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G = C4○D4×C25order 400 = 24·52

Direct product of C25 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C4○D4×C25, D42C50, Q82C50, C50.13C23, C100.21C22, (C2×C4)⋊3C50, (C2×C100)⋊7C2, (D4×C25)⋊5C2, C4.6(C2×C50), (Q8×C25)⋊5C2, C22.(C2×C50), (C5×D4).6C10, (C5×Q8).6C10, (C2×C20).13C10, C20.22(C2×C10), (C2×C50).2C22, C2.3(C22×C50), C10.13(C22×C10), C5.(C5×C4○D4), (C5×C4○D4).C5, (C2×C10).3(C2×C10), SmallGroup(400,48)

Series: Derived Chief Lower central Upper central

C1C2 — C4○D4×C25
C1C5C10C50C2×C50D4×C25 — C4○D4×C25
C1C2 — C4○D4×C25
C1C100 — C4○D4×C25

Generators and relations for C4○D4×C25
 G = < a,b,c,d | a25=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 69 in 60 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C5, C2×C4 [×3], D4 [×3], Q8, C10, C10 [×3], C4○D4, C20, C20 [×3], C2×C10 [×3], C25, C2×C20 [×3], C5×D4 [×3], C5×Q8, C50, C50 [×3], C5×C4○D4, C100, C100 [×3], C2×C50 [×3], C2×C100 [×3], D4×C25 [×3], Q8×C25, C4○D4×C25
Quotients: C1, C2 [×7], C22 [×7], C5, C23, C10 [×7], C4○D4, C2×C10 [×7], C25, C22×C10, C50 [×7], C5×C4○D4, C2×C50 [×7], C22×C50, C4○D4×C25

Smallest permutation representation of C4○D4×C25
On 200 points
Generators in S200
(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 55 47 129)(2 56 48 130)(3 57 49 131)(4 58 50 132)(5 59 26 133)(6 60 27 134)(7 61 28 135)(8 62 29 136)(9 63 30 137)(10 64 31 138)(11 65 32 139)(12 66 33 140)(13 67 34 141)(14 68 35 142)(15 69 36 143)(16 70 37 144)(17 71 38 145)(18 72 39 146)(19 73 40 147)(20 74 41 148)(21 75 42 149)(22 51 43 150)(23 52 44 126)(24 53 45 127)(25 54 46 128)(76 177 124 174)(77 178 125 175)(78 179 101 151)(79 180 102 152)(80 181 103 153)(81 182 104 154)(82 183 105 155)(83 184 106 156)(84 185 107 157)(85 186 108 158)(86 187 109 159)(87 188 110 160)(88 189 111 161)(89 190 112 162)(90 191 113 163)(91 192 114 164)(92 193 115 165)(93 194 116 166)(94 195 117 167)(95 196 118 168)(96 197 119 169)(97 198 120 170)(98 199 121 171)(99 200 122 172)(100 176 123 173)
(1 105 47 82)(2 106 48 83)(3 107 49 84)(4 108 50 85)(5 109 26 86)(6 110 27 87)(7 111 28 88)(8 112 29 89)(9 113 30 90)(10 114 31 91)(11 115 32 92)(12 116 33 93)(13 117 34 94)(14 118 35 95)(15 119 36 96)(16 120 37 97)(17 121 38 98)(18 122 39 99)(19 123 40 100)(20 124 41 76)(21 125 42 77)(22 101 43 78)(23 102 44 79)(24 103 45 80)(25 104 46 81)(51 151 150 179)(52 152 126 180)(53 153 127 181)(54 154 128 182)(55 155 129 183)(56 156 130 184)(57 157 131 185)(58 158 132 186)(59 159 133 187)(60 160 134 188)(61 161 135 189)(62 162 136 190)(63 163 137 191)(64 164 138 192)(65 165 139 193)(66 166 140 194)(67 167 141 195)(68 168 142 196)(69 169 143 197)(70 170 144 198)(71 171 145 199)(72 172 146 200)(73 173 147 176)(74 174 148 177)(75 175 149 178)
(76 124)(77 125)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 121)(99 122)(100 123)(151 179)(152 180)(153 181)(154 182)(155 183)(156 184)(157 185)(158 186)(159 187)(160 188)(161 189)(162 190)(163 191)(164 192)(165 193)(166 194)(167 195)(168 196)(169 197)(170 198)(171 199)(172 200)(173 176)(174 177)(175 178)

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

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

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

250 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B5C5D10A10B10C10D10E···10P20A···20H20I···20T25A···25T50A···50T50U···50CB100A···100AN100AO···100CV
order122224444455551010101010···1020···2020···2025···2550···5050···50100···100100···100
size1122211222111111112···21···12···21···11···12···21···12···2

250 irreducible representations

dim111111111111222
type++++
imageC1C2C2C2C5C10C10C10C25C50C50C50C4○D4C5×C4○D4C4○D4×C25
kernelC4○D4×C25C2×C100D4×C25Q8×C25C5×C4○D4C2×C20C5×D4C5×Q8C4○D4C2×C4D4Q8C25C5C1
# reps1331412124206060202840

Matrix representation of C4○D4×C25 in GL2(𝔽101) generated by

780
078
,
100
010
,
1565
6886
,
156
0100
G:=sub<GL(2,GF(101))| [78,0,0,78],[10,0,0,10],[15,68,65,86],[1,0,56,100] >;

C4○D4×C25 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{25}
% in TeX

G:=Group("C4oD4xC25");
// GroupNames label

G:=SmallGroup(400,48);
// by ID

G=gap.SmallGroup(400,48);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,505,194,261]);
// Polycyclic

G:=Group<a,b,c,d|a^25=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations

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