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G = C10.55(C4×D4)  order 320 = 26·5

7th non-split extension by C10 of C4×D4 acting via C4×D4/C22⋊C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2.7(C4×D20), C10.55(C4×D4), D10⋊C46C4, (C2×C20).235D4, (C2×C4).113D20, C22.64(D4×D5), C2.2(C4⋊D20), (C22×C4).18D10, C2.C429D5, C22.26(C2×D20), C10.34(C4⋊D4), (C2×Dic5).130D4, (C23×D5).2C22, C10.10C424C2, C10.19(C4.4D4), C22.38(C4○D20), (C22×C20).15C22, C53(C24.C22), C23.260(C22×D5), C10.21(C422C2), C10.49(C42⋊C2), C2.10(Dic54D4), C22.39(D42D5), (C22×C10).295C23, C2.4(Dic5.5D4), C22.20(Q82D5), C2.2(C22.D20), C10.10(C22.D4), (C22×Dic5).200C22, (C2×C4⋊Dic5)⋊2C2, (C2×C4×Dic5)⋊18C2, (C2×C4).27(C4×D5), C22.93(C2×C4×D5), (C2×C20).208(C2×C4), C2.4(C4⋊C4⋊D5), C2.8(C4⋊C47D5), (C2×C10).204(C2×D4), (C2×D10⋊C4).7C2, (C2×Dic5).89(C2×C4), (C22×D5).13(C2×C4), (C2×C10).134(C4○D4), (C5×C2.C42)⋊16C2, (C2×C10).155(C22×C4), SmallGroup(320,297)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.55(C4×D4)
C1C5C10C2×C10C22×C10C23×D5C2×D10⋊C4 — C10.55(C4×D4)
C5C2×C10 — C10.55(C4×D4)
C1C23C2.C42

Generators and relations for C10.55(C4×D4)
 G = < a,b,c,d | a10=b4=c4=1, d2=a5, bab-1=cac-1=a-1, ad=da, bc=cb, dbd-1=a5b, dcd-1=a5c-1 >

Subgroups: 766 in 190 conjugacy classes, 69 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C4 [×10], C22 [×7], C22 [×10], C5, C2×C4 [×4], C2×C4 [×18], C23, C23 [×8], D5 [×2], C10 [×7], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×3], C22×C4 [×3], C24, Dic5 [×5], C20 [×5], D10 [×10], C2×C10 [×7], C2.C42, C2.C42, C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4, C2×Dic5 [×4], C2×Dic5 [×7], C2×C20 [×4], C2×C20 [×7], C22×D5 [×2], C22×D5 [×6], C22×C10, C24.C22, C4×Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×4], D10⋊C4 [×4], C22×Dic5 [×3], C22×C20 [×3], C23×D5, C10.10C42, C5×C2.C42, C2×C4×Dic5, C2×C4⋊Dic5, C2×D10⋊C4 [×3], C10.55(C4×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22×C4, C2×D4 [×2], C4○D4 [×4], D10 [×3], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C4×D5 [×2], D20 [×2], C22×D5, C24.C22, C2×C4×D5, C2×D20, C4○D20, D4×D5, D42D5 [×2], Q82D5, C4×D20, Dic54D4, Dic5.5D4, C22.D20, C4⋊C47D5, C4⋊D20, C4⋊C4⋊D5, C10.55(C4×D4)

