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G = C56.22D4order 448 = 26·7

22nd non-split extension by C56 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C56.22D4, C7⋊C8.19D4, (C2×D8).4D7, C4.22(D4×D7), (C8×Dic7)⋊6C2, (C14×D8).5C2, (C2×D4).63D14, C28.165(C2×D4), (C2×C8).237D14, C73(C8.12D4), C8.15(C7⋊D4), (C2×Dic28)⋊18C2, C14.33(C4○D8), C28.17D45C2, (C2×C56).89C22, C22.255(D4×D7), C2.17(D83D7), C14.28(C41D4), C2.19(C28⋊D4), (C2×C28).432C23, (C2×Dic7).111D4, (D4×C14).82C22, (C4×Dic7).239C22, (C2×Dic14).121C22, C4.6(C2×C7⋊D4), (C2×D4.D7)⋊18C2, (C2×C14).345(C2×D4), (C2×C7⋊C8).270C22, (C2×C4).522(C22×D7), SmallGroup(448,689)

Series: Derived Chief Lower central Upper central

C1C2×C28 — C56.22D4
C1C7C14C28C2×C28C4×Dic7C28.17D4 — C56.22D4
C7C14C2×C28 — C56.22D4
C1C22C2×C4C2×D8

Generators and relations for C56.22D4
 G = < a,b,c | a56=b4=1, c2=a28, bab-1=a41, cac-1=a-1, cbc-1=a28b-1 >

Subgroups: 612 in 130 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4 [×4], Q8 [×4], C23 [×2], C14, C14 [×2], C14 [×2], C42, C22⋊C4 [×4], C2×C8, C2×C8, D8 [×2], SD16 [×4], Q16 [×2], C2×D4 [×2], C2×Q8 [×2], Dic7 [×4], C28 [×2], C2×C14, C2×C14 [×6], C4×C8, C4.4D4 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C7⋊C8 [×2], C56 [×2], Dic14 [×4], C2×Dic7 [×2], C2×Dic7 [×2], C2×C28, C7×D4 [×4], C22×C14 [×2], C8.12D4, Dic28 [×2], C2×C7⋊C8, C4×Dic7, D4.D7 [×4], C23.D7 [×4], C2×C56, C7×D8 [×2], C2×Dic14 [×2], D4×C14 [×2], C8×Dic7, C2×Dic28, C2×D4.D7 [×2], C28.17D4 [×2], C14×D8, C56.22D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D7, C2×D4 [×3], D14 [×3], C41D4, C4○D8 [×2], C7⋊D4 [×2], C22×D7, C8.12D4, D4×D7 [×2], C2×C7⋊D4, D83D7 [×2], C28⋊D4, C56.22D4

Smallest permutation representation of C56.22D4
On 224 points
Generators in S224
(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)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 218 70 148)(2 203 71 133)(3 188 72 118)(4 173 73 159)(5 214 74 144)(6 199 75 129)(7 184 76 114)(8 169 77 155)(9 210 78 140)(10 195 79 125)(11 180 80 166)(12 221 81 151)(13 206 82 136)(14 191 83 121)(15 176 84 162)(16 217 85 147)(17 202 86 132)(18 187 87 117)(19 172 88 158)(20 213 89 143)(21 198 90 128)(22 183 91 113)(23 224 92 154)(24 209 93 139)(25 194 94 124)(26 179 95 165)(27 220 96 150)(28 205 97 135)(29 190 98 120)(30 175 99 161)(31 216 100 146)(32 201 101 131)(33 186 102 116)(34 171 103 157)(35 212 104 142)(36 197 105 127)(37 182 106 168)(38 223 107 153)(39 208 108 138)(40 193 109 123)(41 178 110 164)(42 219 111 149)(43 204 112 134)(44 189 57 119)(45 174 58 160)(46 215 59 145)(47 200 60 130)(48 185 61 115)(49 170 62 156)(50 211 63 141)(51 196 64 126)(52 181 65 167)(53 222 66 152)(54 207 67 137)(55 192 68 122)(56 177 69 163)
(1 120 29 148)(2 119 30 147)(3 118 31 146)(4 117 32 145)(5 116 33 144)(6 115 34 143)(7 114 35 142)(8 113 36 141)(9 168 37 140)(10 167 38 139)(11 166 39 138)(12 165 40 137)(13 164 41 136)(14 163 42 135)(15 162 43 134)(16 161 44 133)(17 160 45 132)(18 159 46 131)(19 158 47 130)(20 157 48 129)(21 156 49 128)(22 155 50 127)(23 154 51 126)(24 153 52 125)(25 152 53 124)(26 151 54 123)(27 150 55 122)(28 149 56 121)(57 203 85 175)(58 202 86 174)(59 201 87 173)(60 200 88 172)(61 199 89 171)(62 198 90 170)(63 197 91 169)(64 196 92 224)(65 195 93 223)(66 194 94 222)(67 193 95 221)(68 192 96 220)(69 191 97 219)(70 190 98 218)(71 189 99 217)(72 188 100 216)(73 187 101 215)(74 186 102 214)(75 185 103 213)(76 184 104 212)(77 183 105 211)(78 182 106 210)(79 181 107 209)(80 180 108 208)(81 179 109 207)(82 178 110 206)(83 177 111 205)(84 176 112 204)

