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G = C42.180D10order 320 = 26·5

180th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.180D10, C10.862+ 1+4, C10.412- 1+4, C4⋊Q818D5, C4⋊C4.127D10, (C2×Q8).90D10, D10⋊Q849C2, D103Q839C2, C422D520C2, Dic5⋊Q828C2, (C2×C20).641C23, (C4×C20).274C22, (C2×C10).279C24, C2.90(D46D10), Dic5.Q843C2, D10.13D4.5C2, C20.23D4.11C2, (C2×D20).180C22, C4⋊Dic5.256C22, (Q8×C10).146C22, C22.300(C23×D5), D10⋊C4.76C22, C56(C22.57C24), (C4×Dic5).176C22, (C2×Dic5).147C23, C10.D4.88C22, (C22×D5).124C23, C2.42(Q8.10D10), (C2×Dic10).199C22, (C5×C4⋊Q8)⋊21C2, C4⋊C4⋊D548C2, (C2×C4×D5).161C22, (C5×C4⋊C4).222C22, (C2×C4).222(C22×D5), SmallGroup(320,1407)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.180D10
C1C5C10C2×C10C22×D5C2×C4×D5D103Q8 — C42.180D10
C5C2×C10 — C42.180D10
C1C22C4⋊Q8

Generators and relations for C42.180D10
 G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b, dcd-1=c9 >

Subgroups: 678 in 196 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4, Q8 [×3], C23 [×2], D5 [×2], C10, C10 [×2], C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×2], C2×D4, C2×Q8 [×2], C2×Q8, Dic5 [×6], C20 [×7], D10 [×6], C2×C10, C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C422C2 [×4], C4⋊Q8, C4⋊Q8, Dic10, C4×D5 [×2], D20, C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×2], C22.57C24, C4×Dic5 [×2], C10.D4 [×10], C4⋊Dic5 [×2], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, Q8×C10 [×2], C422D5 [×2], Dic5.Q8 [×2], D10.13D4 [×2], D10⋊Q8 [×2], C4⋊C4⋊D5 [×2], Dic5⋊Q8, D103Q8 [×2], C20.23D4, C5×C4⋊Q8, C42.180D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4, 2- 1+4 [×2], C22×D5 [×7], C22.57C24, C23×D5, D46D10, Q8.10D10 [×2], C42.180D10

Smallest permutation representation of C42.180D10
On 160 points
Generators in S160
(1 50 11 60)(2 41 12 51)(3 52 13 42)(4 43 14 53)(5 54 15 44)(6 45 16 55)(7 56 17 46)(8 47 18 57)(9 58 19 48)(10 49 20 59)(21 121 31 131)(22 132 32 122)(23 123 33 133)(24 134 34 124)(25 125 35 135)(26 136 36 126)(27 127 37 137)(28 138 38 128)(29 129 39 139)(30 140 40 130)(61 81 71 91)(62 92 72 82)(63 83 73 93)(64 94 74 84)(65 85 75 95)(66 96 76 86)(67 87 77 97)(68 98 78 88)(69 89 79 99)(70 100 80 90)(101 147 111 157)(102 158 112 148)(103 149 113 159)(104 160 114 150)(105 151 115 141)(106 142 116 152)(107 153 117 143)(108 144 118 154)(109 155 119 145)(110 146 120 156)
(1 25 84 112)(2 113 85 26)(3 27 86 114)(4 115 87 28)(5 29 88 116)(6 117 89 30)(7 31 90 118)(8 119 91 32)(9 33 92 120)(10 101 93 34)(11 35 94 102)(12 103 95 36)(13 37 96 104)(14 105 97 38)(15 39 98 106)(16 107 99 40)(17 21 100 108)(18 109 81 22)(19 23 82 110)(20 111 83 24)(41 159 75 136)(42 137 76 160)(43 141 77 138)(44 139 78 142)(45 143 79 140)(46 121 80 144)(47 145 61 122)(48 123 62 146)(49 147 63 124)(50 125 64 148)(51 149 65 126)(52 127 66 150)(53 151 67 128)(54 129 68 152)(55 153 69 130)(56 131 70 154)(57 155 71 132)(58 133 72 156)(59 157 73 134)(60 135 74 158)
(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)(21 115 31 105)(22 104 32 114)(23 113 33 103)(24 102 34 112)(25 111 35 101)(26 120 36 110)(27 109 37 119)(28 118 38 108)(29 107 39 117)(30 116 40 106)(41 72 51 62)(42 61 52 71)(43 70 53 80)(44 79 54 69)(45 68 55 78)(46 77 56 67)(47 66 57 76)(48 75 58 65)(49 64 59 74)(50 73 60 63)(81 86 91 96)(82 95 92 85)(83 84 93 94)(87 100 97 90)(88 89 98 99)(121 128 131 138)(122 137 132 127)(123 126 133 136)(124 135 134 125)(129 140 139 130)(141 144 151 154)(142 153 152 143)(145 160 155 150)(146 149 156 159)(147 158 157 148)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A···4G4H···4M5A5B10A···10F20A···20L20M···20T
order1222224···44···45510···1020···2020···20
size111120204···420···20222···24···48···8

47 irreducible representations

dim111111111122224444
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D5D10D10D102+ 1+42- 1+4D46D10Q8.10D10
kernelC42.180D10C422D5Dic5.Q8D10.13D4D10⋊Q8C4⋊C4⋊D5Dic5⋊Q8D103Q8C20.23D4C5×C4⋊Q8C4⋊Q8C42C4⋊C4C2×Q8C10C10C2C2
# reps122222121122841248

Matrix representation of C42.180D10 in GL10(𝔽41)

1000000000
0100000000
0000100000
0000010000
00400000000
00040000000
0000000010
0000000001
00000040000
00000004000
,
40000000000
04000000000
0001000000
00400000000
0000010000
00004000000
0000000100
0000001000
0000000001
0000000010
,
03500000000
73400000000
00032000000
00320000000
0000090000
0000900000
00000000320
00000000032
00000032000
00000003200
,
73500000000
83400000000
00032000000
00320000000
00000320000
00003200000
00000000320
0000000009
00000032000
0000000900

G:=sub<GL(10,GF(41))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,7,0,0,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,32,0,0],[7,8,0,0,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,9,0,0] >;

C42.180D10 in GAP, Magma, Sage, TeX

C_4^2._{180}D_{10}
% in TeX

G:=Group("C4^2.180D10");
// GroupNames label

G:=SmallGroup(320,1407);
// by ID

G=gap.SmallGroup(320,1407);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,100,1571,570,136,12550]);
// Polycyclic

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

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