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G = C4⋊C4.9F5order 320 = 26·5

7th non-split extension by C4⋊C4 of F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.9F5, C20.5(C4⋊C4), (C4×D5).7Q8, C20⋊C814C2, (C4×D5).28D4, C4.20(C4⋊F5), D10.9(C4⋊C4), C4⋊Dic5.12C4, C2.7(Q8.F5), D10⋊C4.4C4, C10.12(C8○D4), Dic5.12(C2×Q8), Dic5.30(C2×D4), C2.13(D4.F5), Dic5⋊C811C2, C22.80(C22×F5), C52(C42.6C22), (C4×Dic5).245C22, (C2×Dic5).335C23, C10.9(C2×C4⋊C4), C2.12(C2×C4⋊F5), (C5×C4⋊C4).12C4, (C2×D5⋊C8).4C2, (C2×C5⋊C8).6C22, (C2×C4).28(C2×F5), (C2×C4.F5).4C2, (C2×C20).86(C2×C4), C4⋊C47D5.19C2, (C2×C4×D5).191C22, (C2×C10).46(C22×C4), (C2×Dic5).61(C2×C4), (C22×D5).51(C2×C4), SmallGroup(320,1046)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4⋊C4.9F5
C1C5C10Dic5C2×Dic5C2×C5⋊C8C2×D5⋊C8 — C4⋊C4.9F5
C5C2×C10 — C4⋊C4.9F5
C1C22C4⋊C4

Generators and relations for C4⋊C4.9F5
 G = < a,b,c,d | a4=b4=c5=1, d4=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 378 in 114 conjugacy classes, 52 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×7], C23, D5 [×2], C10 [×3], C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8 [×6], M4(2) [×2], C22×C4, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4⋊C8 [×4], C42⋊C2, C22×C8, C2×M4(2), C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C42.6C22, C4×Dic5 [×2], C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4, D5⋊C8 [×2], C4.F5 [×2], C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C2×C4×D5, C20⋊C8 [×2], Dic5⋊C8 [×2], C4⋊C47D5, C2×D5⋊C8, C2×C4.F5, C4⋊C4.9F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C8○D4 [×2], C2×F5 [×3], C42.6C22, C4⋊F5 [×2], C22×F5, C2×C4⋊F5, D4.F5, Q8.F5, C4⋊C4.9F5