Smallest permutation representation of C10.55(C4×D4)
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)
(1 138 38 11)(2 137 39 20)(3 136 40 19)(4 135 31 18)(5 134 32 17)(6 133 33 16)(7 132 34 15)(8 131 35 14)(9 140 36 13)(10 139 37 12)(21 147 43 160)(22 146 44 159)(23 145 45 158)(24 144 46 157)(25 143 47 156)(26 142 48 155)(27 141 49 154)(28 150 50 153)(29 149 41 152)(30 148 42 151)(51 115 88 102)(52 114 89 101)(53 113 90 110)(54 112 81 109)(55 111 82 108)(56 120 83 107)(57 119 84 106)(58 118 85 105)(59 117 86 104)(60 116 87 103)(61 129 74 92)(62 128 75 91)(63 127 76 100)(64 126 77 99)(65 125 78 98)(66 124 79 97)(67 123 80 96)(68 122 71 95)(69 121 72 94)(70 130 73 93)
(1 110 23 93)(2 109 24 92)(3 108 25 91)(4 107 26 100)(5 106 27 99)(6 105 28 98)(7 104 29 97)(8 103 30 96)(9 102 21 95)(10 101 22 94)(11 90 158 73)(12 89 159 72)(13 88 160 71)(14 87 151 80)(15 86 152 79)(16 85 153 78)(17 84 154 77)(18 83 155 76)(19 82 156 75)(20 81 157 74)(31 120 48 127)(32 119 49 126)(33 118 50 125)(34 117 41 124)(35 116 42 123)(36 115 43 122)(37 114 44 121)(38 113 45 130)(39 112 46 129)(40 111 47 128)(51 147 68 140)(52 146 69 139)(53 145 70 138)(54 144 61 137)(55 143 62 136)(56 142 63 135)(57 141 64 134)(58 150 65 133)(59 149 66 132)(60 148 67 131)
(1 110 6 105)(2 101 7 106)(3 102 8 107)(4 103 9 108)(5 104 10 109)(11 85 16 90)(12 86 17 81)(13 87 18 82)(14 88 19 83)(15 89 20 84)(21 91 26 96)(22 92 27 97)(23 93 28 98)(24 94 29 99)(25 95 30 100)(31 116 36 111)(32 117 37 112)(33 118 38 113)(34 119 39 114)(35 120 40 115)(41 126 46 121)(42 127 47 122)(43 128 48 123)(44 129 49 124)(45 130 50 125)(51 136 56 131)(52 137 57 132)(53 138 58 133)(54 139 59 134)(55 140 60 135)(61 146 66 141)(62 147 67 142)(63 148 68 143)(64 149 69 144)(65 150 70 145)(71 156 76 151)(72 157 77 152)(73 158 78 153)(74 159 79 154)(75 160 80 155)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G4H4I···4P4Q4R5A5B10A···10N20A···20X
order12···222444444444···4445510···1020···20
size11···120202222444410···102020222···24···4

68 irreducible representations

dim111111122222222444
type++++++++++++-+
imageC1C2C2C2C2C2C4D4D4D5C4○D4D10C4×D5D20C4○D20D4×D5D42D5Q82D5
kernelC10.55(C4×D4)C10.10C42C5×C2.C42C2×C4×Dic5C2×C4⋊Dic5C2×D10⋊C4D10⋊C4C2×Dic5C2×C20C2.C42C2×C10C22×C4C2×C4C2×C4C22C22C22C22
# reps111113822286888242

Matrix representation of C10.55(C4×D4) in GL5(𝔽41)

10000
01000
00100
00066
000351
,
320000
040000
004000
000236
0002118
,
10000
09200
003200
0003928
000162
,
400000
0323900
040900
0003928
000132

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,35,0,0,0,6,1],[32,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,23,21,0,0,0,6,18],[1,0,0,0,0,0,9,0,0,0,0,2,32,0,0,0,0,0,39,16,0,0,0,28,2],[40,0,0,0,0,0,32,40,0,0,0,39,9,0,0,0,0,0,39,13,0,0,0,28,2] >;

C10.55(C4×D4) in GAP, Magma, Sage, TeX

C_{10}._{55}(C_4\times D_4)
% in TeX

G:=Group("C10.55(C4xD4)");
// GroupNames label

G:=SmallGroup(320,297);
// by ID

G=gap.SmallGroup(320,297);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,758,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=c^4=1,d^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=a^5*c^-1>;
// generators/relations

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