metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.63D10, (C2×C20).75D4, C22⋊Q8.2D5, (C2×Q8).25D10, Q8⋊Dic5⋊12C2, C20.Q8⋊37C2, C10.71(C2×SD16), (C2×C10).16SD16, (C22×C10).89D4, C22.6(Q8⋊D5), C20.187(C4○D4), C4.93(D4⋊2D5), (C2×C20).362C23, C20.55D4.6C2, (C22×C4).124D10, C23.60(C5⋊D4), C5⋊5(C23.47D4), (Q8×C10).43C22, C4⋊Dic5.337C22, C2.14(D4.9D10), C10.116(C8.C22), (C22×C20).166C22, C10.80(C22.D4), C2.14(C23.18D10), C2.8(C2×Q8⋊D5), (C5×C22⋊Q8).1C2, (C2×C10).493(C2×D4), (C2×C4).53(C5⋊D4), (C2×C4⋊Dic5).38C2, (C5×C4⋊C4).110C22, (C2×C4).462(C22×D5), C22.168(C2×C5⋊D4), (C2×C5⋊2C8).112C22, SmallGroup(320,670)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊Q8.D5
G = < a,b,c,d,e,f | a2=b2=c4=e5=1, d2=c2, f2=b, dad-1=ab=ba, ac=ca, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fcf-1=c-1, ce=ec, de=ed, fdf-1=c-1d, fef-1=e-1 >
Subgroups: 334 in 104 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C4.Q8, C2×C4⋊C4, C22⋊Q8, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×C10, C23.47D4, C2×C5⋊2C8, C4⋊Dic5, C4⋊Dic5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, Q8×C10, C20.Q8, C20.55D4, Q8⋊Dic5, C2×C4⋊Dic5, C5×C22⋊Q8, C22⋊Q8.D5
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C22.D4, C2×SD16, C8.C22, C5⋊D4, C22×D5, C23.47D4, Q8⋊D5, D4⋊2D5, C2×C5⋊D4, C23.18D10, C2×Q8⋊D5, D4.9D10, C22⋊Q8.D5
►On 160 points
Generators in S160
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)(81 86)(82 87)(83 88)(84 89)(85 90)(91 96)(92 97)(93 98)(94 99)(95 100)(101 106)(102 107)(103 108)(104 109)(105 110)(111 116)(112 117)(113 118)(114 119)(115 120)(121 136)(122 137)(123 138)(124 139)(125 140)(126 131)(127 132)(128 133)(129 134)(130 135)(141 156)(142 157)(143 158)(144 159)(145 160)(146 151)(147 152)(148 153)(149 154)(150 155)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 61 46 66)(42 62 47 67)(43 63 48 68)(44 64 49 69)(45 65 50 70)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 60 80)(81 101 86 106)(82 102 87 107)(83 103 88 108)(84 104 89 109)(85 105 90 110)(91 111 96 116)(92 112 97 117)(93 113 98 118)(94 114 99 119)(95 115 100 120)(121 146 126 141)(122 147 127 142)(123 148 128 143)(124 149 129 144)(125 150 130 145)(131 156 136 151)(132 157 137 152)(133 158 138 153)(134 159 139 154)(135 160 140 155)
(1 49 9 44)(2 50 10 45)(3 46 6 41)(4 47 7 42)(5 48 8 43)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)(81 146 86 141)(82 147 87 142)(83 148 88 143)(84 149 89 144)(85 150 90 145)(91 156 96 151)(92 157 97 152)(93 158 98 153)(94 159 99 154)(95 160 100 155)(101 121 106 126)(102 122 107 127)(103 123 108 128)(104 124 109 129)(105 125 110 130)(111 131 116 136)(112 132 117 137)(113 133 118 138)(114 134 119 139)(115 135 120 140)
(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)
