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G = C20.76(C4⋊C4)  order 320 = 26·5

23rd non-split extension by C20 of C4⋊C4 acting via C4⋊C4/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.76(C4⋊C4), (C2×C20).22Q8, C20.65(C2×Q8), C4⋊C4.234D10, (C2×C20).495D4, C10.84(C4○D8), C10.D844C2, C20.Q843C2, (C2×C4).32Dic10, C4.30(C2×Dic10), C42⋊C2.6D5, (C22×C10).73D4, C20.122(C22×C4), (C2×C20).327C23, (C22×C4).337D10, C54(C23.25D4), C23.37(C5⋊D4), C2.1(D4.8D10), C4.32(C10.D4), C4⋊Dic5.325C22, (C22×C20).148C22, C22.9(C10.D4), C23.21D10.13C2, C4.88(C2×C4×D5), (C2×C52C8)⋊10C4, C10.63(C2×C4⋊C4), C52C8.41(C2×C4), (C2×C4).155(C4×D5), (C2×C10).40(C4⋊C4), (C2×C20).262(C2×C4), (C2×C10).456(C2×D4), (C22×C52C8).7C2, C22.71(C2×C5⋊D4), (C2×C4).273(C5⋊D4), (C5×C4⋊C4).265C22, (C5×C42⋊C2).7C2, C2.14(C2×C10.D4), (C2×C4).427(C22×D5), (C2×C52C8).250C22, SmallGroup(320,625)

Series: Derived Chief Lower central Upper central

C1C20 — C20.76(C4⋊C4)
C1C5C10C2×C10C2×C20C2×C52C8C22×C52C8 — C20.76(C4⋊C4)
C5C10C20 — C20.76(C4⋊C4)
C1C2×C4C22×C4C42⋊C2

Generators and relations for C20.76(C4⋊C4)
 G = < a,b,c | a20=c4=1, b4=a10, bab-1=a9, ac=ca, cbc-1=a10b3 >

Subgroups: 302 in 114 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×6], C22×C4, Dic5 [×2], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C42⋊C2, C22×C8, C52C8 [×4], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×C10, C23.25D4, C2×C52C8 [×2], C2×C52C8 [×4], C4×Dic5, C4⋊Dic5 [×2], C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4 [×2], C22×C20, C10.D8 [×2], C20.Q8 [×2], C22×C52C8, C23.21D10, C5×C42⋊C2, C20.76(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C4○D8 [×2], Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, C23.25D4, C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×C10.D4, D4.8D10 [×2], C20.76(C4⋊C4)

Smallest permutation representation of C20.76(C4⋊C4)
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)
(1 138 25 108 11 128 35 118)(2 127 26 117 12 137 36 107)(3 136 27 106 13 126 37 116)(4 125 28 115 14 135 38 105)(5 134 29 104 15 124 39 114)(6 123 30 113 16 133 40 103)(7 132 31 102 17 122 21 112)(8 121 32 111 18 131 22 101)(9 130 33 120 19 140 23 110)(10 139 34 109 20 129 24 119)(41 155 67 94 51 145 77 84)(42 144 68 83 52 154 78 93)(43 153 69 92 53 143 79 82)(44 142 70 81 54 152 80 91)(45 151 71 90 55 141 61 100)(46 160 72 99 56 150 62 89)(47 149 73 88 57 159 63 98)(48 158 74 97 58 148 64 87)(49 147 75 86 59 157 65 96)(50 156 76 95 60 146 66 85)
(1 50 40 61)(2 51 21 62)(3 52 22 63)(4 53 23 64)(5 54 24 65)(6 55 25 66)(7 56 26 67)(8 57 27 68)(9 58 28 69)(10 59 29 70)(11 60 30 71)(12 41 31 72)(13 42 32 73)(14 43 33 74)(15 44 34 75)(16 45 35 76)(17 46 36 77)(18 47 37 78)(19 48 38 79)(20 49 39 80)(81 119 157 134)(82 120 158 135)(83 101 159 136)(84 102 160 137)(85 103 141 138)(86 104 142 139)(87 105 143 140)(88 106 144 121)(89 107 145 122)(90 108 146 123)(91 109 147 124)(92 110 148 125)(93 111 149 126)(94 112 150 127)(95 113 151 128)(96 114 152 129)(97 115 153 130)(98 116 154 131)(99 117 155 132)(100 118 156 133)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B8A···8H10A···10F10G10H10I10J20A···20H20I···20AB
order12222244444444444444558···810···101010101020···2020···20
size1111221111224444202020202210···102···244442···24···4

68 irreducible representations

dim1111111222222222224
type+++++++-++++-
imageC1C2C2C2C2C2C4D4Q8D4D5D10D10C4○D8Dic10C4×D5C5⋊D4C5⋊D4D4.8D10
kernelC20.76(C4⋊C4)C10.D8C20.Q8C22×C52C8C23.21D10C5×C42⋊C2C2×C52C8C2×C20C2×C20C22×C10C42⋊C2C4⋊C4C22×C4C10C2×C4C2×C4C2×C4C23C2
# reps1221118121242888448

Matrix representation of C20.76(C4⋊C4) in GL5(𝔽41)

10000
034800
040100
000320
000032
,
400000
014900
0102700
000270
0001138
,
90000
017700
0352400
00055
0002836

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,34,40,0,0,0,8,1,0,0,0,0,0,32,0,0,0,0,0,32],[40,0,0,0,0,0,14,10,0,0,0,9,27,0,0,0,0,0,27,11,0,0,0,0,38],[9,0,0,0,0,0,17,35,0,0,0,7,24,0,0,0,0,0,5,28,0,0,0,5,36] >;

C20.76(C4⋊C4) in GAP, Magma, Sage, TeX

C_{20}._{76}(C_4\rtimes C_4)
% in TeX

G:=Group("C20.76(C4:C4)");
// GroupNames label

G:=SmallGroup(320,625);
// by ID

G=gap.SmallGroup(320,625);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,387,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=c^4=1,b^4=a^10,b*a*b^-1=a^9,a*c=c*a,c*b*c^-1=a^10*b^3>;
// generators/relations

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