Copied to
clipboard

G = C42.241D10order 320 = 26·5

61st non-split extension by C42 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.241D10, C4⋊Q820D5, (C4×D5)⋊5Q8, C4.40(Q8×D5), D10.5(C2×Q8), C20.54(C2×Q8), C4⋊C4.219D10, C202Q836C2, (Q8×Dic5)⋊22C2, (C2×Q8).147D10, C4.Dic1042C2, Dic5.34(C2×Q8), (D5×C42).10C2, Dic53Q842C2, C20.136(C4○D4), C4.41(D42D5), C10.48(C22×Q8), (C2×C20).105C23, (C4×C20).213C22, (C2×C10).272C24, D103Q8.12C2, D102Q8.14C2, C4.22(Q82D5), C4⋊Dic5.251C22, (Q8×C10).139C22, C22.293(C23×D5), D10⋊C4.51C22, C56(C23.37C23), (C2×Dic5).143C23, (C4×Dic5).169C22, C10.D4.61C22, (C22×D5).243C23, (C2×Dic10).196C22, C2.31(C2×Q8×D5), (C5×C4⋊Q8)⋊14C2, C4⋊C47D5.14C2, C10.100(C2×C4○D4), C2.64(C2×D42D5), C2.29(C2×Q82D5), (C2×C4×D5).322C22, (C5×C4⋊C4).215C22, (C2×C4).600(C22×D5), SmallGroup(320,1400)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.241D10
C1C5C10C2×C10C22×D5C2×C4×D5D5×C42 — C42.241D10
C5C2×C10 — C42.241D10
C1C22C4⋊Q8

Generators and relations for C42.241D10
 G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2c9 >

Subgroups: 654 in 222 conjugacy classes, 111 normal (33 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×12], C22, C22 [×4], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×15], Q8 [×8], C23, D5 [×2], C10 [×3], C42, C42 [×7], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×3], C2×Q8 [×2], C2×Q8 [×2], Dic5 [×2], Dic5 [×6], C20 [×6], C20 [×4], D10 [×2], D10 [×2], C2×C10, C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8, C4⋊Q8, Dic10 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5 [×3], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×4], C22×D5, C23.37C23, C4×Dic5 [×3], C4×Dic5 [×4], C10.D4 [×4], C4⋊Dic5 [×8], D10⋊C4 [×4], C4×C20, C5×C4⋊C4 [×4], C2×Dic10 [×2], C2×C4×D5 [×3], Q8×C10 [×2], C202Q8, D5×C42, Dic53Q8 [×2], C4.Dic10 [×2], C4⋊C47D5 [×2], D102Q8 [×2], Q8×Dic5 [×2], D103Q8 [×2], C5×C4⋊Q8, C42.241D10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×4], C24, D10 [×7], C22×Q8, C2×C4○D4 [×2], C22×D5 [×7], C23.37C23, D42D5 [×2], Q8×D5 [×2], Q82D5 [×2], C23×D5, C2×D42D5, C2×Q8×D5, C2×Q82D5, C42.241D10

Smallest permutation representation of C42.241D10
On 160 points
Generators in S160
(1 77 97 29)(2 30 98 78)(3 79 99 31)(4 32 100 80)(5 61 81 33)(6 34 82 62)(7 63 83 35)(8 36 84 64)(9 65 85 37)(10 38 86 66)(11 67 87 39)(12 40 88 68)(13 69 89 21)(14 22 90 70)(15 71 91 23)(16 24 92 72)(17 73 93 25)(18 26 94 74)(19 75 95 27)(20 28 96 76)(41 132 116 154)(42 155 117 133)(43 134 118 156)(44 157 119 135)(45 136 120 158)(46 159 101 137)(47 138 102 160)(48 141 103 139)(49 140 104 142)(50 143 105 121)(51 122 106 144)(52 145 107 123)(53 124 108 146)(54 147 109 125)(55 126 110 148)(56 149 111 127)(57 128 112 150)(58 151 113 129)(59 130 114 152)(60 153 115 131)
(1 114 11 104)(2 105 12 115)(3 116 13 106)(4 107 14 117)(5 118 15 108)(6 109 16 119)(7 120 17 110)(8 111 18 101)(9 102 19 112)(10 113 20 103)(21 122 31 132)(22 133 32 123)(23 124 33 134)(24 135 34 125)(25 126 35 136)(26 137 36 127)(27 128 37 138)(28 139 38 129)(29 130 39 140)(30 121 40 131)(41 89 51 99)(42 100 52 90)(43 91 53 81)(44 82 54 92)(45 93 55 83)(46 84 56 94)(47 95 57 85)(48 86 58 96)(49 97 59 87)(50 88 60 98)(61 156 71 146)(62 147 72 157)(63 158 73 148)(64 149 74 159)(65 160 75 150)(66 151 76 141)(67 142 77 152)(68 153 78 143)(69 144 79 154)(70 155 80 145)
(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 10 87 96)(2 95 88 9)(3 8 89 94)(4 93 90 7)(5 6 91 92)(11 20 97 86)(12 85 98 19)(13 18 99 84)(14 83 100 17)(15 16 81 82)(21 26 79 64)(22 63 80 25)(23 24 61 62)(27 40 65 78)(28 77 66 39)(29 38 67 76)(30 75 68 37)(31 36 69 74)(32 73 70 35)(33 34 71 72)(41 46 106 111)(42 110 107 45)(43 44 108 109)(47 60 112 105)(48 104 113 59)(49 58 114 103)(50 102 115 57)(51 56 116 101)(52 120 117 55)(53 54 118 119)(121 160 153 128)(122 127 154 159)(123 158 155 126)(124 125 156 157)(129 152 141 140)(130 139 142 151)(131 150 143 138)(132 137 144 149)(133 148 145 136)(134 135 146 147)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H4I4J4K4L4M4N4O4P4Q4R4S4T4U4V5A5B10A···10F20A···20L20M···20T
order1222224···444444444444444445510···1020···2020···20
size111110102···2444455551010101020202020222···24···48···8

56 irreducible representations

dim1111111111222222444
type++++++++++-++++--+
imageC1C2C2C2C2C2C2C2C2C2Q8D5C4○D4D10D10D10D42D5Q8×D5Q82D5
kernelC42.241D10C202Q8D5×C42Dic53Q8C4.Dic10C4⋊C47D5D102Q8Q8×Dic5D103Q8C5×C4⋊Q8C4×D5C4⋊Q8C20C42C4⋊C4C2×Q8C4C4C4
# reps1112222221428284444

Matrix representation of C42.241D10 in GL6(𝔽41)

900000
0320000
001000
000100
0000320
000009
,
100000
010000
001000
000100
0000320
000009
,
010000
100000
00353500
0064000
000001
0000400
,
0400000
100000
00353500
0040600
0000040
0000400

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;

C42.241D10 in GAP, Magma, Sage, TeX

C_4^2._{241}D_{10}
% in TeX

G:=Group("C4^2.241D10");
// GroupNames label

G:=SmallGroup(320,1400);
// by ID

G=gap.SmallGroup(320,1400);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,1123,570,185,80,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^9>;
// generators/relations

׿
×
𝔽