(1 95 14 85)(2 94 15 84)(3 93 11 83)(4 92 12 82)(5 91 13 81)(6 98 16 88)(7 97 17 87)(8 96 18 86)(9 100 19 90)(10 99 20 89)(21 113 31 103)(22 112 32 102)(23 111 33 101)(24 115 34 105)(25 114 35 104)(26 118 36 108)(27 117 37 107)(28 116 38 106)(29 120 39 110)(30 119 40 109)(41 133 51 123)(42 132 52 122)(43 131 53 121)(44 135 54 125)(45 134 55 124)(46 138 56 128)(47 137 57 127)(48 136 58 126)(49 140 59 130)(50 139 60 129)(61 153 71 143)(62 152 72 142)(63 151 73 141)(64 155 74 145)(65 154 75 144)(66 158 76 148)(67 157 77 147)(68 156 78 146)(69 160 79 150)(70 159 80 149)
G:=sub<Sym(160)| (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75)(81,86)(82,87)(83,88)(84,89)(85,90)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,116)(112,117)(113,118)(114,119)(115,120)(121,136)(122,137)(123,138)(124,139)(125,140)(126,131)(127,132)(128,133)(129,134)(130,135)(141,156)(142,157)(143,158)(144,159)(145,160)(146,151)(147,152)(148,153)(149,154)(150,155), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,29,9,24)(2,30,10,25)(3,26,6,21)(4,27,7,22)(5,28,8,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,61,46,66)(42,62,47,67)(43,63,48,68)(44,64,49,69)(45,65,50,70)(51,71,56,76)(52,72,57,77)(53,73,58,78)(54,74,59,79)(55,75,60,80)(81,101,86,106)(82,102,87,107)(83,103,88,108)(84,104,89,109)(85,105,90,110)(91,111,96,116)(92,112,97,117)(93,113,98,118)(94,114,99,119)(95,115,100,120)(121,146,126,141)(122,147,127,142)(123,148,128,143)(124,149,129,144)(125,150,130,145)(131,156,136,151)(132,157,137,152)(133,158,138,153)(134,159,139,154)(135,160,140,155), (1,49,9,44)(2,50,10,45)(3,46,6,41)(4,47,7,42)(5,48,8,43)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75)(81,146,86,141)(82,147,87,142)(83,148,88,143)(84,149,89,144)(85,150,90,145)(91,156,96,151)(92,157,97,152)(93,158,98,153)(94,159,99,154)(95,160,100,155)(101,121,106,126)(102,122,107,127)(103,123,108,128)(104,124,109,129)(105,125,110,130)(111,131,116,136)(112,132,117,137)(113,133,118,138)(114,134,119,139)(115,135,120,140), (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), (1,95,14,85)(2,94,15,84)(3,93,11,83)(4,92,12,82)(5,91,13,81)(6,98,16,88)(7,97,17,87)(8,96,18,86)(9,100,19,90)(10,99,20,89)(21,113,31,103)(22,112,32,102)(23,111,33,101)(24,115,34,105)(25,114,35,104)(26,118,36,108)(27,117,37,107)(28,116,38,106)(29,120,39,110)(30,119,40,109)(41,133,51,123)(42,132,52,122)(43,131,53,121)(44,135,54,125)(45,134,55,124)(46,138,56,128)(47,137,57,127)(48,136,58,126)(49,140,59,130)(50,139,60,129)(61,153,71,143)(62,152,72,142)(63,151,73,141)(64,155,74,145)(65,154,75,144)(66,158,76,148)(67,157,77,147)(68,156,78,146)(69,160,79,150)(70,159,80,149)>;
G:=Group( (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75)(81,86)(82,87)(83,88)(84,89)(85,90)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,116)(112,117)(113,118)(114,119)(115,120)(121,136)(122,137)(123,138)(124,139)(125,140)(126,131)(127,132)(128,133)(129,134)(130,135)(141,156)(142,157)(143,158)(144,159)(145,160)(146,151)(147,152)(148,153)(149,154)(150,155), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,29,9,24)(2,30,10,25)(3,26,6,21)(4,27,7,22)(5,28,