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G = C42.125D10order 320 = 26·5

125th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.125D10, C10.92- 1+4, (C4×Q8)⋊6D5, (Q8×D5)⋊5C4, (Q8×C20)⋊7C2, (Q8×Dic5)⋊8C2, Q8.12(C4×D5), C4⋊C4.323D10, (C4×Dic10)⋊38C2, C10.46(C23×C4), C20.70(C22×C4), (C2×Q8).200D10, C42⋊D5.3C2, Dic53Q818C2, (C2×C10).116C24, (C2×C20).495C23, (C4×C20).168C22, Dic10.35(C2×C4), D10.41(C22×C4), C22.35(C23×D5), C4⋊Dic5.366C22, (Q8×C10).216C22, Dic5.19(C22×C4), (C4×Dic5).92C22, C2.4(D4.10D10), C2.2(Q8.10D10), C53(C23.32C23), (C2×Dic5).222C23, (C22×D5).185C23, D10⋊C4.124C22, (C2×Dic10).298C22, C10.D4.137C22, C4.35(C2×C4×D5), (C2×Q8×D5).6C2, (C4×D5).9(C2×C4), C2.27(D5×C22×C4), (C5×Q8).31(C2×C4), (C2×C4×D5).78C22, C4⋊C47D5.10C2, (C5×C4⋊C4).344C22, (C2×C4).288(C22×D5), SmallGroup(320,1244)

Series: Derived Chief Lower central Upper central

C1C10 — C42.125D10
C1C5C10C2×C10C22×D5C2×C4×D5C2×Q8×D5 — C42.125D10
C5C10 — C42.125D10
C1C22C4×Q8

Generators and relations for C42.125D10
 G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c9 >

Subgroups: 718 in 266 conjugacy classes, 151 normal (22 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×14], C22, C22 [×4], C5, C2×C4, C2×C4 [×6], C2×C4 [×19], Q8 [×4], Q8 [×12], C23, D5 [×2], C10 [×3], C42 [×3], C42 [×9], C22⋊C4 [×4], C4⋊C4 [×3], C4⋊C4 [×9], C22×C4 [×3], C2×Q8, C2×Q8 [×11], Dic5 [×6], Dic5 [×4], C20 [×6], C20 [×4], D10 [×2], D10 [×2], C2×C10, C42⋊C2 [×6], C4×Q8, C4×Q8 [×7], C22×Q8, Dic10 [×12], C4×D5 [×12], C2×Dic5, C2×Dic5 [×6], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5, C23.32C23, C4×Dic5 [×9], C10.D4 [×6], C4⋊Dic5 [×3], D10⋊C4, D10⋊C4 [×3], C4×C20 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×3], C2×C4×D5 [×3], Q8×D5 [×8], Q8×C10, C4×Dic10 [×3], C42⋊D5 [×3], Dic53Q8 [×3], C4⋊C47D5 [×3], Q8×Dic5, Q8×C20, C2×Q8×D5, C42.125D10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, 2- 1+4 [×2], C4×D5 [×4], C22×D5 [×7], C23.32C23, C2×C4×D5 [×6], C23×D5, D5×C22×C4, Q8.10D10, D4.10D10, C42.125D10

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

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E4A···4N4O···4AB5A5B10A···10F20A···20H20I···20AF
order1222224···44···45510···1020···2020···20
size111110102···210···10222···22···24···4

74 irreducible representations

dim11111111122222444
type++++++++++++--
imageC1C2C2C2C2C2C2C2C4D5D10D10D10C4×D52- 1+4Q8.10D10D4.10D10
kernelC42.125D10C4×Dic10C42⋊D5Dic53Q8C4⋊C47D5Q8×Dic5Q8×C20C2×Q8×D5Q8×D5C4×Q8C42C4⋊C4C2×Q8Q8C10C2C2
# reps1333311116266216244

Matrix representation of C42.125D10 in GL6(𝔽41)

4000000
0400000
0011010
0001101
0010300
0001030
,
900000
090000
00244000
0011700
00002440
0000117
,
160000
3560000
0000407
0000347
0013400
0073400
,
100000
35400000
0000740
0000734
0034100
0034700

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,11,0,1,0,0,0,0,11,0,1,0,0,1,0,30,0,0,0,0,1,0,30],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[1,35,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,7,0,0,0,0,34,34,0,0,40,34,0,0,0,0,7,7,0,0],[1,35,0,0,0,0,0,40,0,0,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,7,7,0,0,0,0,40,34,0,0] >;

C42.125D10 in GAP, Magma, Sage, TeX

C_4^2._{125}D_{10}
% in TeX

G:=Group("C4^2.125D10");
// GroupNames label

G:=SmallGroup(320,1244);
// by ID

G=gap.SmallGroup(320,1244);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,1123,80,12550]);
// 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=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations

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×
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