C56:8D4 - GroupNames

G = C₅₆⋊₈D₄ order 448 = 2⁶·7

8^th semidirect product of C₅₆ and D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₅₆⋊₈D₄, C₈⋊₅(C₇⋊D₄), C₇⋊₇(C₈⋊D₄), (C₂×SD₁₆)⋊₁D₇, C₅₆⋊₁C₄⋊₂₇C₂, (C₂×C₈).₉₀D₁₄, D₁₄⋊₃Q₈⋊₆C₂, (C₁₄×SD₁₆)⋊₃C₂, (C₂×D₄).₇₅D₁₄, C₂₈⋊₂D₄.₉C₂, C₂₈.₁₇₈(C₂×D₄), (C₂×Q₈).₅₆D₁₄, D₄⋊Dic₇⋊₃₆C₂, Q₈⋊Dic₇⋊₃₁C₂, (C₂×Dic₇).₇₃D₄, (C₂²×D₇).₃₉D₄, C₂².₂₇₁(D₄×D₇), C₂₈.₁₀₂(C₄○D₄), C₄.₃₃(D₄⋊₂D₇), C₂.₂₀(C₂₈⋊₂D₄), C₂.₃₀(D₅₆⋊C₂), C₁₄.₈₀(C₈⋊C₂²), (C₂×C₂₈).₄₅₁C₂³, (C₂×C₅₆).₁₁₅C₂², (Q₈×C₁₄).₈₀C₂², C₁₄.₁₁₇(C₄⋊D₄), C₂.₃₁(SD₁₆⋊D₇), (D₄×C₁₄).₁₀₀C₂², C₁₄.₅₁(C₈.C₂²), C₄⋊Dic₇.₁₇₈C₂², (C₂×C₈⋊D₇)⋊₃C₂, C₄.₈₃(C₂×C₇⋊D₄), (C₂×C₄×D₇).₅₁C₂², (C₂×C₁₄).₃₆₃(C₂×D₄), (C₂×C₇⋊C₈).₁₆₀C₂², (C₂×C₄).₅₄₀(C₂²×D₇), SmallGroup(448,708)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — C₅₆⋊₈D₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×C₂₈ — C₂×C₄×D₇ — C₂×C₈⋊D₇ — C₅₆⋊₈D₄

Lower central

C₇ — C₁₄ — C₂×C₂₈ — C₅₆⋊₈D₄

Upper central

C₁ — C₂² — C₂×C₄ — C₂×SD₁₆

Generators and relations for C₅₆⋊₈D₄
G = < a,b,c | a⁵⁶=b⁴=c²=1, bab^-1=a^-1, cac=a¹³, cbc=b^-1 >

Subgroups: 612 in 120 conjugacy classes, 41 normal (37 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, D₇, C₁₄, C₁₄, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), SD₁₆, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, Dic₇, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, D₄⋊C₄, Q₈⋊C₄, C₂.D₈, C₄⋊D₄, C₂²⋊Q₈, C₂×M₄(2), C₂×SD₁₆, C₇⋊C₈, C₅₆, C₄×D₇, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×Q₈, C₂²×D₇, C₂²×C₁₄, C₈⋊D₄, C₈⋊D₇, C₂×C₇⋊C₈, Dic₇⋊C₄, C₄⋊Dic₇, D₁₄⋊C₄, C₂³.D₇, C₂×C₅₆, C₇×SD₁₆, C₂×C₄×D₇, C₂×C₇⋊D₄, D₄×C₁₄, Q₈×C₁₄, C₅₆⋊₁C₄, D₄⋊Dic₇, Q₈⋊Dic₇, C₂×C₈⋊D₇, C₂₈⋊₂D₄, D₁₄⋊₃Q₈, C₁₄×SD₁₆, C₅₆⋊₈D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₄○D₄, D₁₄, C₄⋊D₄, C₈⋊C₂², C₈.C₂², C₇⋊D₄, C₂²×D₇, C₈⋊D₄, D₄×D₇, D₄⋊₂D₇, C₂×C₇⋊D₄, D₅₆⋊C₂, SD₁₆⋊D₇, C₂₈⋊₂D₄, C₅₆⋊₈D₄

Smallest permutation representation of C₅₆⋊₈D₄
►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)

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

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

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

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

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

58 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	7A	7B	7C	8A	8B	8C	8D	14A	···	14I	14J	···	14O	28A	···	28F	28G	···	28L	56A	···	56L
order	1	2	2	2	2	2	4	4	4	4	4	4	7	7	7	8	8	8	8	14	···	14	14	···	14	28	···	28	28	···	28	56	···	56
size	1	1	1	1	8	28	2	2	8	28	56	56	2	2	2	4	4	28	28	2	···	2	8	···	8	4	···	4	8	···	8	4	···	4

58 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+		+	+	+		+	-	-	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₄	D₇	C₄○D₄	D₁₄	D₁₄	D₁₄	C₇⋊D₄	C₈⋊C₂²	C₈.C₂²	D₄⋊₂D₇	D₄×D₇	D₅₆⋊C₂	SD₁₆⋊D₇
kernel	C₅₆⋊₈D₄	C₅₆⋊₁C₄	D₄⋊Dic₇	Q₈⋊Dic₇	C₂×C₈⋊D₇	C₂₈⋊₂D₄	D₁₄⋊₃Q₈	C₁₄×SD₁₆	C₅₆	C₂×Dic₇	C₂²×D₇	C₂×SD₁₆	C₂₈	C₂×C₈	C₂×D₄	C₂×Q₈	C₈	C₁₄	C₁₄	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	2	1	1	3	2	3	3	3	12	1	1	3	3	6	6

Matrix representation of C₅₆⋊₈D₄ ►in GL₆(𝔽₁₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	34	24
0	0	0	0	88	10
0	0	97	28	96	20
0	0	97	69	93	50

1	77	0	0	0	0
44	112	0	0	0	0
0	0	0	10	51	55
0	0	110	0	4	77
0	0	61	21	14	0
0	0	57	36	13	99

1	0	0	0	0	0
44	112	0	0	0	0
0	0	0	80	0	0
0	0	89	0	0	0
0	0	0	0	1	0
0	0	0	0	9	112

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,97,97,0,0,0,0,28,69,0,0,34,88,96,93,0,0,24,10,20,50],[1,44,0,0,0,0,77,112,0,0,0,0,0,0,0,110,61,57,0,0,10,0,21,36,0,0,51,4,14,13,0,0,55,77,0,99],[1,44,0,0,0,0,0,112,0,0,0,0,0,0,0,89,0,0,0,0,80,0,0,0,0,0,0,0,1,9,0,0,0,0,0,112] >;

C₅₆⋊₈D₄ in GAP, Magma, Sage, TeX

C_{56}\rtimes_8D_4

% in TeX

G:=Group("C56:8D4");

// GroupNames label

G:=SmallGroup(448,708);

// by ID

G=gap.SmallGroup(448,708);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,120,254,555,438,102,18822]);

// Polycyclic

G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^13,c*b*c=b^-1>;

// generators/relations

G = C56⋊8D4 order 448 = 26·7

8th semidirect product of C56 and D4 acting via D4/C2=C22

G = C₅₆⋊₈D₄ order 448 = 2⁶·7

8^th semidirect product of C₅₆ and D₄ acting via D₄/C₂=C₂²