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G = C4.89(C2×D20)  order 320 = 26·5

16th central extension by C4 of C2×D20

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4.89(C2×D20), (C2×D20).26C4, (C2×C20).171D4, (C2×C8).189D10, (C2×C4).152D20, C20.444(C2×D4), (C2×M4(2))⋊9D5, D101C840C2, C23.29(C4×D5), C10.58(C8○D4), C20.73(C22⋊C4), (C10×M4(2))⋊17C2, (C2×C20).869C23, (C2×C40).319C22, (C2×Dic10).27C4, (C22×C4).350D10, C4.12(D10⋊C4), C2.19(D20.2C4), C22.2(D10⋊C4), (C22×C20).186C22, (C2×C4).84(C4×D5), (C22×C52C8)⋊6C2, (C2×C5⋊D4).21C4, C4.135(C2×C5⋊D4), C22.148(C2×C4×D5), (C2×C20).279(C2×C4), C56((C22×C8)⋊C2), (C2×C4○D20).11C2, C10.97(C2×C22⋊C4), (C2×C4×D5).237C22, C2.28(C2×D10⋊C4), (C2×C4).141(C5⋊D4), (C2×Dic5).37(C2×C4), (C22×D5).31(C2×C4), (C2×C4).811(C22×D5), (C2×C10).84(C22⋊C4), (C2×C10).240(C22×C4), (C22×C10).137(C2×C4), (C2×C52C8).333C22, SmallGroup(320,756)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4.89(C2×D20)
C1C5C10C20C2×C20C2×C4×D5C2×C4○D20 — C4.89(C2×D20)
C5C2×C10 — C4.89(C2×D20)
C1C2×C4C2×M4(2)

Generators and relations for C4.89(C2×D20)
 G = < a,b,c,d | a4=b2=1, c20=a2, d2=a, ab=ba, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=a-1c19 >

Subgroups: 574 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×8], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×2], C2×C8 [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8 [×4], C22×C8, C2×M4(2), C2×C4○D4, C52C8 [×2], C40 [×2], Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×C10, (C22×C8)⋊C2, C2×C52C8 [×2], C2×C52C8 [×2], C2×C40 [×2], C5×M4(2) [×2], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, D101C8 [×4], C22×C52C8, C10×M4(2), C2×C4○D20, C4.89(C2×D20)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C8○D4 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, (C22×C8)⋊C2, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, D20.2C4 [×2], C2×D10⋊C4, C4.89(C2×D20)

Smallest permutation representation of C4.89(C2×D20)
On 160 points
Generators in S160
(1 117 21 97)(2 118 22 98)(3 119 23 99)(4 120 24 100)(5 81 25 101)(6 82 26 102)(7 83 27 103)(8 84 28 104)(9 85 29 105)(10 86 30 106)(11 87 31 107)(12 88 32 108)(13 89 33 109)(14 90 34 110)(15 91 35 111)(16 92 36 112)(17 93 37 113)(18 94 38 114)(19 95 39 115)(20 96 40 116)(41 142 61 122)(42 143 62 123)(43 144 63 124)(44 145 64 125)(45 146 65 126)(46 147 66 127)(47 148 67 128)(48 149 68 129)(49 150 69 130)(50 151 70 131)(51 152 71 132)(52 153 72 133)(53 154 73 134)(54 155 74 135)(55 156 75 136)(56 157 76 137)(57 158 77 138)(58 159 78 139)(59 160 79 140)(60 121 80 141)
(1 130)(2 151)(3 132)(4 153)(5 134)(6 155)(7 136)(8 157)(9 138)(10 159)(11 140)(12 121)(13 142)(14 123)(15 144)(16 125)(17 146)(18 127)(19 148)(20 129)(21 150)(22 131)(23 152)(24 133)(25 154)(26 135)(27 156)(28 137)(29 158)(30 139)(31 160)(32 141)(33 122)(34 143)(35 124)(36 145)(37 126)(38 147)(39 128)(40 149)(41 109)(42 90)(43 111)(44 92)(45 113)(46 94)(47 115)(48 96)(49 117)(50 98)(51 119)(52 100)(53 81)(54 102)(55 83)(56 104)(57 85)(58 106)(59 87)(60 108)(61 89)(62 110)(63 91)(64 112)(65 93)(66 114)(67 95)(68 116)(69 97)(70 118)(71 99)(72 120)(73 101)(74 82)(75 103)(76 84)(77 105)(78 86)(79 107)(80 88)
(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)
(1 116 117 20 21 96 97 40)(2 19 118 95 22 39 98 115)(3 94 119 38 23 114 99 18)(4 37 120 113 24 17 100 93)(5 112 81 16 25 92 101 36)(6 15 82 91 26 35 102 111)(7 90 83 34 27 110 103 14)(8 33 84 109 28 13 104 89)(9 108 85 12 29 88 105 32)(10 11 86 87 30 31 106 107)(41 137 142 56 61 157 122 76)(42 55 143 156 62 75 123 136)(43 155 144 74 63 135 124 54)(44 73 145 134 64 53 125 154)(45 133 146 52 65 153 126 72)(46 51 147 152 66 71 127 132)(47 151 148 70 67 131 128 50)(48 69 149 130 68 49 129 150)(57 121 158 80 77 141 138 60)(58 79 159 140 78 59 139 160)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E···8L10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222222444444445588888···810···101010101020···202020202040···40
size1111222020111122202022444410···102···244442···244444···4

68 irreducible representations

dim111111112222222224
type++++++++++
imageC1C2C2C2C2C4C4C4D4D5D10D10C8○D4C4×D5D20C5⋊D4C4×D5D20.2C4
kernelC4.89(C2×D20)D101C8C22×C52C8C10×M4(2)C2×C4○D20C2×Dic10C2×D20C2×C5⋊D4C2×C20C2×M4(2)C2×C8C22×C4C10C2×C4C2×C4C2×C4C23C2
# reps141112244242848848

Matrix representation of C4.89(C2×D20) in GL4(𝔽41) generated by

40000
04000
0090
0009
,
1000
0100
0092
00132
,
211700
31800
00380
00273
,
63800
263500
00380
00038
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,9,1,0,0,2,32],[21,3,0,0,17,18,0,0,0,0,38,27,0,0,0,3],[6,26,0,0,38,35,0,0,0,0,38,0,0,0,0,38] >;

C4.89(C2×D20) in GAP, Magma, Sage, TeX

C_4._{89}(C_2\times D_{20})
% in TeX

G:=Group("C4.89(C2xD20)");
// GroupNames label

G:=SmallGroup(320,756);
// by ID

G=gap.SmallGroup(320,756);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,422,387,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^20=a^2,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^-1*c^19>;
// generators/relations

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