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G = C20.51(C4⋊C4)  order 320 = 26·5

20th non-split extension by C20 of C4⋊C4 acting via C4⋊C4/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.51(C4⋊C4), (C2×C20).25Q8, C20.88(C2×Q8), (C2×C20).165D4, (C2×C8).186D10, C20.439(C2×D4), C4⋊Dic5.32C4, C23.27(C4×D5), C10.54(C8○D4), C20.8Q839C2, C4.53(C2×Dic10), (C2×C4).35Dic10, C23.D5.17C4, (C2×C40).316C22, (C2×C20).864C23, (C22×C4).346D10, (C2×M4(2)).13D5, C4.19(C10.D4), (C10×M4(2)).24C2, C56(C42.6C22), C2.16(D20.2C4), (C22×C20).178C22, (C4×Dic5).208C22, C23.21D10.15C2, C22.11(C10.D4), C10.72(C2×C4⋊C4), (C2×C4).82(C4×D5), C4.129(C2×C5⋊D4), (C2×C10).43(C4⋊C4), C22.144(C2×C4×D5), (C2×C20).271(C2×C4), (C2×C4).193(C5⋊D4), (C22×C52C8).10C2, (C2×Dic5).35(C2×C4), C2.17(C2×C10.D4), (C2×C4).806(C22×D5), (C2×C10).235(C22×C4), (C22×C10).132(C2×C4), (C2×C52C8).330C22, SmallGroup(320,746)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20.51(C4⋊C4)
C1C5C10C20C2×C20C4×Dic5C23.21D10 — C20.51(C4⋊C4)
C5C2×C10 — C20.51(C4⋊C4)
C1C2×C4C2×M4(2)

Generators and relations for C20.51(C4⋊C4)
 G = < a,b,c | a20=b4=1, c4=a10, bab-1=a-1, cac-1=a11, cbc-1=a10b-1 >

Subgroups: 286 in 114 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×4], M4(2) [×2], C22×C4, Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4⋊C8 [×4], C42⋊C2, C22×C8, C2×M4(2), C52C8 [×2], C40 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, C42.6C22, C2×C52C8 [×2], C2×C52C8 [×2], C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×2], C2×C40 [×2], C5×M4(2) [×2], C22×C20, C20.8Q8 [×4], C22×C52C8, C23.21D10, C10×M4(2), C20.51(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C8○D4 [×2], Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, C42.6C22, C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, D20.2C4 [×2], C2×C10.D4, C20.51(C4⋊C4)

Smallest permutation representation of C20.51(C4⋊C4)
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 92 37 46)(2 91 38 45)(3 90 39 44)(4 89 40 43)(5 88 21 42)(6 87 22 41)(7 86 23 60)(8 85 24 59)(9 84 25 58)(10 83 26 57)(11 82 27 56)(12 81 28 55)(13 100 29 54)(14 99 30 53)(15 98 31 52)(16 97 32 51)(17 96 33 50)(18 95 34 49)(19 94 35 48)(20 93 36 47)(61 141 108 123)(62 160 109 122)(63 159 110 121)(64 158 111 140)(65 157 112 139)(66 156 113 138)(67 155 114 137)(68 154 115 136)(69 153 116 135)(70 152 117 134)(71 151 118 133)(72 150 119 132)(73 149 120 131)(74 148 101 130)(75 147 102 129)(76 146 103 128)(77 145 104 127)(78 144 105 126)(79 143 106 125)(80 142 107 124)
(1 68 6 63 11 78 16 73)(2 79 7 74 12 69 17 64)(3 70 8 65 13 80 18 75)(4 61 9 76 14 71 19 66)(5 72 10 67 15 62 20 77)(21 119 26 114 31 109 36 104)(22 110 27 105 32 120 37 115)(23 101 28 116 33 111 38 106)(24 112 29 107 34 102 39 117)(25 103 30 118 35 113 40 108)(41 149 56 154 51 159 46 144)(42 160 57 145 52 150 47 155)(43 151 58 156 53 141 48 146)(44 142 59 147 54 152 49 157)(45 153 60 158 55 143 50 148)(81 125 96 130 91 135 86 140)(82 136 97 121 92 126 87 131)(83 127 98 132 93 137 88 122)(84 138 99 123 94 128 89 133)(85 129 100 134 95 139 90 124)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E···8L10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222244444444445588888···810···101010101020···202020202040···40
size1111221111222020202022444410···102···244442···244444···4

68 irreducible representations

dim111111122222222224
type++++++-+++-
imageC1C2C2C2C2C4C4D4Q8D5D10D10C8○D4Dic10C4×D5C5⋊D4C4×D5D20.2C4
kernelC20.51(C4⋊C4)C20.8Q8C22×C52C8C23.21D10C10×M4(2)C4⋊Dic5C23.D5C2×C20C2×C20C2×M4(2)C2×C8C22×C4C10C2×C4C2×C4C2×C4C23C2
# reps141114422242884848

Matrix representation of C20.51(C4⋊C4) in GL4(𝔽41) generated by

9000
83200
00310
00354
,
342600
17700
0052
002836
,
16500
82500
0090
003732
G:=sub<GL(4,GF(41))| [9,8,0,0,0,32,0,0,0,0,31,35,0,0,0,4],[34,17,0,0,26,7,0,0,0,0,5,28,0,0,2,36],[16,8,0,0,5,25,0,0,0,0,9,37,0,0,0,32] >;

C20.51(C4⋊C4) in GAP, Magma, Sage, TeX

C_{20}._{51}(C_4\rtimes C_4)
% in TeX

G:=Group("C20.51(C4:C4)");
// GroupNames label

G:=SmallGroup(320,746);
// by ID

G=gap.SmallGroup(320,746);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,422,387,58,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^4=1,c^4=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b^-1>;
// generators/relations

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