C2xC5^2:6C4 - GroupNames

G = C₂×C₅²⋊₆C₄ order 200 = 2³·5²

Direct product of C₂ and C₅²⋊₆C₄

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₂×C₅²⋊₆C₄, C₁₀⋊₂Dic₅, C₁₀.₁₅D₁₀, C₁₀².₂C₂, (C₅×C₁₀)⋊₆C₄, C₅²⋊₁₂(C₂×C₄), C₅⋊₃(C₂×Dic₅), (C₂×C₁₀).₄D₅, C₂².(C₅⋊D₅), (C₅×C₁₀).₁₄C₂², C₂.₂(C₂×C₅⋊D₅), SmallGroup(200,35)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅² — C₂×C₅²⋊₆C₄

Chief series

C₁ — C₅ — C₅² — C₅×C₁₀ — C₅²⋊₆C₄ — C₂×C₅²⋊₆C₄

Lower central

C₅² — C₂×C₅²⋊₆C₄

Upper central

C₁ — C₂²

Generators and relations for C₂×C₅²⋊₆C₄
G = < a,b,c,d | a²=b⁵=c⁵=d⁴=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b^-1, dcd^-1=c^-1 >

Subgroups: 208 in 64 conjugacy classes, 43 normal (7 characteristic)
C₁, C₂, C₂, C₄, C₂², C₅, C₂×C₄, C₁₀, Dic₅, C₂×C₁₀, C₅², C₂×Dic₅, C₅×C₁₀, C₅×C₁₀, C₅²⋊₆C₄, C₁₀², C₂×C₅²⋊₆C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₅, Dic₅, D₁₀, C₂×Dic₅, C₅⋊D₅, C₅²⋊₆C₄, C₂×C₅⋊D₅, C₂×C₅²⋊₆C₄

Smallest permutation representation of C₂×C₅²⋊₆C₄
►Regular action on 200 points

Generators in S₂₀₀

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

(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)

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

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

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

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

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

C₂×C₅²⋊₆C₄ is a maximal subgroup of
Dic₅² D₁₀⋊Dic₅ Dic₅⋊Dic₅ C₁₀.Dic₁₀ C₁₀².₂₂C₂² C₂₀⋊₃Dic₅ C₁₀.₁₁D₂₀ C₁₀²⋊₁₁C₄ C₅²⋊₁₃M₄(2) C₅²⋊₁₄M₄(2) C₂×D₅×Dic₅ D₁₀.₄D₁₀ C₂×C₄×C₅⋊D₅ C₂₀.D₁₀
C₂×C₅²⋊₆C₄ is a maximal quotient of
C₂₀.₅₉D₁₀ C₂₀⋊₃Dic₅ C₁₀²⋊₁₁C₄

56 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	4D	5A	···	5L	10A	···	10AJ
order	1	2	2	2	4	4	4	4	5	···	5	10	···	10
size	1	1	1	1	25	25	25	25	2	···	2	2	···	2

56 irreducible representations

dim	1	1	1	1	2	2	2
type	+	+	+		+	-	+
image	C₁	C₂	C₂	C₄	D₅	Dic₅	D₁₀
kernel	C₂×C₅²⋊₆C₄	C₅²⋊₆C₄	C₁₀²	C₅×C₁₀	C₂×C₁₀	C₁₀	C₁₀
# reps	1	2	1	4	12	24	12

Matrix representation of C₂×C₅²⋊₆C₄ ►in GL₅(𝔽₄₁)

1	0	0	0	0
0	40	0	0	0
0	0	40	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	37	0	0	0
0	0	10	0	0
0	0	0	1	0
0	0	0	0	1

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

32	0	0	0	0
0	0	40	0	0
0	40	0	0	0
0	0	0	36	20
0	0	0	7	5

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,37,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,5,0,0,0,1,35],[32,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,36,7,0,0,0,20,5] >;

C₂×C₅²⋊₆C₄ in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes_6C_4

% in TeX

G:=Group("C2xC5^2:6C4");

// GroupNames label

G:=SmallGroup(200,35);

// by ID

G=gap.SmallGroup(200,35);

# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,20,643,4004]);

// Polycyclic

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

// generators/relations

G = C2×C52⋊6C4 order 200 = 23·52

Direct product of C2 and C52⋊6C4

G = C₂×C₅²⋊₆C₄ order 200 = 2³·5²

Direct product of C₂ and C₅²⋊₆C₄