C5^2:8SD16 - GroupNames

G = C₅²⋊₈SD₁₆ order 400 = 2⁴·5²

2^nd semidirect product of C₅² and SD₁₆ acting via SD₁₆/D₄=C₂

Aliases: C₅²⋊₈SD₁₆, C₂₀.₁₇D₁₀, D₄.(C₅⋊D₅), (C₅×D₄).₅D₅, C₅⋊₃(D₄.D₅), C₅²⋊₇C₈⋊₄C₂, (C₅×C₁₀).₃₅D₄, C₅²⋊₄Q₈⋊₃C₂, (D₄×C₅²).₂C₂, C₁₀.₂₃(C₅⋊D₄), (C₅×C₂₀).₁₃C₂², C₂.₅(C₅²⋊₇D₄), C₄.₂(C₂×C₅⋊D₅), SmallGroup(400,104)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅×C₂₀ — C₅²⋊₈SD₁₆

Chief series

C₁ — C₅ — C₅² — C₅×C₁₀ — C₅×C₂₀ — C₅²⋊₄Q₈ — C₅²⋊₈SD₁₆

Lower central

C₅² — C₅×C₁₀ — C₅×C₂₀ — C₅²⋊₈SD₁₆

Upper central

C₁ — C₂ — C₄ — D₄

Generators and relations for C₅²⋊₈SD₁₆
G = < a,b,c,d | a⁵=b⁵=c⁸=d²=1, ab=ba, cac^-1=a^-1, ad=da, cbc^-1=b^-1, bd=db, dcd=c³ >

Subgroups: 360 in 80 conjugacy classes, 35 normal (11 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₅, C₈, D₄, Q₈, C₁₀, C₁₀, SD₁₆, Dic₅, C₂₀, C₂×C₁₀, C₅², C₅⋊₂C₈, Dic₁₀, C₅×D₄, C₅×C₁₀, C₅×C₁₀, D₄.D₅, C₅²⋊₆C₄, C₅×C₂₀, C₁₀², C₅²⋊₇C₈, C₅²⋊₄Q₈, D₄×C₅², C₅²⋊₈SD₁₆
Quotients: C₁, C₂, C₂², D₄, D₅, SD₁₆, D₁₀, C₅⋊D₄, C₅⋊D₅, D₄.D₅, C₂×C₅⋊D₅, C₅²⋊₇D₄, C₅²⋊₈SD₁₆

Smallest permutation representation of C₅²⋊₈SD₁₆
►On 200 points

Generators in S₂₀₀

(1 90 116 189 111)(2 112 190 117 91)(3 92 118 191 105)(4 106 192 119 93)(5 94 120 185 107)(6 108 186 113 95)(7 96 114 187 109)(8 110 188 115 89)(9 171 144 33 68)(10 69 34 137 172)(11 173 138 35 70)(12 71 36 139 174)(13 175 140 37 72)(14 65 38 141 176)(15 169 142 39 66)(16 67 40 143 170)(17 78 177 99 87)(18 88 100 178 79)(19 80 179 101 81)(20 82 102 180 73)(21 74 181 103 83)(22 84 104 182 75)(23 76 183 97 85)(24 86 98 184 77)(25 159 54 44 150)(26 151 45 55 160)(27 153 56 46 152)(28 145 47 49 154)(29 155 50 48 146)(30 147 41 51 156)(31 157 52 42 148)(32 149 43 53 158)(57 125 164 130 193)(58 194 131 165 126)(59 127 166 132 195)(60 196 133 167 128)(61 121 168 134 197)(62 198 135 161 122)(63 123 162 136 199)(64 200 129 163 124)

(1 48 168 72 100)(2 101 65 161 41)(3 42 162 66 102)(4 103 67 163 43)(5 44 164 68 104)(6 97 69 165 45)(7 46 166 70 98)(8 99 71 167 47)(9 182 94 150 130)(10 131 151 95 183)(11 184 96 152 132)(12 133 145 89 177)(13 178 90 146 134)(14 135 147 91 179)(15 180 92 148 136)(16 129 149 93 181)(17 139 60 154 188)(18 189 155 61 140)(19 141 62 156 190)(20 191 157 63 142)(21 143 64 158 192)(22 185 159 57 144)(23 137 58 160 186)(24 187 153 59 138)(25 193 171 75 120)(26 113 76 172 194)(27 195 173 77 114)(28 115 78 174 196)(29 197 175 79 116)(30 117 80 176 198)(31 199 169 73 118)(32 119 74 170 200)(33 84 107 54 125)(34 126 55 108 85)(35 86 109 56 127)(36 128 49 110 87)(37 88 111 50 121)(38 122 51 112 81)(39 82 105 52 123)(40 124 53 106 83)

