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G = C42.14F5order 320 = 26·5

11st non-split extension by C42 of F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.14F5, Dic57M4(2), (C4×C20).6C4, (C4×D5).79D4, C4.10(C4⋊F5), C20.17(C4⋊C4), (C4×D5).21Q8, D10.26(C4⋊C4), Dic5⋊C84C2, C52(C4⋊M4(2)), Dic5.28(C2×D4), Dic5.10(C2×Q8), (C4×Dic5).36C4, (D5×C42).19C2, C10.7(C2×M4(2)), C2.8(D5⋊M4(2)), C22.63(C22×F5), (C4×Dic5).321C22, (C2×Dic5).318C23, C2.7(C2×C4⋊F5), C10.3(C2×C4⋊C4), (C2×C4×D5).29C4, (C2×C5⋊C8).2C22, (C2×C4).98(C2×F5), (C2×C4.F5).8C2, (C2×C20).122(C2×C4), (C2×C4×D5).391C22, (C2×C10).20(C22×C4), (C2×Dic5).168(C2×C4), (C22×D5).120(C2×C4), SmallGroup(320,1020)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.14F5
C1C5C10Dic5C2×Dic5C2×C5⋊C8C2×C4.F5 — C42.14F5
C5C2×C10 — C42.14F5
C1C22C42

Generators and relations for C42.14F5
 G = < a,b,c,d | a4=b4=c5=1, d4=a2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c3 >

Subgroups: 426 in 126 conjugacy classes, 56 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, D5 [×2], C10, C10 [×2], C42, C42 [×3], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4⋊C8 [×4], C2×C42, C2×M4(2) [×2], C5⋊C8 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C4⋊M4(2), C4×Dic5, C4×Dic5 [×2], C4×C20, C4.F5 [×4], C2×C5⋊C8 [×4], C2×C4×D5, C2×C4×D5 [×2], Dic5⋊C8 [×4], D5×C42, C2×C4.F5 [×2], C42.14F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C2×M4(2) [×2], C2×F5 [×3], C4⋊M4(2), C4⋊F5 [×2], C22×F5, D5⋊M4(2) [×2], C2×C4⋊F5, C42.14F5

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

dim11111112224444
type+++++-++
imageC1C2C2C2C4C4C4D4Q8M4(2)F5C2×F5C4⋊F5D5⋊M4(2)
kernelC42.14F5Dic5⋊C8D5×C42C2×C4.F5C4×Dic5C4×C20C2×C4×D5C4×D5C4×D5Dic5C42C2×C4C4C2
# reps14122242281348

Matrix representation of C42.14F5 in GL6(𝔽41)

3200000
2390000
001000
000100
000010
000001
,
100000
2400000
00701414
002734270
000273427
00141407
,
100000
010000
0040404040
001000
000100
000010
,
10310000
5310000
0040000
0000040
0004000
001111

G:=sub<GL(6,GF(41))| [32,23,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,40,0,0,0,0,0,0,7,27,0,14,0,0,0,34,27,14,0,0,14,27,34,0,0,0,14,0,27,7],[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],[10,5,0,0,0,0,31,31,0,0,0,0,0,0,40,0,0,1,0,0,0,0,40,1,0,0,0,0,0,1,0,0,0,40,0,1] >;

C42.14F5 in GAP, Magma, Sage, TeX

C_4^2._{14}F_5
% in TeX

G:=Group("C4^2.14F5");
// GroupNames label

G:=SmallGroup(320,1020);
// by ID

G=gap.SmallGroup(320,1020);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,232,422,268,136,6278,1595]);
// 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^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^3>;
// generators/relations

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