Smallest permutation representation of C4⋊C4.9F5
On 160 points
Generators in S160
(1 63 5 59)(2 64 6 60)(3 57 7 61)(4 58 8 62)(9 154 13 158)(10 155 14 159)(11 156 15 160)(12 157 16 153)(17 95 21 91)(18 96 22 92)(19 89 23 93)(20 90 24 94)(25 72 29 68)(26 65 30 69)(27 66 31 70)(28 67 32 71)(33 76 37 80)(34 77 38 73)(35 78 39 74)(36 79 40 75)(41 136 45 132)(42 129 46 133)(43 130 47 134)(44 131 48 135)(49 97 53 101)(50 98 54 102)(51 99 55 103)(52 100 56 104)(81 114 85 118)(82 115 86 119)(83 116 87 120)(84 117 88 113)(105 123 109 127)(106 124 110 128)(107 125 111 121)(108 126 112 122)(137 151 141 147)(138 152 142 148)(139 145 143 149)(140 146 144 150)
(1 98 57 52)(2 53 58 99)(3 100 59 54)(4 55 60 101)(5 102 61 56)(6 49 62 103)(7 104 63 50)(8 51 64 97)(9 126 80 26)(10 27 73 127)(11 128 74 28)(12 29 75 121)(13 122 76 30)(14 31 77 123)(15 124 78 32)(16 25 79 125)(17 150 119 131)(18 132 120 151)(19 152 113 133)(20 134 114 145)(21 146 115 135)(22 136 116 147)(23 148 117 129)(24 130 118 149)(33 69 154 108)(34 109 155 70)(35 71 156 110)(36 111 157 72)(37 65 158 112)(38 105 159 66)(39 67 160 106)(40 107 153 68)(41 87 141 92)(42 93 142 88)(43 81 143 94)(44 95 144 82)(45 83 137 96)(46 89 138 84)(47 85 139 90)(48 91 140 86)
(1 12 132 143 38)(2 144 13 39 133)(3 40 137 134 14)(4 135 33 15 138)(5 16 136 139 34)(6 140 9 35 129)(7 36 141 130 10)(8 131 37 11 142)(17 65 128 88 51)(18 81 66 52 121)(19 53 82 122 67)(20 123 54 68 83)(21 69 124 84 55)(22 85 70 56 125)(23 49 86 126 71)(24 127 50 72 87)(25 116 90 109 102)(26 110 117 103 91)(27 104 111 92 118)(28 93 97 119 112)(29 120 94 105 98)(30 106 113 99 95)(31 100 107 96 114)(32 89 101 115 108)(41 149 73 63 157)(42 64 150 158 74)(43 159 57 75 151)(44 76 160 152 58)(45 145 77 59 153)(46 60 146 154 78)(47 155 61 79 147)(48 80 156 148 62)
(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)| (1,63,5,59)(2,64,6,60)(3,57,7,61)(4,58,8,62)(9,154,13,158)(10,155,14,159)(11,156,15,160)(12,157,16,153)(17,95,21,91)(18,96,22,92)(19,89,23,93)(20,90,24,94)(25,72,29,68)(26,65,30,69)(27,66,31,70)(28,67,32,71)(33,76,37,80)(34,77,38,73)(35,78,39,74)(36,79,40,75)(41,136,45,132)(42,129,46,133)(43,130,47,134)(44,131,48,135)(49,97,53,101)(50,98,54,102)(51,99,55,103)(52,100,56,104)(81,114,85,118)(82,115,86,119)(83,116,87,120)(84,117,88,113)(105,123,109,127)(106,124,110,128)(107,125,111,121)(108,126,112,122)(137,151,141,147)(138,152,142,148)(139,145,143,149)(140,146,144,150), (1,98,57,52)(2,53,58,99)(3,100,59,54)(4,55,60,101)(5,102,61,56)(6,49,62,103)(7,104,63,50)(8,51,64,97)(9,126,80,26)(10,27,73,127)(11,128,74,28)(12,29,75,121)(13,122,76,30)(14,31,77,123)(15,124,78,32)(16,25,79,125)(17,150,119,131)(18,132,120,151)(19,152,113,133)(20,134,114,145)(21,146,115,135)(22,136,116,147)(23,148,117,129)(24,130,118,149)(33,69,154,108)(34,109,155,70)(35,71,156,110)(36,111,157,72)(37,65,158,112)(38,105,159,66)(39,67,160,106)(40,107,153,68)(41,87,141,92)(42,93,142,88)(43,81,143,94)(44,95,144,82)(45,83,137,96)(46,89,138,84)(47,85,139,90)(48,91,140,86), (1,12,132,143,38)(2,144,13,39,133)(3,40,137,134,14)(4,135,33,15,138)(5,16,136,139,34)(6,140,9,35,129)(7,36,141,130,10)(8,131,37,11,142)(17,65,128,88,51)(18,81,66,52,121)(19,53,82,122,67)(20,123,54,68,83)(21,69,124,84,55)(22,85,70,56,125)(23,49,86,126,71)(24,127,50,72,87)(25,116,90,109,102)(26,110,117,103,91)(27,104,111,92,118)(28,93,97,119,112)(29,120,94,105,98)(30,106,113,99,95)(31,100,107,96,114)(32,89,101,115,108)(41,149,73,63,157)(42,64,150,158,74)(43,159,57,75,151)(44,76,160,152,58)(45,145,77,59,153)(46,60,146,154,78)(47,155,61,79,147)(48,80,156,148,62), (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( (1,63,5,59)(2,64,6,60)(3,57,7,61)(4,58,8,62)(9,154,13,158)(10,155,14,159)(11,156,15,160)(12,157,16,153)(17,95,21,91)(18,96,22,92)(19,89,23,93)(20,90,24,94)(25,72,29,68)(26,65,30,69)(27,66,31,70)(28,67,32,71)(33,76,37,80)(34,77,38,73)(35,78,39,74)(36,79,40,75)(41,136,45,132)(42,129,46,133)(43,130,47,134)(44,131,48,135)(49,97,53,101)(50,98,54,102)(51,99,55,103)(52,100,56,104)(81,114,85,118)(82,115,86,119)(83,116,87,120)(84,117,88,113)(105,123,109,127)(106,124,110,128)(107,125,111,121)(108,126,112,122)(137,151,141,147)(138,152,142,148)(139,145,143,149)(140,146,144,150), (1,98,57,52)(2,53,58,99)(3,100,59,54)(4,55,60,101)(5,102,61,56)(6,49,62,103)(7,104,63,50)(8,51,64,97)(9,126,80,26)(10,27,73,127)(11,128,74,28)(12,29,75,121)(13,122,76,30)(14,31,77,123)(15,124,78,32)(16,25,79,125)(17,150,119,131)(18,132,120,151)(19,152,113,133)(20,134,114,145)(21,146,115,135)(22,136,116,147)(23,148,117,129)(24,130,118,149)(33,69,154,108)(34,109,155,70)(35,71,156,110)(36,111,157,72)(37,65,158,112)(38,105,159,66)(39,67,160,106)(40,107,153,68)(41,87,141,92)(42,93,142,88)(43,81,143,94)(44,95,144,82)(45,83,137,96)(46,89,138,84)(47,85,139,90)(48,91,140,86), (1,12,132,143,38)(2,144,13,39,133)(3,40,137,134,14)(4,135,33,15,138)(5,16,136,139,34)(6,140,9,35,129)(7,36,141,130,10)(8,131,37,11,142)(17,65,128,88,51)(18,81,66,52,121)(19,53,82,122,67)(20,123,54,68,83)(21,69,124,84,55)(22,85,70,56,125)(23,49,86,126,71)(24,127,50,72,87)(25,116,90,109,102)(26,110,117,103,91)(27,104,111,92,118)(28,93,97,119,112)(29,120,94,105,98)(30,106,113,99,95)(31,100,107,96,114)(32,89,101,115,108)(41,149,73,63,157)(42,64,150,158,74)(43,159,57,75,151)(44,76,160,152,58)(45,145,77,59,153)(46,60,146,154,78)(47,155,61,79,147)(48,80,156,148,62), (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([(1,63,5,59),(2,64,6,60),(3,57,7,61),(4,58,8,62),(9,154,13,158),(10,155,14,159),(11,156,15,160),(12,157,16,153),(17,95,21,91),(18,96,22,92),(19,89,23,93),(20,90,24,94),(25,72,29,68),(26,65,30,69),(27,66,31,70),(28,67,32,71),(33,76,37,80),(34,77,38,73),(35,78,39,74),(36,79,40,75),(41,136,45,132),(42,129,46,133),(43,130,47,134),(44,131,48,135),(49,97,53,101),(50,98,54,102),(51,99,55,103),(52,100,56,104),(81,114,85,118),(82,115,86,119),(83,116,87,120),(84,117,88,113),(105,123,109,127),(106,124,110,128),(107,125,111,121),(108,126,112,122),(137,151,141,147),(138,152,142,148),(139,145,143,149),(140,146,144,150)], [(1,98,57,52),(2,53,58,99),(3,100,59,54),(4,55,60,101),(5,102,61,56),(6,49,62,103),(7,104,63,50),(8,51,64,97),(9,126,80,26),(10,27,73,127),(11,128,74,28),(12,29,75,121),(13,122,76,30),(14,31,77,123),(15,124,78,32),(16,25,79,125),(17,150,119,131),(18,132,120,151),(19,152,113,133),(20,134,114,145),(21,146,115,135),(22,136,116,147),(23,148,117,129),(24,130,118,149),(33,69,154,108),(34,109,155,70),(35,71,156,110),(36,111,157,72),(37,65,158,112),(38,105,159,66),(39,67,160,106),(40,107,153,68),(41,87,141,92),(42,93,142,88),(43,81,143,94),(44,95,144,82),(45,83,137,96),(46,89,138,84),(47,85,139,90),(48,91,140,86)], [(1,12,132,143,38),(2,144,13,39,133),(3,40,137,134,14),(4,135,33,15,138),(5,16,136,139,34),(6,140,9,35,129),(7,36,141,130,10),(8,131,37,11,142),(17,65,128,88,51),(18,81,66,52,121),(19,53,82,122,67),(20,123,54,68,83),(21,69,124,84,55),(22,85,70,56,125),(23,49,86,126,71),(24,127,50,72,87),(25,116,90,109,102),(26,110,117,103,91),(27,104,111,92,118),(28,93,97,119,112),(29,120,94,105,98),(30,106,113,99,95),(31,100,107,96,114),(32,89,101,115,108),(41,149,73,63,157),(42,64,150,158,74),(43,159,57,75,151),(44,76,160,152,58),(45,145,77,59,153),(46,60,146,154,78),(47,155,61,79,147),(48,80,156,148,62)], [(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)])