(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)(161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176)(177 178 179 180 181 182 183 184)(185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200)

(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 23)(19 21)(20 24)(26 28)(27 31)(30 32)(34 36)(35 39)(38 40)(41 43)(42 46)(45 47)(49 55)(51 53)(52 56)(58 60)(59 63)(62 64)(65 67)(66 70)(69 71)(73 77)(74 80)(76 78)(81 83)(82 86)(85 87)(89 95)(91 93)(92 96)(97 99)(98 102)(101 103)(105 109)(106 112)(108 110)(113 115)(114 118)(117 119)(122 124)(123 127)(126 128)(129 135)(131 133)(132 136)(137 139)(138 142)(141 143)(145 151)(147 149)(148 152)(153 157)(154 160)(156 158)(161 163)(162 166)(165 167)(169 173)(170 176)(172 174)(177 183)(179 181)(180 184)(186 188)(187 191)(190 192)(194 196)(195 199)(198 200)

G:=sub<Sym(200)| (1,90,116,189,111)(2,112,190,117,91)(3,92,118,191,105)(4,106,192,119,93)(5,94,120,185,107)(6,108,186,113,95)(7,96,114,187,109)(8,110,188,115,89)(9,171,144,33,68)(10,69,34,137,172)(11,173,138,35,70)(12,71,36,139,174)(13,175,140,37,72)(14,65,38,141,176)(15,169,142,39,66)(16,67,40,143,170)(17,78,177,99,87)(18,88,100,178,79)(19,80,179,101,81)(20,82,102,180,73)(21,74,181,103,83)(22,84,104,182,75)(23,76,183,97,85)(24,86,98,184,77)(25,159,54,44,150)(26,151,45,55,160)(27,153,56,46,152)(28,145,47,49,154)(29,155,50,48,146)(30,147,41,51,156)(31,157,52,42,148)(32,149,43,53,158)(57,125,164,130,193)(58,194,131,165,126)(59,127,166,132,195)(60,196,133,167,128)(61,121,168,134,197)(62,198,135,161,122)(63,123,162,136,199)(64,200,129,163,124), (1,48,168,72,100)(2,101,65,161,41)(3,42,162,66,102)(4,103,67,163,43)(5,44,164,68,104)(6,97,69,165,45)(7,46,166,70,98)(8,99,71,167,47)(9,182,94,150,130)(10,131,151,95,183)(11,184,96,152,132)(12,133,145,89,177)(13,178,90,146,134)(14,135,147,91,179)(15,180,92,148,136)(16,129,149,93,181)(17,139,60,154,188)(18,189,155,61,140)(19,141,62,156,190)(20,191,157,63,142)(21,143,64,158,192)(22,185,159,57,144)(23,137,58,160,186)(24,187,153,59,138)(25,193,171,75,120)(26,113,76,172,194)(27,195,173,77,114)(28,115,78,174,196)(29,197,175,79,116)(30,117,80,176,198)(31,199,169,73,118)(32,119,74,170,200)(33,84,107,54,125)(34,126,55,108,85)(35,86,109,56,127)(36,128,49,110,87)(37,88,111,50,121)(38,122,51,112,81)(39,82,105,52,123)(40,124,53,106,83), (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)(161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(34,36)(35,39)(38,40)(41,43)(42,46)(45,47)(49,55)(51,53)(52,56)(58,60)(59,63)(62,64)(65,67)(66,70)(69,71)(73,77)(74,80)(76,78)(81,83)(82,86)(85,87)(89,95)(91,93)(92,96)(97,99)(98,102)(101,103)(105,109)(106,112)(108,110)(113,115)(114,118)(117,119)(122,124)(123,127)(126,128)(129,135)(131,133)(132,136)(137,139)(138,142)(141,143)(145,151)(147,149)(148,152)(153,157)(154,160)(156,158)(161,163)(162,166)(165,167)(169,173)(170,176)(172,174)(177,183)(179,181)(180,184)(186,188)(187,191)(190,192)(194,196)(195,199)(198,200)>;

