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G = C42.135D10order 320 = 26·5

135th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.135D10, C10.672- 1+4, (C4×Q8)⋊17D5, (Q8×C20)⋊19C2, C4⋊C4.302D10, C202Q829C2, (C4×D20).23C2, D103Q811C2, (C4×Dic10)⋊41C2, C4.69(C4○D20), (C2×Q8).183D10, C20.122(C4○D4), (C4×C20).180C22, (C2×C10).128C24, (C2×C20).591C23, C4.50(Q82D5), C4.D20.13C2, (C2×D20).227C22, C4⋊Dic5.400C22, (Q8×C10).228C22, (C22×D5).50C23, C22.149(C23×D5), D10⋊C4.56C22, C54(C22.50C24), (C4×Dic5).230C22, (C2×Dic5).228C23, C2.25(D4.10D10), (C2×Dic10).251C22, C10.D4.157C22, C4⋊C4⋊D511C2, C4⋊C47D518C2, C2.67(C2×C4○D20), (C2×C4×D5).87C22, C10.113(C2×C4○D4), C2.13(C2×Q82D5), (C5×C4⋊C4).356C22, (C2×C4).172(C22×D5), SmallGroup(320,1256)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.135D10
C1C5C10C2×C10C22×D5C2×D20C4×D20 — C42.135D10
C5C2×C10 — C42.135D10
C1C22C4×Q8

Generators and relations for C42.135D10
 G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=c9 >

Subgroups: 694 in 212 conjugacy classes, 99 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×11], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×10], D4 [×2], Q8 [×6], C23 [×2], D5 [×2], C10 [×3], C42, C42 [×2], C42 [×4], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×6], C20 [×4], C20 [×5], D10 [×6], C2×C10, C42⋊C2 [×2], C4×D4, C4×Q8, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C422C2 [×4], C4⋊Q8, Dic10 [×4], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×2], C22.50C24, C4×Dic5 [×4], C10.D4 [×4], C4⋊Dic5, C4⋊Dic5 [×4], D10⋊C4 [×10], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, Q8×C10, C4×Dic10 [×2], C202Q8, C4×D20, C4.D20 [×2], C4⋊C47D5 [×2], C4⋊C4⋊D5 [×4], D103Q8 [×2], Q8×C20, C42.135D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- 1+4, C22×D5 [×7], C22.50C24, C4○D20 [×2], Q82D5 [×2], C23×D5, C2×C4○D20, C2×Q82D5, D4.10D10, C42.135D10

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

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J4K4L4M4N4O4P4Q4R4S5A5B10A···10F20A···20H20I···20AF
order1222224···4444444444445510···1020···2020···20
size111120202···24441010101020202020222···22···24···4

65 irreducible representations

dim111111111222222444
type+++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2D5C4○D4D10D10D10C4○D202- 1+4Q82D5D4.10D10
kernelC42.135D10C4×Dic10C202Q8C4×D20C4.D20C4⋊C47D5C4⋊C4⋊D5D103Q8Q8×C20C4×Q8C20C42C4⋊C4C2×Q8C4C10C4C2
# reps1211224212866216144

Matrix representation of C42.135D10 in GL4(𝔽41) generated by

23200
373900
0010
0001
,
9000
0900
004023
00321
,
353400
6000
0090
004032
,
64000
353500
0090
0009
G:=sub<GL(4,GF(41))| [2,37,0,0,32,39,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,40,32,0,0,23,1],[35,6,0,0,34,0,0,0,0,0,9,40,0,0,0,32],[6,35,0,0,40,35,0,0,0,0,9,0,0,0,0,9] >;

C42.135D10 in GAP, Magma, Sage, TeX

C_4^2._{135}D_{10}
% in TeX

G:=Group("C4^2.135D10");
// GroupNames label

G:=SmallGroup(320,1256);
// by ID

G=gap.SmallGroup(320,1256);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,185,192,12550]);
// Polycyclic

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

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