G:=sub<Sym(224)| (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,218,70,148)(2,203,71,133)(3,188,72,118)(4,173,73,159)(5,214,74,144)(6,199,75,129)(7,184,76,114)(8,169,77,155)(9,210,78,140)(10,195,79,125)(11,180,80,166)(12,221,81,151)(13,206,82,136)(14,191,83,121)(15,176,84,162)(16,217,85,147)(17,202,86,132)(18,187,87,117)(19,172,88,158)(20,213,89,143)(21,198,90,128)(22,183,91,113)(23,224,92,154)(24,209,93,139)(25,194,94,124)(26,179,95,165)(27,220,96,150)(28,205,97,135)(29,190,98,120)(30,175,99,161)(31,216,100,146)(32,201,101,131)(33,186,102,116)(34,171,103,157)(35,212,104,142)(36,197,105,127)(37,182,106,168)(38,223,107,153)(39,208,108,138)(40,193,109,123)(41,178,110,164)(42,219,111,149)(43,204,112,134)(44,189,57,119)(45,174,58,160)(46,215,59,145)(47,200,60,130)(48,185,61,115)(49,170,62,156)(50,211,63,141)(51,196,64,126)(52,181,65,167)(53,222,66,152)(54,207,67,137)(55,192,68,122)(56,177,69,163), (1,120,29,148)(2,119,30,147)(3,118,31,146)(4,117,32,145)(5,116,33,144)(6,115,34,143)(7,114,35,142)(8,113,36,141)(9,168,37,140)(10,167,38,139)(11,166,39,138)(12,165,40,137)(13,164,41,136)(14,163,42,135)(15,162,43,134)(16,161,44,133)(17,160,45,132)(18,159,46,131)(19,158,47,130)(20,157,48,129)(21,156,49,128)(22,155,50,127)(23,154,51,126)(24,153,52,125)(25,152,53,124)(26,151,54,123)(27,150,55,122)(28,149,56,121)(57,203,85,175)(58,202,86,174)(59,201,87,173)(60,200,88,172)(61,199,89,171)(62,198,90,170)(63,197,91,169)(64,196,92,224)(65,195,93,223)(66,194,94,222)(67,193,95,221)(68,192,96,220)(69,191,97,219)(70,190,98,218)(71,189,99,217)(72,188,100,216)(73,187,101,215)(74,186,102,214)(75,185,103,213)(76,184,104,212)(77,183,105,211)(78,182,106,210)(79,181,107,209)(80,180,108,208)(81,179,109,207)(82,178,110,206)(83,177,111,205)(84,176,112,204)>;

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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,218,70,148)(2,203,71,133)(3,188,72,118)(4,173,73,159)(5,214,74,144)(6,199,75,129)(7,184,76,114)(8,169,77,155)(9,210,78,140)(10,195,79,125)(11,180,80,166)(12,221,81,151)(13,206,82,136)(14,191,83,121)(15,176,84,162)(16,217,85,147)(17,202,86,132)(18,187,87,117)(19,172,88,158)(20,213,89,143)(21,198,90,128)(22,183,91,113)(23,224,92,154)(24,209,93,139)(25,194,94,124)(26,179,95,165)(27,220,96,150)(28,205,97,135)(29,190,98,120)(30,175,99,161)(31,216,100,146)(32,201,101,131)(33,186,102,116)(34,171,103,157)(35,212,104,142)(36,197,105,127)(37,182,106,168)(38,223,107,153)(39,208,108,138)(40,193,109,123)(41,178,110,164)(42,219,111,149)(43,204,112,134)(44,189,57,119)(45,174,58,160)(46,215,59,145)(47,200,60,130)(48,185,61,115)(49,170,62,156)(50,211,63,141)(51,196,64,126)(52,181,65,167)(53,222,66,152)(54,207,67,137)(55,192,68,122)(56,177,69,163), (1,120,29,148)(2,119,30,147)(3,118,31,146)(4,117,32,145)(5,116,33,144)(6,115,34,143)(7,114,35,142)(8,113,36,141)(9,168,37,140)(10,167,38,139)(11,166,39,138)(12,165,40,137)(13,164,41,136)(14,163,42,135)(15,162,43,134)(16,161,44,133)(17,160,45,132)(18,159,46,131)(19,158,47,130)(20,157,48,129)(21,156,49,128)(22,155,50,127)(23,154,51,126)(24,153,52,125)(25,152,53,124)(26,151,54,123)(27,150,55,122)(28,149,56,121)(57,203,85,175)(58,202,86,174)(59,201,87,173)(60,200,88,172)(61,199,89,171)(62,198,90,170)(63,197,91,169)(64,196,92,224)(65,195,93,223)(66,194,94,222)(67,193,95,221)(68,192,96,220)(69,191,97,219)(70,190,98,218)(71,189,99,217)(72,188,100,216)(73,187,101,215)(74,186,102,214)(75,185,103,213)(76,184,104,212)(77,183,105,211)(78,182,106,210)(79,181,107,209)(80,180,108,208)(81,179,109,207)(82,178,110,206)(83,177,111,205)(84,176,112,204) );

