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G = C42.72D10order 320 = 26·5

72nd non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.72D10, C41D4.4D5, (C2×D4).54D10, (C2×C20).291D4, C20.75(C4○D4), D4⋊Dic521C2, C20.6Q813C2, C4.23(D42D5), C10.93(C8⋊C22), (C2×C20).389C23, (C4×C20).119C22, (D4×C10).70C22, C42.D512C2, C10.44(C4.4D4), C4⋊Dic5.155C22, C2.14(D4.D10), C2.11(C20.17D4), C54(C42.29C22), (C5×C41D4).3C2, (C2×C10).520(C2×D4), (C2×C4).69(C5⋊D4), (C2×C4).487(C22×D5), C22.193(C2×C5⋊D4), (C2×C52C8).129C22, SmallGroup(320,698)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.72D10
C1C5C10C2×C10C2×C20C2×C52C8C42.D5 — C42.72D10
C5C10C2×C20 — C42.72D10
C1C22C42C41D4

Generators and relations for C42.72D10
 G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >

Subgroups: 366 in 110 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×8], C23 [×2], C10, C10 [×2], C10 [×2], C42, C4⋊C4 [×4], C2×C8 [×2], C2×D4 [×2], C2×D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], C8⋊C4, D4⋊C4 [×4], C42.C2, C41D4, C52C8 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×D4 [×8], C22×C10 [×2], C42.29C22, C2×C52C8 [×2], C10.D4 [×2], C4⋊Dic5 [×2], C4×C20, D4×C10 [×2], D4×C10 [×2], C42.D5, D4⋊Dic5 [×4], C20.6Q8, C5×C41D4, C42.72D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C8⋊C22 [×2], C5⋊D4 [×2], C22×D5, C42.29C22, D42D5 [×2], C2×C5⋊D4, D4.D10 [×2], C20.17D4, C42.72D10

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D10A···10F10G···10N20A···20L
order12222244444455888810···1010···1020···20
size1111882244404022202020202···28···84···4

44 irreducible representations

dim11111222222444
type++++++++++-
imageC1C2C2C2C2D4D5C4○D4D10D10C5⋊D4C8⋊C22D42D5D4.D10
kernelC42.72D10C42.D5D4⋊Dic5C20.6Q8C5×C41D4C2×C20C41D4C20C42C2×D4C2×C4C10C4C2
# reps11411224248248

Matrix representation of C42.72D10 in GL6(𝔽41)

1230000
32400000
00403900
001100
000012
00004040
,
4000000
0400000
00403900
001100
00004039
000011
,
100000
32400000
00373300
000400
0000100
00003131
,
900000
1320000
00001020
0000031
004800
0003700

G:=sub<GL(6,GF(41))| [1,32,0,0,0,0,23,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,1,40,0,0,0,0,2,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,37,0,0,0,0,0,33,4,0,0,0,0,0,0,10,31,0,0,0,0,0,31],[9,1,0,0,0,0,0,32,0,0,0,0,0,0,0,0,4,0,0,0,0,0,8,37,0,0,10,0,0,0,0,0,20,31,0,0] >;

C42.72D10 in GAP, Magma, Sage, TeX

C_4^2._{72}D_{10}
% in TeX

G:=Group("C4^2.72D10");
// GroupNames label

G:=SmallGroup(320,698);
// by ID

G=gap.SmallGroup(320,698);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,64,590,135,438,102,12550]);
// Polycyclic

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

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