G:=Group( (1,90,116,189,111)(2,112,190,117,91)(3,92,118,191,105)(4,106,192,119,93)(5,94,120,185,107)(6,108,186,113,95)(7,96,114,187,109)(8,110,188,115,89)(9,171,144,33,68)(10,69,34,137,172)(11,173,138,35,70)(12,71,36,139,174)(13,175,140,37,72)(14,65,38,141,176)(15,169,142,39,66)(16,67,40,143,170)(17,78,177,99,87)(18,88,100,178,79)(19,80,179,101,81)(20,82,102,180,73)(21,74,181,103,83)(22,84,104,182,75)(23,76,183,97,85)(24,86,98,184,77)(25,159,54,44,150)(26,151,45,55,160)(27,153,56,46,152)(28,145,47,49,154)(29,155,50,48,146)(30,147,41,51,156)(31,157,52,42,148)(32,149,43,53,158)(57,125,164,130,193)(58,194,131,165,126)(59,127,166,132,195)(60,196,133,167,128)(61,121,168,134,197)(62,198,135,161,122)(63,123,162,136,199)(64,200,129,163,124), (1,48,168,72,100)(2,101,65,161,41)(3,42,162,66,102)(4,103,67,163,43)(5,44,164,68,104)(6,97,69,165,45)(7,46,166,70,98)(8,99,71,167,47)(9,182,94,150,130)(10,131,151,95,183)(11,184,96,152,132)(12,133,145,89,177)(13,178,90,146,134)(14,135,147,91,179)(15,180,92,148,136)(16,129,149,93,181)(17,139,60,154,188)(18,189,155,61,140)(19,141,62,156,190)(20,191,157,63,142)(21,143,64,158,192)(22,185,159,57,144)(23,137,58,160,186)(24,187,153,59,138)(25,193,171,75,120)(26,113,76,172,194)(27,195,173,77,114)(28,115,78,174,196)(29,197,175,79,116)(30,117,80,176,198)(31,199,169,73,118)(32,119,74,170,200)(33,84,107,54,125)(34,126,55,108,85)(35,86,109,56,127)(36,128,49,110,87)(37,88,111,50,121)(38,122,51,112,81)(39,82,105,52,123)(40,124,53,106,83), (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)(161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(34,36)(35,39)(38,40)(41,43)(42,46)(45,47)(49,55)(51,53)(52,56)(58,60)(59,63)(62,64)(65,67)(66,70)(69,71)(73,77)(74,80)(76,78)(81,83)(82,86)(85,87)(89,95)(91,93)(92,96)(97,99)(98,102)(101,103)(105,109)(106,112)(108,110)(113,115)(114,118)(117,119)(122,124)(123,127)(126,128)(129,135)(131,133)(132,136)(137,139)(138,142)(141,143)(145,151)(147,149)(148,152)(153,157)(154,160)(156,158)(161,163)(162,166)(165,167)(169,173)(170,176)(172,174)(177,183)(179,181)(180,184)(186,188)(187,191)(190,192)(194,196)(195,199)(198,200) );

