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G = C4015D4order 320 = 26·5

15th semidirect product of C40 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4015D4, Dic51SD16, C52C815D4, C88(C5⋊D4), C54(C85D4), C4.25(D4×D5), C20.50(C2×D4), (C10×SD16)⋊9C2, (C8×Dic5)⋊11C2, (C2×SD16)⋊15D5, (C2×D4).76D10, (C2×C8).264D10, C20⋊D4.8C2, (C2×Q8).57D10, C2.31(D5×SD16), Dic5⋊Q85C2, C10.48(C2×SD16), C22.272(D4×D5), C2.22(C20⋊D4), C10.31(C41D4), (C2×C20).452C23, (C2×C40).165C22, (C2×Dic5).160D4, (Q8×C10).81C22, (D4×C10).101C22, (C2×D20).126C22, (C4×Dic5).273C22, (C2×Dic10).133C22, C4.9(C2×C5⋊D4), (C2×Q8⋊D5)⋊18C2, (C2×C40⋊C2)⋊30C2, (C2×D4.D5)⋊21C2, (C2×C10).364(C2×D4), (C2×C4).541(C22×D5), (C2×C52C8).283C22, SmallGroup(320,802)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4015D4
C1C5C10C2×C10C2×C20C2×D20C2×C40⋊C2 — C4015D4
C5C10C2×C20 — C4015D4
C1C22C2×C4C2×SD16

Generators and relations for C4015D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a9, cac=a19, cbc=b-1 >

Subgroups: 654 in 142 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4 [×8], Q8 [×4], C23 [×2], D5, C10, C10 [×2], C10, C42, C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×8], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, Dic5 [×4], Dic5, C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C4×C8, C41D4, C4⋊Q8, C2×SD16, C2×SD16 [×3], C52C8 [×2], C40 [×2], Dic10 [×2], D20 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10, C85D4, C40⋊C2 [×2], C2×C52C8, C4×Dic5, C10.D4 [×2], D4.D5 [×2], Q8⋊D5 [×2], C2×C40, C5×SD16 [×2], C2×Dic10, C2×D20, C2×C5⋊D4 [×2], D4×C10, Q8×C10, C8×Dic5, C2×C40⋊C2, C2×D4.D5, C20⋊D4, C2×Q8⋊D5, Dic5⋊Q8, C10×SD16, C4015D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×4], C2×D4 [×3], D10 [×3], C41D4, C2×SD16 [×2], C5⋊D4 [×2], C22×D5, C85D4, D4×D5 [×2], C2×C5⋊D4, D5×SD16 [×2], C20⋊D4, C4015D4

Smallest permutation representation of C4015D4
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)
(1 104 149 54)(2 113 150 63)(3 82 151 72)(4 91 152 41)(5 100 153 50)(6 109 154 59)(7 118 155 68)(8 87 156 77)(9 96 157 46)(10 105 158 55)(11 114 159 64)(12 83 160 73)(13 92 121 42)(14 101 122 51)(15 110 123 60)(16 119 124 69)(17 88 125 78)(18 97 126 47)(19 106 127 56)(20 115 128 65)(21 84 129 74)(22 93 130 43)(23 102 131 52)(24 111 132 61)(25 120 133 70)(26 89 134 79)(27 98 135 48)(28 107 136 57)(29 116 137 66)(30 85 138 75)(31 94 139 44)(32 103 140 53)(33 112 141 62)(34 81 142 71)(35 90 143 80)(36 99 144 49)(37 108 145 58)(38 117 146 67)(39 86 147 76)(40 95 148 45)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 97)(42 116)(43 95)(44 114)(45 93)(46 112)(47 91)(48 110)(49 89)(50 108)(51 87)(52 106)(53 85)(54 104)(55 83)(56 102)(57 81)(58 100)(59 119)(60 98)(61 117)(62 96)(63 115)(64 94)(65 113)(66 92)(67 111)(68 90)(69 109)(70 88)(71 107)(72 86)(73 105)(74 84)(75 103)(76 82)(77 101)(78 120)(79 99)(80 118)(121 137)(122 156)(123 135)(124 154)(125 133)(126 152)(127 131)(128 150)(130 148)(132 146)(134 144)(136 142)(138 140)(139 159)(141 157)(143 155)(145 153)(147 151)(158 160)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222244444444558888888810···1010101010202020202020202040···40
size11118402281010101040222222101010102···28888444488884···4

50 irreducible representations

dim11111111222222222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5SD16D10D10D10C5⋊D4D4×D5D4×D5D5×SD16
kernelC4015D4C8×Dic5C2×C40⋊C2C2×D4.D5C20⋊D4C2×Q8⋊D5Dic5⋊Q8C10×SD16C52C8C40C2×Dic5C2×SD16Dic5C2×C8C2×D4C2×Q8C8C4C22C2
# reps11111111222282228228

Matrix representation of C4015D4 in GL6(𝔽41)

0350000
770000
00152600
00151500
00001526
00001515
,
34350000
870000
001000
000100
000001
0000400
,
34350000
870000
001000
0004000
000010
0000040

G:=sub<GL(6,GF(41))| [0,7,0,0,0,0,35,7,0,0,0,0,0,0,15,15,0,0,0,0,26,15,0,0,0,0,0,0,15,15,0,0,0,0,26,15],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

C4015D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_{15}D_4
% in TeX

G:=Group("C40:15D4");
// GroupNames label

G:=SmallGroup(320,802);
// by ID

G=gap.SmallGroup(320,802);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,555,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^19,c*b*c=b^-1>;
// generators/relations

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