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G = C203M4(2)  order 320 = 26·5

1st semidirect product of C20 and M4(2) acting via M4(2)/C4=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C203M4(2), C42.13F5, (C4×C20).5C4, C20⋊C87C2, C42(C4.F5), C4.9(C4⋊F5), (C4×D5).78D4, C20.16(C4⋊C4), (C4×D5).20Q8, Dic5.9(C2×Q8), D10.25(C4⋊C4), C51(C4⋊M4(2)), (C4×Dic5).35C4, Dic5.27(C2×D4), (D5×C42).18C2, C10.6(C2×M4(2)), C22.62(C22×F5), (C2×Dic5).317C23, (C4×Dic5).345C22, C2.6(C2×C4⋊F5), C10.2(C2×C4⋊C4), (C2×C4×D5).36C4, C2.6(C2×C4.F5), (C2×C5⋊C8).1C22, (C2×C4.F5).7C2, (C2×C4).132(C2×F5), (C2×C20).121(C2×C4), (C2×C4×D5).390C22, (C2×C10).19(C22×C4), (C2×Dic5).167(C2×C4), (C22×D5).119(C2×C4), SmallGroup(320,1019)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C203M4(2)
C1C5C10Dic5C2×Dic5C2×C5⋊C8C20⋊C8 — C203M4(2)
C5C2×C10 — C203M4(2)
C1C22C42

Generators and relations for C203M4(2)
 G = < a,b,c | a20=b8=c2=1, bab-1=a7, cac=a9, cbc=b5 >

Subgroups: 426 in 126 conjugacy classes, 60 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×6], C4 [×4], C22, C22 [×4], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, D5 [×2], C10, C10 [×2], C42, C42 [×3], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×6], D10 [×2], D10 [×2], C2×C10, C4⋊C8 [×4], C2×C42, C2×M4(2) [×2], C5⋊C8 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C4⋊M4(2), C4×Dic5, C4×Dic5 [×2], C4×C20, C4.F5 [×4], C2×C5⋊C8 [×4], C2×C4×D5, C2×C4×D5 [×2], C20⋊C8 [×4], D5×C42, C2×C4.F5 [×2], C203M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C2×M4(2) [×2], C2×F5 [×3], C4⋊M4(2), C4.F5 [×4], C4⋊F5 [×2], C22×F5, C2×C4.F5 [×2], C2×C4⋊F5, C203M4(2)

Smallest permutation representation of C203M4(2)
On 160 points
Generators in S160
(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)
(1 111 41 129 160 65 40 82)(2 114 50 136 141 68 29 89)(3 117 59 123 142 71 38 96)(4 120 48 130 143 74 27 83)(5 103 57 137 144 77 36 90)(6 106 46 124 145 80 25 97)(7 109 55 131 146 63 34 84)(8 112 44 138 147 66 23 91)(9 115 53 125 148 69 32 98)(10 118 42 132 149 72 21 85)(11 101 51 139 150 75 30 92)(12 104 60 126 151 78 39 99)(13 107 49 133 152 61 28 86)(14 110 58 140 153 64 37 93)(15 113 47 127 154 67 26 100)(16 116 56 134 155 70 35 87)(17 119 45 121 156 73 24 94)(18 102 54 128 157 76 33 81)(19 105 43 135 158 79 22 88)(20 108 52 122 159 62 31 95)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 29)(22 38)(23 27)(24 36)(26 34)(28 32)(31 39)(33 37)(42 50)(43 59)(44 48)(45 57)(47 55)(49 53)(52 60)(54 58)(61 115)(62 104)(63 113)(64 102)(65 111)(66 120)(67 109)(68 118)(69 107)(70 116)(71 105)(72 114)(73 103)(74 112)(75 101)(76 110)(77 119)(78 108)(79 117)(80 106)(81 140)(82 129)(83 138)(84 127)(85 136)(86 125)(87 134)(88 123)(89 132)(90 121)(91 130)(92 139)(93 128)(94 137)(95 126)(96 135)(97 124)(98 133)(99 122)(100 131)(141 149)(142 158)(143 147)(144 156)(146 154)(148 152)(151 159)(153 157)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H4I4J4K4L4M4N 5 8A···8H10A10B10C20A···20L
order1222224···44444444458···810101020···20
size111110102···2555510101010420···204444···4

44 irreducible representations

dim11111112224444
type+++++-++
imageC1C2C2C2C4C4C4D4Q8M4(2)F5C2×F5C4.F5C4⋊F5
kernelC203M4(2)C20⋊C8D5×C42C2×C4.F5C4×Dic5C4×C20C2×C4×D5C4×D5C4×D5C20C42C2×C4C4C4
# reps14122242281384

Matrix representation of C203M4(2) in GL6(𝔽41)

120000
40400000
000100
00403500
000016
0000356
,
3990000
2720000
000016
0000040
0032000
0003200
,
4000000
0400000
001600
0004000
00004035
000001

G:=sub<GL(6,GF(41))| [1,40,0,0,0,0,2,40,0,0,0,0,0,0,0,40,0,0,0,0,1,35,0,0,0,0,0,0,1,35,0,0,0,0,6,6],[39,27,0,0,0,0,9,2,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,1,0,0,0,0,0,6,40,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,6,40,0,0,0,0,0,0,40,0,0,0,0,0,35,1] >;

C203M4(2) in GAP, Magma, Sage, TeX

C_{20}\rtimes_3M_4(2)
% in TeX

G:=Group("C20:3M4(2)");
// GroupNames label

G:=SmallGroup(320,1019);
// by ID

G=gap.SmallGroup(320,1019);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,120,758,184,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^7,c*a*c=a^9,c*b*c=b^5>;
// generators/relations

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