G=PermutationGroup([[(1,90,116,189,111),(2,112,190,117,91),(3,92,118,191,105),(4,106,192,119,93),(5,94,120,185,107),(6,108,186,113,95),(7,96,114,187,109),(8,110,188,115,89),(9,171,144,33,68),(10,69,34,137,172),(11,173,138,35,70),(12,71,36,139,174),(13,175,140,37,72),(14,65,38,141,176),(15,169,142,39,66),(16,67,40,143,170),(17,78,177,99,87),(18,88,100,178,79),(19,80,179,101,81),(20,82,102,180,73),(21,74,181,103,83),(22,84,104,182,75),(23,76,183,97,85),(24,86,98,184,77),(25,159,54,44,150),(26,151,45,55,160),(27,153,56,46,152),(28,145,47,49,154),(29,155,50,48,146),(30,147,41,51,156),(31,157,52,42,148),(32,149,43,53,158),(57,125,164,130,193),(58,194,131,165,126),(59,127,166,132,195),(60,196,133,167,128),(61,121,168,134,197),(62,198,135,161,122),(63,123,162,136,199),(64,200,129,163,124)], [(1,48,168,72,100),(2,101,65,161,41),(3,42,162,66,102),(4,103,67,163,43),(5,44,164,68,104),(6,97,69,165,45),(7,46,166,70,98),(8,99,71,167,47),(9,182,94,150,130),(10,131,151,95,183),(11,184,96,152,132),(12,133,145,89,177),(13,178,90,146,134),(14,135,147,91,179),(15,180,92,148,136),(16,129,149,93,181),(17,139,60,154,188),(18,189,155,61,140),(19,141,62,156,190),(20,191,157,63,142),(21,143,64,158,192),(22,185,159,57,144),(23,137,58,160,186),(24,187,153,59,138),(25,193,171,75,120),(26,113,76,172,194),(27,195,173,77,114),(28,115,78,174,196),(29,197,175,79,116),(30,117,80,176,198),(31,199,169,73,118),(32,119,74,170,200),(33,84,107,54,125),(34,126,55,108,85),(35,86,109,56,127),(36,128,49,110,87),(37,88,111,50,121),(38,122,51,112,81),(39,82,105,52,123),(40,124,53,106,83)], [(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),(161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176),(177,178,179,180,181,182,183,184),(185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(17,23),(19,21),(20,24),(26,28),(27,31),(30,32),(34,36),(35,39),(38,40),(41,43),(42,46),(45,47),(49,55),(51,53),(52,56),(58,60),(59,63),(62,64),(65,67),(66,70),(69,71),(73,77),(74,80),(76,78),(81,83),(82,86),(85,87),(89,95),(91,93),(92,96),(97,99),(98,102),(101,103),(105,109),(106,112),(108,110),(113,115),(114,118),(117,119),(122,124),(123,127),(126,128),(129,135),(131,133),(132,136),(137,139),(138,142),(141,143),(145,151),(147,149),(148,152),(153,157),(154,160),(156,158),(161,163),(162,166),(165,167),(169,173),(170,176),(172,174),(177,183),(179,181),(180,184),(186,188),(187,191),(190,192),(194,196),(195,199),(198,200)]])

67 conjugacy classes

class	1	2A	2B	4A	4B	5A	···	5L	8A	8B	10A	···	10L	10M	···	10AJ	20A	···	20L
order	1	2	2	4	4	5	···	5	8	8	10	···	10	10	···	10	20	···	20
size	1	1	4	2	100	2	···	2	50	50	2	···	2	4	···	4	4	···	4

67 irreducible representations

dim	1	1	1	1	2	2	2	2	2	4
type	+	+	+	+	+	+		+		-
image	C₁	C₂	C₂	C₂	D₄	D₅	SD₁₆	D₁₀	C₅⋊D₄	D₄.D₅
kernel	C₅²⋊₈SD₁₆	C₅²⋊₇C₈	C₅²⋊₄Q₈	D₄×C₅²	C₅×C₁₀	C₅×D₄	C₅²	C₂₀	C₁₀	C₅
# reps	1	1	1	1	1	12	2	12	24	12

Matrix representation of C₅²⋊₈SD₁₆ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	7	7	0	0
0	0	34	40	0	0
0	0	0	0	1	0
0	0	0	0	0	1

40	1	0	0	0	0
5	35	0	0	0	0
0	0	0	1	0	0
0	0	40	34	0	0
0	0	0	0	1	0
0	0	0	0	0	1

25	27	0	0	0	0
27	16	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	13	7
0	0	0	0	20	17

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	24	40

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,5,0,0,0,0,1,35,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,27,0,0,0,0,27,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,20,0,0,0,0,7,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,24,0,0,0,0,0,40] >;

C₅²⋊₈SD₁₆ in GAP, Magma, Sage, TeX

C_5^2\rtimes_8{\rm SD}_{16}

% in TeX

G:=Group("C5^2:8SD16");

// GroupNames label

G:=SmallGroup(400,104);

// by ID

G=gap.SmallGroup(400,104);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,73,218,116,50,1924,11525]);

// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^3>;

// generators/relations

G = C52⋊8SD16 order 400 = 24·52

2nd semidirect product of C52 and SD16 acting via SD16/D4=C2

G = C₅²⋊₈SD₁₆ order 400 = 2⁴·5²

2^nd semidirect product of C₅² and SD₁₆ acting via SD₁₆/D₄=C₂