C28:2D8 - GroupNames

G = C₂₈⋊₂D₈ order 448 = 2⁶·7

2^nd semidirect product of C₂₈ and D₈ acting via D₈/C₄=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₈⋊₂D₈, D₂₈⋊₉D₄, C₄².₇₃D₁₄, C₄⋊₂(D₄⋊D₇), C₄⋊₁D₄⋊₁D₇, C₇⋊₄(C₄⋊D₈), C₄.₅₃(D₄×D₇), (C₄×D₂₈)⋊₂₂C₂, C₂₈⋊C₈⋊₃₁C₂, C₂₈.₃₀(C₂×D₄), C₁₄.₅₇(C₂×D₈), (C₂×D₄).₅₅D₁₄, (C₂×C₂₈).₁₄₇D₄, C₂₈.₇₆(C₄○D₄), D₄⋊Dic₇⋊₂₂C₂, C₄.₃(D₄⋊₂D₇), C₂.₁₂(C₂₈⋊₂D₄), C₁₄.₉₄(C₈⋊C₂²), (C₂×C₂₈).₃₉₀C₂³, (C₄×C₂₈).₁₂₀C₂², (D₄×C₁₄).₇₁C₂², C₁₄.₁₀₃(C₄⋊D₄), (C₂×D₂₈).₂₄₆C₂², C₄⋊Dic₇.₃₄₄C₂², C₂.₁₅(D₄.D₁₄), (C₂×D₄⋊D₇)⋊₁₃C₂, (C₇×C₄⋊₁D₄)⋊₁C₂, C₂.₁₂(C₂×D₄⋊D₇), (C₂×C₁₄).₅₂₁(C₂×D₄), (C₂×C₇⋊C₈).₁₃₀C₂², (C₂×C₄).₁₈₅(C₇⋊D₄), (C₂×C₄).₄₈₈(C₂²×D₇), C₂².₁₉₄(C₂×C₇⋊D₄), SmallGroup(448,606)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — C₂₈⋊₂D₈

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₂×D₂₈ — C₄×D₂₈ — C₂₈⋊₂D₈

Lower central

C₇ — C₁₄ — C₂×C₂₈ — C₂₈⋊₂D₈

Upper central

C₁ — C₂² — C₄² — C₄⋊₁D₄

Generators and relations for C₂₈⋊₂D₈
G = < a,b,c | a²⁸=b⁸=c²=1, bab^-1=a^-1, cac=a¹³, cbc=b^-1 >

Subgroups: 716 in 140 conjugacy classes, 45 normal (29 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, D₇, C₁₄, C₁₄, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, D₈, C₂²×C₄, C₂×D₄, C₂×D₄, Dic₇, C₂₈, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, D₄⋊C₄, C₄⋊C₈, C₄×D₄, C₄⋊₁D₄, C₂×D₈, C₇⋊C₈, C₄×D₇, D₂₈, D₂₈, C₂×Dic₇, C₂×C₂₈, C₇×D₄, C₂²×D₇, C₂²×C₁₄, C₄⋊D₈, C₂×C₇⋊C₈, C₄⋊Dic₇, D₁₄⋊C₄, D₄⋊D₇, C₄×C₂₈, C₂×C₄×D₇, C₂×D₂₈, D₄×C₁₄, D₄×C₁₄, C₂₈⋊C₈, D₄⋊Dic₇, C₄×D₂₈, C₂×D₄⋊D₇, C₇×C₄⋊₁D₄, C₂₈⋊₂D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, D₈, C₂×D₄, C₄○D₄, D₁₄, C₄⋊D₄, C₂×D₈, C₈⋊C₂², C₇⋊D₄, C₂²×D₇, C₄⋊D₈, D₄⋊D₇, D₄×D₇, D₄⋊₂D₇, C₂×C₇⋊D₄, C₂×D₄⋊D₇, D₄.D₁₄, C₂₈⋊₂D₄, C₂₈⋊₂D₈

Smallest permutation representation of C₂₈⋊₂D₈
►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)

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

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

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

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

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

61 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	7A	7B	7C	8A	8B	8C	8D	14A	···	14I	14J	···	14U	28A	···	28R
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	7	7	7	8	8	8	8	14	···	14	14	···	14	28	···	28
size	1	1	1	1	8	8	28	28	2	2	2	2	4	28	28	2	2	2	28	28	28	28	2	···	2	8	···	8	4	···	4

61 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+		+	+		+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₇	D₈	C₄○D₄	D₁₄	D₁₄	C₇⋊D₄	C₈⋊C₂²	D₄⋊D₇	D₄×D₇	D₄⋊₂D₇	D₄.D₁₄
kernel	C₂₈⋊₂D₈	C₂₈⋊C₈	D₄⋊Dic₇	C₄×D₂₈	C₂×D₄⋊D₇	C₇×C₄⋊₁D₄	D₂₈	C₂×C₂₈	C₄⋊₁D₄	C₂₈	C₂₈	C₄²	C₂×D₄	C₂×C₄	C₁₄	C₄	C₄	C₄	C₂
# reps	1	1	2	1	2	1	2	2	3	4	2	3	6	12	1	6	3	3	6

Matrix representation of C₂₈⋊₂D₈ ►in GL₆(𝔽₁₁₃)

1	89	0	0	0	0
24	103	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	98	0
0	0	0	0	0	15

112	0	0	0	0	0
89	1	0	0	0	0
0	0	31	31	0	0
0	0	82	31	0	0
0	0	0	0	0	112
0	0	0	0	1	0

1	0	0	0	0	0
24	112	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	112

G:=sub<GL(6,GF(113))| [1,24,0,0,0,0,89,103,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,98,0,0,0,0,0,0,15],[112,89,0,0,0,0,0,1,0,0,0,0,0,0,31,82,0,0,0,0,31,31,0,0,0,0,0,0,0,1,0,0,0,0,112,0],[1,24,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112] >;

C₂₈⋊₂D₈ in GAP, Magma, Sage, TeX

C_{28}\rtimes_2D_8

% in TeX

G:=Group("C28:2D8");

// GroupNames label

G:=SmallGroup(448,606);

// by ID

G=gap.SmallGroup(448,606);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,219,1123,297,136,18822]);

// Polycyclic

G:=Group<a,b,c|a^28=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^13,c*b*c=b^-1>;

// generators/relations

G = C28⋊2D8 order 448 = 26·7

2nd semidirect product of C28 and D8 acting via D8/C4=C22

G = C₂₈⋊₂D₈ order 448 = 2⁶·7

2^nd semidirect product of C₂₈ and D₈ acting via D₈/C₄=C₂²