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),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,218,70,148),(2,203,71,133),(3,188,72,118),(4,173,73,159),(5,214,74,144),(6,199,75,129),(7,184,76,114),(8,169,77,155),(9,210,78,140),(10,195,79,125),(11,180,80,166),(12,221,81,151),(13,206,82,136),(14,191,83,121),(15,176,84,162),(16,217,85,147),(17,202,86,132),(18,187,87,117),(19,172,88,158),(20,213,89,143),(21,198,90,128),(22,183,91,113),(23,224,92,154),(24,209,93,139),(25,194,94,124),(26,179,95,165),(27,220,96,150),(28,205,97,135),(29,190,98,120),(30,175,99,161),(31,216,100,146),(32,201,101,131),(33,186,102,116),(34,171,103,157),(35,212,104,142),(36,197,105,127),(37,182,106,168),(38,223,107,153),(39,208,108,138),(40,193,109,123),(41,178,110,164),(42,219,111,149),(43,204,112,134),(44,189,57,119),(45,174,58,160),(46,215,59,145),(47,200,60,130),(48,185,61,115),(49,170,62,156),(50,211,63,141),(51,196,64,126),(52,181,65,167),(53,222,66,152),(54,207,67,137),(55,192,68,122),(56,177,69,163)], [(1,120,29,148),(2,119,30,147),(3,118,31,146),(4,117,32,145),(5,116,33,144),(6,115,34,143),(7,114,35,142),(8,113,36,141),(9,168,37,140),(10,167,38,139),(11,166,39,138),(12,165,40,137),(13,164,41,136),(14,163,42,135),(15,162,43,134),(16,161,44,133),(17,160,45,132),(18,159,46,131),(19,158,47,130),(20,157,48,129),(21,156,49,128),(22,155,50,127),(23,154,51,126),(24,153,52,125),(25,152,53,124),(26,151,54,123),(27,150,55,122),(28,149,56,121),(57,203,85,175),(58,202,86,174),(59,201,87,173),(60,200,88,172),(61,199,89,171),(62,198,90,170),(63,197,91,169),(64,196,92,224),(65,195,93,223),(66,194,94,222),(67,193,95,221),(68,192,96,220),(69,191,97,219),(70,190,98,218),(71,189,99,217),(72,188,100,216),(73,187,101,215),(74,186,102,214),(75,185,103,213),(76,184,104,212),(77,183,105,211),(78,182,106,210),(79,181,107,209),(80,180,108,208),(81,179,109,207),(82,178,110,206),(83,177,111,205),(84,176,112,204)])

64 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H7A7B7C8A8B8C8D8E8F8G8H14A···14I14J···14U28A···28F56A···56L
order122222444444447778888888814···1414···1428···2856···56
size111188221414141456562222222141414142···28···84···44···4

64 irreducible representations

dim11111122222222444
type++++++++++++++-
imageC1C2C2C2C2C2D4D4D4D7D14D14C4○D8C7⋊D4D4×D7D4×D7D83D7
kernelC56.22D4C8×Dic7C2×Dic28C2×D4.D7C28.17D4C14×D8C7⋊C8C56C2×Dic7C2×D8C2×C8C2×D4C14C8C4C22C2
# reps1112212223368123312

Matrix representation of C56.22D4 in GL4(𝔽113) generated by

971700
0700
00015
001562
,
12800
811200
001562
006298
,
112000
105100
001562
00098
G:=sub<GL(4,GF(113))| [97,0,0,0,17,7,0,0,0,0,0,15,0,0,15,62],[1,8,0,0,28,112,0,0,0,0,15,62,0,0,62,98],[112,105,0,0,0,1,0,0,0,0,15,0,0,0,62,98] >;

C56.22D4 in GAP, Magma, Sage, TeX

C_{56}._{22}D_4
% in TeX

G:=Group("C56.22D4");
// GroupNames label

G:=SmallGroup(448,689);
// by ID

G=gap.SmallGroup(448,689);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,701,1094,135,570,297,136,18822]);
// Polycyclic

G:=Group<a,b,c|a^56=b^4=1,c^2=a^28,b*a*b^-1=a^41,c*a*c^-1=a^-1,c*b*c^-1=a^28*b^-1>;
// generators/relations

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