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G = C42.15F5order 320 = 26·5

12nd non-split extension by C42 of F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.15F5, D10.9M4(2), C20.11M4(2), (C4×C20).7C4, C20⋊C88C2, C4.8(C4.F5), D10⋊C8.4C2, C51(C42.6C4), (C4×Dic5).27C4, (D5×C42).20C2, C10.8(C2×M4(2)), C10.C425C2, C2.9(D5⋊M4(2)), C10.3(C42⋊C2), Dic5.25(C4○D4), C22.64(C22×F5), (C4×Dic5).322C22, (C2×Dic5).319C23, C2.8(D10.C23), (C2×C4×D5).30C4, C2.7(C2×C4.F5), (C2×C5⋊C8).3C22, (C2×C4).99(C2×F5), (C2×C20).123(C2×C4), (C2×C4×D5).358C22, (C2×C10).21(C22×C4), (C2×Dic5).169(C2×C4), (C22×D5).121(C2×C4), SmallGroup(320,1021)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.15F5
C1C5C10Dic5C2×Dic5C2×C5⋊C8D10⋊C8 — C42.15F5
C5C2×C10 — C42.15F5
C1C22C42

Generators and relations for C42.15F5
 G = < a,b,c,d | a4=b4=c5=1, d4=a2, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 378 in 110 conjugacy classes, 48 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×4], C2×C4 [×3], C2×C4 [×9], C23, D5 [×2], C10 [×3], C42, C42 [×3], C2×C8 [×4], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C8⋊C4 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C5⋊C8 [×4], C4×D5 [×6], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.6C4, C4×Dic5 [×3], C4×C20, C2×C5⋊C8 [×4], C2×C4×D5 [×3], C20⋊C8 [×2], C10.C42 [×2], D10⋊C8 [×2], D5×C42, C42.15F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×4], C22×C4, C4○D4 [×2], F5, C42⋊C2, C2×M4(2) [×2], C2×F5 [×3], C42.6C4, C4.F5 [×2], C22×F5, C2×C4.F5, D5⋊M4(2), D10.C23, C42.15F5

Smallest permutation representation of C42.15F5
On 160 points
Generators in S160
(1 7 5 3)(2 144 6 140)(4 138 8 142)(9 154 13 158)(10 117 14 113)(11 156 15 160)(12 119 16 115)(17 60 21 64)(18 105 22 109)(19 62 23 58)(20 107 24 111)(25 100 29 104)(26 95 30 91)(27 102 31 98)(28 89 32 93)(33 145 37 149)(34 135 38 131)(35 147 39 151)(36 129 40 133)(41 81 45 85)(42 48 46 44)(43 83 47 87)(49 152 53 148)(50 134 54 130)(51 146 55 150)(52 136 56 132)(57 123 61 127)(59 125 63 121)(65 90 69 94)(66 97 70 101)(67 92 71 96)(68 99 72 103)(73 155 77 159)(74 118 78 114)(75 157 79 153)(76 120 80 116)(82 88 86 84)(106 128 110 124)(108 122 112 126)(137 143 141 139)
(1 86 137 44)(2 83 138 41)(3 88 139 46)(4 85 140 43)(5 82 141 48)(6 87 142 45)(7 84 143 42)(8 81 144 47)(9 17 80 122)(10 22 73 127)(11 19 74 124)(12 24 75 121)(13 21 76 126)(14 18 77 123)(15 23 78 128)(16 20 79 125)(25 134 69 145)(26 131 70 150)(27 136 71 147)(28 133 72 152)(29 130 65 149)(30 135 66 146)(31 132 67 151)(32 129 68 148)(33 104 50 90)(34 101 51 95)(35 98 52 92)(36 103 53 89)(37 100 54 94)(38 97 55 91)(39 102 56 96)(40 99 49 93)(57 117 109 155)(58 114 110 160)(59 119 111 157)(60 116 112 154)(61 113 105 159)(62 118 106 156)(63 115 107 153)(64 120 108 158)
(1 12 52 130 113)(2 131 13 114 53)(3 115 132 54 14)(4 55 116 15 133)(5 16 56 134 117)(6 135 9 118 49)(7 119 136 50 10)(8 51 120 11 129)(17 106 93 87 66)(18 88 107 67 94)(19 68 81 95 108)(20 96 69 109 82)(21 110 89 83 70)(22 84 111 71 90)(23 72 85 91 112)(24 92 65 105 86)(25 57 48 125 102)(26 126 58 103 41)(27 104 127 42 59)(28 43 97 60 128)(29 61 44 121 98)(30 122 62 99 45)(31 100 123 46 63)(32 47 101 64 124)(33 73 143 157 147)(34 158 74 148 144)(35 149 159 137 75)(36 138 150 76 160)(37 77 139 153 151)(38 154 78 152 140)(39 145 155 141 79)(40 142 146 80 156)
(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)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H4I4J4K4L4M4N 5 8A···8H10A10B10C20A···20L
order1222224···44444444458···810101020···20
size111110102···2555510101010420···204444···4

44 irreducible representations

dim1111111122244444
type+++++++
imageC1C2C2C2C2C4C4C4C4○D4M4(2)M4(2)F5C2×F5C4.F5D5⋊M4(2)D10.C23
kernelC42.15F5C20⋊C8C10.C42D10⋊C8D5×C42C4×Dic5C4×C20C2×C4×D5Dic5C20D10C42C2×C4C4C2C2
# reps1222122444413444

Matrix representation of C42.15F5 in GL6(𝔽41)

3200000
0320000
0022033
003819380
000381938
0033022
,
0400000
4000000
009000
000900
000090
000009
,
100000
010000
0040404040
001000
000100
000010
,
6390000
2350000
007352413
003019613
0028352211
002817634

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

C42.15F5 in GAP, Magma, Sage, TeX

C_4^2._{15}F_5
% in TeX

G:=Group("C4^2.15F5");
// GroupNames label

G:=SmallGroup(320,1021);
// by ID

G=gap.SmallGroup(320,1021);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,232,758,100,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^5=1,d^4=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations

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