 142)(46 78)(47 150)(48 76)(49 148)(50 74)(51 126)(53 124)(54 60)(55 122)(56 58)(57 130)(59 128)(62 105)(64 103)(66 101)(68 109)(70 107)(81 112)(83 120)(85 118)(87 116)(89 114)(91 99)(93 97)(102 138)(104 136)(106 134)(108 132)(110 140)(111 159)(113 157)(115 155)(117 153)(119 151)(121 129)(123 127)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222244444455888810···1010101010202020202020202040···40
size1111840228202040224420202···28888444488884···4

44 irreducible representations

dim11111111222222244444
type++++++++++++++-++
imageC1C2C2C2C2C2C2C2D4D5C4○D4D10D10D10C4○D20C8⋊C22D42D5D4×D5D8⋊D5D40⋊C2
kernelC4⋊C4.D10D206C4C408C4D205C4D4⋊Dic5C5×D4⋊C4C4.Dic10C20⋊D4C2×Dic5D4⋊C4C20C4⋊C4C2×C8C2×D4C4C10C4C22C2C2
# reps11111111224222822244

Matrix representation of C4⋊C4.D10 in GL8(𝔽41)

10000000
01000000
00100000
00010000
0000392800
000013200
000000213
0000002839
,
30323900000
9110390000
001190000
0032300000
00000010
00000001
00001000
00000100
,
173813280000
33813220000
32192430000
22223830000
0000003927
0000001437
000014200
0000392600
,
77000000
4034000000
1427770000
112740340000
0000353500
000040600
0000002516
000000216

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,39,13,0,0,0,0,0,0,28,2,0,0,0,0,0,0,0,0,2,28,0,0,0,0,0,0,13,39],[30,9,0,0,0,0,0,0,32,11,0,0,0,0,0,0,39,0,11,32,0,0,0,0,0,39,9,30,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[17,3,32,22,0,0,0,0,38,38,19,22,0,0,0,0,13,13,24,38,0,0,0,0,28,22,3,3,0,0,0,0,0,0,0,0,0,0,14,39,0,0,0,0,0,0,2,26,0,0,0,0,39,14,0,0,0,0,0,0,27,37,0,0],[7,40,14,11,0,0,0,0,7,34,27,27,0,0,0,0,0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,0,0,0,0,35,40,0,0,0,0,0,0,35,6,0,0,0,0,0,0,0,0,25,2,0,0,0,0,0,0,16,16] >;

C4⋊C4.D10 in GAP, Magma, Sage, TeX

C_4\rtimes C_4.D_{10}
% in TeX

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

G:=SmallGroup(320,391);
// by ID

G=gap.SmallGroup(320,391);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,1094,135,100,570,297,136,12550]);
// Polycyclic

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

׿
×
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