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G = C5×M4(2)⋊C4order 320 = 26·5

Direct product of C5 and M4(2)⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×M4(2)⋊C4, M4(2)⋊1C20, C81(C2×C20), C4029(C2×C4), C4.Q82C10, C4.4(Q8×C10), C2.D810C10, C20.95(C4⋊C4), C20.93(C2×Q8), (C2×C20).43Q8, (C2×C20).522D4, C23.40(C5×D4), (C5×M4(2))⋊10C4, C4.27(C22×C20), C22.50(D4×C10), C20.244(C22×C4), (C2×C20).901C23, (C2×C40).269C22, C42⋊C2.7C10, (C22×C10).162D4, (C10×M4(2)).7C2, (C2×M4(2)).1C10, C10.128(C8⋊C22), C10.128(C8.C22), (C22×C20).414C22, C4.15(C5×C4⋊C4), C10.93(C2×C4⋊C4), C2.14(C10×C4⋊C4), (C2×C4).6(C5×Q8), (C5×C4.Q8)⋊11C2, (C5×C2.D8)⋊25C2, C2.3(C5×C8⋊C22), (C2×C4⋊C4).15C10, (C10×C4⋊C4).44C2, C4⋊C4.44(C2×C10), (C2×C4).25(C2×C20), (C2×C8).16(C2×C10), (C2×C4).125(C5×D4), C22.10(C5×C4⋊C4), (C2×C10).55(C4⋊C4), C2.3(C5×C8.C22), (C2×C20).371(C2×C4), (C2×C10).626(C2×D4), (C5×C4⋊C4).365C22, (C2×C4).76(C22×C10), (C22×C4).33(C2×C10), (C5×C42⋊C2).21C2, SmallGroup(320,929)

Series: Derived Chief Lower central Upper central

C1C4 — C5×M4(2)⋊C4
C1C2C22C2×C4C2×C20C5×C4⋊C4C5×C4.Q8 — C5×M4(2)⋊C4
C1C2C4 — C5×M4(2)⋊C4
C1C2×C10C22×C20 — C5×M4(2)⋊C4

Generators and relations for C5×M4(2)⋊C4
 G = < a,b,c,d | a5=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, cd=dc >

Subgroups: 178 in 118 conjugacy classes, 82 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4, C20 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C2×M4(2), C40 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×6], C22×C10, M4(2)⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C5×C4⋊C4, C2×C40 [×2], C5×M4(2) [×4], C22×C20, C22×C20, C5×C4.Q8 [×2], C5×C2.D8 [×2], C10×C4⋊C4, C5×C42⋊C2, C10×M4(2), C5×M4(2)⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], Q8 [×2], C23, C10 [×7], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C20 [×4], C2×C10 [×7], C2×C4⋊C4, C8⋊C22, C8.C22, C2×C20 [×6], C5×D4 [×2], C5×Q8 [×2], C22×C10, M4(2)⋊C4, C5×C4⋊C4 [×4], C22×C20, D4×C10, Q8×C10, C10×C4⋊C4, C5×C8⋊C22, C5×C8.C22, C5×M4(2)⋊C4

Smallest permutation representation of C5×M4(2)⋊C4
On 160 points
Generators in S160
(1 85 75 33 67)(2 86 76 34 68)(3 87 77 35 69)(4 88 78 36 70)(5 81 79 37 71)(6 82 80 38 72)(7 83 73 39 65)(8 84 74 40 66)(9 27 57 54 19)(10 28 58 55 20)(11 29 59 56 21)(12 30 60 49 22)(13 31 61 50 23)(14 32 62 51 24)(15 25 63 52 17)(16 26 64 53 18)(41 154 116 146 108)(42 155 117 147 109)(43 156 118 148 110)(44 157 119 149 111)(45 158 120 150 112)(46 159 113 151 105)(47 160 114 152 106)(48 153 115 145 107)(89 127 143 97 135)(90 128 144 98 136)(91 121 137 99 129)(92 122 138 100 130)(93 123 139 101 131)(94 124 140 102 132)(95 125 141 103 133)(96 126 142 104 134)
(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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)(34 38)(36 40)(41 45)(43 47)(49 53)(51 55)(58 62)(60 64)(66 70)(68 72)(74 78)(76 80)(82 86)(84 88)(89 93)(91 95)(97 101)(99 103)(106 110)(108 112)(114 118)(116 120)(121 125)(123 127)(129 133)(131 135)(137 141)(139 143)(146 150)(148 152)(154 158)(156 160)
(1 126 13 111)(2 125 14 110)(3 124 15 109)(4 123 16 108)(5 122 9 107)(6 121 10 106)(7 128 11 105)(8 127 12 112)(17 147 69 94)(18 146 70 93)(19 145 71 92)(20 152 72 91)(21 151 65 90)(22 150 66 89)(23 149 67 96)(24 148 68 95)(25 42 87 140)(26 41 88 139)(27 48 81 138)(28 47 82 137)(29 46 83 144)(30 45 84 143)(31 44 85 142)(32 43 86 141)(33 134 50 119)(34 133 51 118)(35 132 52 117)(36 131 53 116)(37 130 54 115)(38 129 55 114)(39 136 56 113)(40 135 49 120)(57 153 79 100)(58 160 80 99)(59 159 73 98)(60 158 74 97)(61 157 75 104)(62 156 76 103)(63 155 77 102)(64 154 78 101)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4L5A5B5C5D8A8B8C8D10A···10L10M···10T20A···20P20Q···20AV40A···40P
order12222244444···45555888810···1010···1020···2020···2040···40
size11112222224···4111144441···12···22···24···44···4

110 irreducible representations

dim111111111111112222224444
type+++++++-++-
imageC1C2C2C2C2C2C4C5C10C10C10C10C10C20D4Q8D4C5×D4C5×Q8C5×D4C8⋊C22C8.C22C5×C8⋊C22C5×C8.C22
kernelC5×M4(2)⋊C4C5×C4.Q8C5×C2.D8C10×C4⋊C4C5×C42⋊C2C10×M4(2)C5×M4(2)M4(2)⋊C4C4.Q8C2.D8C2×C4⋊C4C42⋊C2C2×M4(2)M4(2)C2×C20C2×C20C22×C10C2×C4C2×C4C23C10C10C2C2
# reps1221118488444321214841144

Matrix representation of C5×M4(2)⋊C4 in GL6(𝔽41)

100000
010000
0010000
0001000
0000100
0000010
,
40390000
110000
000010
000001
0004000
001000
,
4000000
0400000
001000
000100
0000400
0000040
,
9180000
0320000
0092400
00243200
0000179
0000924

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[40,1,0,0,0,0,39,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,18,32,0,0,0,0,0,0,9,24,0,0,0,0,24,32,0,0,0,0,0,0,17,9,0,0,0,0,9,24] >;

C5×M4(2)⋊C4 in GAP, Magma, Sage, TeX

C_5\times M_4(2)\rtimes C_4
% in TeX

G:=Group("C5xM4(2):C4");
// GroupNames label

G:=SmallGroup(320,929);
// by ID

G=gap.SmallGroup(320,929);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,288,1731,7004,172]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b^-1,c*d=d*c>;
// generators/relations

׿
×
𝔽