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G = C4⋊C4.228D10order 320 = 26·5

6th non-split extension by C4⋊C4 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.228D10, (C2×C20).284D4, D206C426C2, C4.87(C4○D20), C20.55D45C2, C207D4.10C2, C20.Q825C2, C10.67(C2×SD16), (C2×C10).41SD16, (C22×C4).95D10, C22.8(Q8⋊D5), C20.175(C4○D4), C10.84(C8⋊C22), (C2×C20).321C23, (C2×D20).95C22, (C22×C10).186D4, C54(C23.46D4), C23.78(C5⋊D4), C2.6(D4.D10), C4⋊Dic5.131C22, (C22×C20).136C22, C2.9(C23.23D10), C10.59(C22.D4), (C2×C4⋊C4)⋊4D5, (C10×C4⋊C4)⋊4C2, C2.5(C2×Q8⋊D5), (C2×C10).441(C2×D4), (C2×C4).32(C5⋊D4), (C5×C4⋊C4).259C22, (C2×C52C8).82C22, (C2×C4).421(C22×D5), C22.131(C2×C5⋊D4), SmallGroup(320,595)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4⋊C4.228D10
C1C5C10C20C2×C20C2×D20C207D4 — C4⋊C4.228D10
C5C10C2×C20 — C4⋊C4.228D10
C1C22C22×C4C2×C4⋊C4

Generators and relations for C4⋊C4.228D10
 G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2b2c-1 >

Subgroups: 462 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×2], Dic5, C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, D4⋊C4 [×2], C4.Q8 [×2], C2×C4⋊C4, C4⋊D4, C52C8 [×2], D20 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×6], C22×D5, C22×C10, C23.46D4, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×D20, C2×C5⋊D4, C22×C20, C22×C20, C20.Q8 [×2], D206C4 [×2], C20.55D4, C207D4, C10×C4⋊C4, C4⋊C4.228D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×SD16, C8⋊C22, C5⋊D4 [×2], C22×D5, C23.46D4, Q8⋊D5 [×2], C4○D20 [×2], C2×C5⋊D4, C23.23D10, D4.D10, C2×Q8⋊D5, C4⋊C4.228D10

Smallest permutation representation of C4⋊C4.228D10
On 160 points
Generators in S160
(1 36 123 70)(2 37 124 61)(3 38 125 62)(4 39 126 63)(5 40 127 64)(6 31 128 65)(7 32 129 66)(8 33 130 67)(9 34 121 68)(10 35 122 69)(11 119 86 143)(12 120 87 144)(13 111 88 145)(14 112 89 146)(15 113 90 147)(16 114 81 148)(17 115 82 149)(18 116 83 150)(19 117 84 141)(20 118 85 142)(21 104 55 138)(22 105 56 139)(23 106 57 140)(24 107 58 131)(25 108 59 132)(26 109 60 133)(27 110 51 134)(28 101 52 135)(29 102 53 136)(30 103 54 137)(41 158 75 98)(42 159 76 99)(43 160 77 100)(44 151 78 91)(45 152 79 92)(46 153 80 93)(47 154 71 94)(48 155 72 95)(49 156 73 96)(50 157 74 97)
(1 75 60 114)(2 76 51 115)(3 77 52 116)(4 78 53 117)(5 79 54 118)(6 80 55 119)(7 71 56 120)(8 72 57 111)(9 73 58 112)(10 74 59 113)(11 31 153 138)(12 32 154 139)(13 33 155 140)(14 34 156 131)(15 35 157 132)(16 36 158 133)(17 37 159 134)(18 38 160 135)(19 39 151 136)(20 40 152 137)(21 143 128 46)(22 144 129 47)(23 145 130 48)(24 146 121 49)(25 147 122 50)(26 148 123 41)(27 149 124 42)(28 150 125 43)(29 141 126 44)(30 142 127 45)(61 99 110 82)(62 100 101 83)(63 91 102 84)(64 92 103 85)(65 93 104 86)(66 94 105 87)(67 95 106 88)(68 96 107 89)(69 97 108 90)(70 98 109 81)
(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 10 26 25)(2 24 27 9)(3 8 28 23)(4 22 29 7)(5 6 30 21)(11 45 93 118)(12 117 94 44)(13 43 95 116)(14 115 96 42)(15 41 97 114)(16 113 98 50)(17 49 99 112)(18 111 100 48)(19 47 91 120)(20 119 92 46)(31 137 104 64)(32 63 105 136)(33 135 106 62)(34 61 107 134)(35 133 108 70)(36 69 109 132)(37 131 110 68)(38 67 101 140)(39 139 102 66)(40 65 103 138)(51 121 124 58)(52 57 125 130)(53 129 126 56)(54 55 127 128)(59 123 122 60)(71 151 144 84)(72 83 145 160)(73 159 146 82)(74 81 147 158)(75 157 148 90)(76 89 149 156)(77 155 150 88)(78 87 141 154)(79 153 142 86)(80 85 143 152)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C···4G4H5A5B8A8B8C8D10A···10N20A···20X
order1222222444···4455888810···1020···20
size11112240224···44022202020202···24···4

59 irreducible representations

dim1111112222222222444
type+++++++++++++
imageC1C2C2C2C2C2D4D4D5C4○D4SD16D10D10C5⋊D4C5⋊D4C4○D20C8⋊C22Q8⋊D5D4.D10
kernelC4⋊C4.228D10C20.Q8D206C4C20.55D4C207D4C10×C4⋊C4C2×C20C22×C10C2×C4⋊C4C20C2×C10C4⋊C4C22×C4C2×C4C23C4C10C22C2
# reps12211111244424416144

Matrix representation of C4⋊C4.228D10 in GL4(𝔽41) generated by

1000
0100
00040
0010
,
32000
03200
002626
002615
,
352600
382000
00400
00040
,
212600
242000
00400
0001
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0],[32,0,0,0,0,32,0,0,0,0,26,26,0,0,26,15],[35,38,0,0,26,20,0,0,0,0,40,0,0,0,0,40],[21,24,0,0,26,20,0,0,0,0,40,0,0,0,0,1] >;

C4⋊C4.228D10 in GAP, Magma, Sage, TeX

C_4\rtimes C_4._{228}D_{10}
% in TeX

G:=Group("C4:C4.228D10");
// GroupNames label

G:=SmallGroup(320,595);
// by ID

G=gap.SmallGroup(320,595);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,268,1123,297,136,12550]);
// Polycyclic

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

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