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G = C10.(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: C10.7(C4⋊D4), (C2×Dic5).21D4, (C22×D5).17D4, C2.C424D5, (C22×C4).20D10, C22.159(D4×D5), C2.6(C422D5), C52(C23.11D4), C2.10(D10⋊D4), (C23×D5).7C22, C10.10C427C2, C10.21(C4.4D4), C22.92(C4○D20), (C22×C20).19C22, C23.363(C22×D5), C10.23(C422C2), C22.90(D42D5), (C22×C10).300C23, C22.47(Q82D5), C2.11(D10.12D4), C2.10(D10.13D4), C2.11(Dic5.5D4), C10.12(C22.D4), (C22×Dic5).22C22, (C2×C10).207(C2×D4), (C2×C10.D4)⋊3C2, C2.11(C4⋊C4⋊D5), (C5×C2.C42)⋊1C2, (C2×D10⋊C4).10C2, (C2×C10).135(C4○D4), SmallGroup(320,302)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C10.(C4⋊D4)
C1C5C10C2×C10C22×C10C23×D5C2×D10⋊C4 — C10.(C4⋊D4)
C5C22×C10 — C10.(C4⋊D4)
C1C23C2.C42

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

Subgroups: 742 in 170 conjugacy classes, 55 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C4 [×7], C22 [×7], C22 [×10], C5, C2×C4 [×19], C23, C23 [×8], D5 [×2], C10 [×7], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×3], C22×C4 [×3], C24, Dic5 [×4], C20 [×3], D10 [×10], C2×C10 [×7], C2.C42, C2.C42 [×2], C2×C22⋊C4 [×3], C2×C4⋊C4, C2×Dic5 [×2], C2×Dic5 [×8], C2×C20 [×9], C22×D5 [×2], C22×D5 [×6], C22×C10, C23.11D4, C10.D4 [×2], D10⋊C4 [×6], C22×Dic5 [×3], C22×C20 [×3], C23×D5, C10.10C42 [×2], C5×C2.C42, C2×C10.D4, C2×D10⋊C4 [×3], C10.(C4⋊D4)
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4 [×5], D10 [×3], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C22×D5, C23.11D4, C4○D20 [×3], D4×D5 [×2], D42D5, Q82D5, C422D5, D10.12D4, D10⋊D4, Dic5.5D4, D10.13D4 [×2], C4⋊C4⋊D5, C10.(C4⋊D4)

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

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

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

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

62 conjugacy classes

class 1 2A···2G2H2I4A···4F4G···4L5A5B10A···10N20A···20X
order12···2224···44···45510···1020···20
size11···120204···420···20222···24···4

62 irreducible representations

dim11111222222444
type++++++++++-+
imageC1C2C2C2C2D4D4D5C4○D4D10C4○D20D4×D5D42D5Q82D5
kernelC10.(C4⋊D4)C10.10C42C5×C2.C42C2×C10.D4C2×D10⋊C4C2×Dic5C22×D5C2.C42C2×C10C22×C4C22C22C22C22
# reps1211322210624422

Matrix representation of C10.(C4⋊D4) in GL6(𝔽41)

660000
3510000
0040700
0034700
000010
000001
,
2130000
25390000
00221900
0091900
000010
000001
,
900000
13320000
003300
00243800
00001537
00003626
,
900000
090000
00244000
0011700
00001537
00001526

G:=sub<GL(6,GF(41))| [6,35,0,0,0,0,6,1,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,25,0,0,0,0,13,39,0,0,0,0,0,0,22,9,0,0,0,0,19,19,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,13,0,0,0,0,0,32,0,0,0,0,0,0,3,24,0,0,0,0,3,38,0,0,0,0,0,0,15,36,0,0,0,0,37,26],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,15,15,0,0,0,0,37,26] >;

C10.(C4⋊D4) in GAP, Magma, Sage, TeX

C_{10}.(C_4\rtimes D_4)
% in TeX

G:=Group("C10.(C4:D4)");
// GroupNames label

G:=SmallGroup(320,302);
// by ID

G=gap.SmallGroup(320,302);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,64,590,387,100,12550]);
// 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,c*b*c^-1=a^5*b^-1,d*b*d^-1=b^-1,d*c*d^-1=a^5*c^-1>;
// generators/relations

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