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G = C42.276D10order 320 = 26·5

35th non-split extension by C42 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.276D10, (C2×C4)⋊8D20, (C4×D20)⋊3C2, (C2×C20)⋊31D4, C45(C4○D20), C4.90(C2×D20), (C2×C42)⋊11D5, C2011(C4○D4), C204D418C2, C207D450C2, C202Q838C2, C20.307(C2×D4), C10.5(C22×D4), C22.6(C2×D20), C2.7(C22×D20), C4.D2033C2, (C2×C10).21C24, (C2×C20).694C23, (C4×C20).315C22, (C22×C4).440D10, (C2×Dic5).5C23, (C22×D5).3C23, C22.64(C23×D5), (C2×D20).211C22, C4⋊Dic5.289C22, C51(C22.26C24), C23.218(C22×D5), D10⋊C4.80C22, (C22×C20).524C22, (C22×C10).383C23, (C2×Dic10).232C22, (C2×C4×C20)⋊13C2, (C2×C4○D20)⋊2C2, C10.8(C2×C4○D4), C2.10(C2×C4○D20), (C2×C10).172(C2×D4), (C2×C4×D5).238C22, (C2×C4).730(C22×D5), (C2×C5⋊D4).93C22, SmallGroup(320,1149)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.276D10
C1C5C10C2×C10C22×D5C2×D20C4×D20 — C42.276D10
C5C2×C10 — C42.276D10
C1C2×C4C2×C42

Generators and relations for C42.276D10
 G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=b2c-1 >

Subgroups: 1182 in 310 conjugacy classes, 119 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×6], C22, C22 [×2], C22 [×14], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×20], Q8 [×4], C23, C23 [×4], D5 [×4], C10, C10 [×2], C10 [×2], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], Dic5 [×4], C20 [×8], C20 [×2], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×8], D20 [×12], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×8], C2×C20 [×4], C22×D5 [×4], C22×C10, C22.26C24, C4⋊Dic5 [×4], D10⋊C4 [×8], C4×C20 [×2], C4×C20 [×2], C2×Dic10 [×2], C2×C4×D5 [×4], C2×D20 [×6], C4○D20 [×8], C2×C5⋊D4 [×4], C22×C20, C22×C20 [×2], C202Q8, C4×D20 [×4], C204D4, C4.D20 [×2], C207D4 [×4], C2×C4×C20, C2×C4○D20 [×2], C42.276D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], D20 [×4], C22×D5 [×7], C22.26C24, C2×D20 [×6], C4○D20 [×4], C23×D5, C22×D20, C2×C4○D20 [×2], C42.276D10

Smallest permutation representation of C42.276D10
On 160 points
Generators in S160
(1 113 33 100)(2 114 34 91)(3 115 35 92)(4 116 36 93)(5 117 37 94)(6 118 38 95)(7 119 39 96)(8 120 40 97)(9 111 31 98)(10 112 32 99)(11 123 43 101)(12 124 44 102)(13 125 45 103)(14 126 46 104)(15 127 47 105)(16 128 48 106)(17 129 49 107)(18 130 50 108)(19 121 41 109)(20 122 42 110)(21 85 143 63)(22 86 144 64)(23 87 145 65)(24 88 146 66)(25 89 147 67)(26 90 148 68)(27 81 149 69)(28 82 150 70)(29 83 141 61)(30 84 142 62)(51 154 74 131)(52 155 75 132)(53 156 76 133)(54 157 77 134)(55 158 78 135)(56 159 79 136)(57 160 80 137)(58 151 71 138)(59 152 72 139)(60 153 73 140)
(1 68 18 60)(2 69 19 51)(3 70 20 52)(4 61 11 53)(5 62 12 54)(6 63 13 55)(7 64 14 56)(8 65 15 57)(9 66 16 58)(10 67 17 59)(21 125 158 118)(22 126 159 119)(23 127 160 120)(24 128 151 111)(25 129 152 112)(26 130 153 113)(27 121 154 114)(28 122 155 115)(29 123 156 116)(30 124 157 117)(31 88 48 71)(32 89 49 72)(33 90 50 73)(34 81 41 74)(35 82 42 75)(36 83 43 76)(37 84 44 77)(38 85 45 78)(39 86 46 79)(40 87 47 80)(91 149 109 131)(92 150 110 132)(93 141 101 133)(94 142 102 134)(95 143 103 135)(96 144 104 136)(97 145 105 137)(98 146 106 138)(99 147 107 139)(100 148 108 140)
(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 99 18 107)(2 106 19 98)(3 97 20 105)(4 104 11 96)(5 95 12 103)(6 102 13 94)(7 93 14 101)(8 110 15 92)(9 91 16 109)(10 108 17 100)(21 77 158 84)(22 83 159 76)(23 75 160 82)(24 81 151 74)(25 73 152 90)(26 89 153 72)(27 71 154 88)(28 87 155 80)(29 79 156 86)(30 85 157 78)(31 114 48 121)(32 130 49 113)(33 112 50 129)(34 128 41 111)(35 120 42 127)(36 126 43 119)(37 118 44 125)(38 124 45 117)(39 116 46 123)(40 122 47 115)(51 146 69 138)(52 137 70 145)(53 144 61 136)(54 135 62 143)(55 142 63 134)(56 133 64 141)(57 150 65 132)(58 131 66 149)(59 148 67 140)(60 139 68 147)

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

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

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

92 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4N4O4P4Q4R5A5B10A···10N20A···20AV
order122222222244444···444445510···1020···20
size1111222020202011112···220202020222···22···2

92 irreducible representations

dim111111112222222
type+++++++++++++
imageC1C2C2C2C2C2C2C2D4D5C4○D4D10D10D20C4○D20
kernelC42.276D10C202Q8C4×D20C204D4C4.D20C207D4C2×C4×C20C2×C4○D20C2×C20C2×C42C20C42C22×C4C2×C4C4
# reps11412412428861632

Matrix representation of C42.276D10 in GL4(𝔽41) generated by

40000
04000
003028
002211
,
32000
03200
00320
00032
,
13700
213800
00321
003740
,
38400
18300
001932
002222
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,30,22,0,0,28,11],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[1,21,0,0,37,38,0,0,0,0,3,37,0,0,21,40],[38,18,0,0,4,3,0,0,0,0,19,22,0,0,32,22] >;

C42.276D10 in GAP, Magma, Sage, TeX

C_4^2._{276}D_{10}
% in TeX

G:=Group("C4^2.276D10");
// GroupNames label

G:=SmallGroup(320,1149);
// by ID

G=gap.SmallGroup(320,1149);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^-1>;
// generators/relations

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