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J 5 8A···8H8I8J8K8L10A10B10C20A···20F
order122222444444444458···8888810101020···20
size11111010224455552020410···10202020204448···8

38 irreducible representations

dim11111111122244488
type+++++++-++-+
imageC1C2C2C2C2C2C4C4C4D4Q8C8○D4F5C2×F5C4⋊F5D4.F5Q8.F5
kernelC4⋊C4.9F5C20⋊C8Dic5⋊C8C4⋊C47D5C2×D5⋊C8C2×C4.F5C4⋊Dic5D10⋊C4C5×C4⋊C4C4×D5C4×D5C10C4⋊C4C2×C4C4C2C2
# reps12211124222813411

Matrix representation of C4⋊C4.9F5 in GL6(𝔽41)

3200000
3890000
001000
000100
000010
000001
,
650000
9350000
003414027
00071427
00271470
002701434
,
100000
010000
0000040
0010040
0001040
0000140
,
1400000
32270000
00347270
00207034
00340720
00027734

G:=sub<GL(6,GF(41))| [32,38,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,9,0,0,0,0,5,35,0,0,0,0,0,0,34,0,27,27,0,0,14,7,14,0,0,0,0,14,7,14,0,0,27,27,0,34],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[14,32,0,0,0,0,0,27,0,0,0,0,0,0,34,20,34,0,0,0,7,7,0,27,0,0,27,0,7,7,0,0,0,34,20,34] >;

C4⋊C4.9F5 in GAP, Magma, Sage, TeX

C_4\rtimes C_4._9F_5
% in TeX

G:=Group("C4:C4.9F5");
// GroupNames label

G:=SmallGroup(320,1046);
// by ID

G=gap.SmallGroup(320,1046);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,100,136,6278,1595]);
// Polycyclic

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

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