8,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,61,46,66)(42,62,47,67)(43,63,48,68)(44,64,49,69)(45,65,50,70)(51,71,56,76)(52,72,57,77)(53,73,58,78)(54,74,59,79)(55,75,60,80)(81,101,86,106)(82,102,87,107)(83,103,88,108)(84,104,89,109)(85,105,90,110)(91,111,96,116)(92,112,97,117)(93,113,98,118)(94,114,99,119)(95,115,100,120)(121,146,126,141)(122,147,127,142)(123,148,128,143)(124,149,129,144)(125,150,130,145)(131,156,136,151)(132,157,137,152)(133,158,138,153)(134,159,139,154)(135,160,140,155), (1,49,9,44)(2,50,10,45)(3,46,6,41)(4,47,7,42)(5,48,8,43)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75)(81,146,86,141)(82,147,87,142)(83,148,88,143)(84,149,89,144)(85,150,90,145)(91,156,96,151)(92,157,97,152)(93,158,98,153)(94,159,99,154)(95,160,100,155)(101,121,106,126)(102,122,107,127)(103,123,108,128)(104,124,109,129)(105,125,110,130)(111,131,116,136)(112,132,117,137)(113,133,118,138)(114,134,119,139)(115,135,120,140), (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), (1,95,14,85)(2,94,15,84)(3,93,11,83)(4,92,12,82)(5,91,13,81)(6,98,16,88)(7,97,17,87)(8,96,18,86)(9,100,19,90)(10,99,20,89)(21,113,31,103)(22,112,32,102)(23,111,33,101)(24,115,34,105)(25,114,35,104)(26,118,36,108)(27,117,37,107)(28,116,38,106)(29,120,39,110)(30,119,40,109)(41,133,51,123)(42,132,52,122)(43,131,53,121)(44,135,54,125)(45,134,55,124)(46,138,56,128)(47,137,57,127)(48,136,58,126)(49,140,59,130)(50,139,60,129)(61,153,71,143)(62,152,72,142)(63,151,73,141)(64,155,74,145)(65,154,75,144)(66,158,76,148)(67,157,77,147)(68,156,78,146)(69,160,79,150)(70,159,80,149) );
G=PermutationGroup([[(1,9),(2,10),(3,6),(4,7),(5,8),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,56),(42,57),(43,58),(44,59),(45,60),(46,51),(47,52),(48,53),(49,54),(50,55),(61,76),(62,77),(63,78),(64,79),(65,80),(66,71),(67,72),(68,73),(69,74),(70,75),(81,86),(82,87),(83,88),(84,89),(85,90),(91,96),(92,97),(93,98),(94,99),(95,100),(101,106),(102,107),(103,108),(104,109),(105,110),(111,116),(112,117),(113,118),(114,119),(115,120),(121,136),(122,137),(123,138),(124,139),(125,140),(126,131),(127,132),(128,133),(129,134),(130,135),(141,156),(142,157),(143,158),(144,159),(145,160),(146,151),(147,152),(148,153),(149,154),(150,155)], [(1,14),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160)], [(1,29,9,24),(2,30,10,25),(3,26,6,21),(4,27,7,22),(5,28,8,23),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,61,46,66),(42,62,47,67),(43,63,48,68),(44,64,49,69),(45,65,50,70),(51,71,56,76),(52,72,57,77),(53,73,58,78),(54,74,59,79),(55,75,60,80),(81,101,86,106),(82,102,87,107),(83,103,88,108),(84,104,89,109),(85,105,90,110),(91,111,96,116),(92,112,97,117),(93,113,98,118),(94,114,99,119),(95,115,100,120),(121,146,126,141),(122,147,127,142),(123,148,128,143),(124,149,129,144),(125,150,130,145),(131,156,136,151),(132,157,137,152),(133,158,138,153),(134,159,139,154),(135,160,140,155)], [(1,49,9,44),(2,50,10,45),(3,46,6,41),(4,47,7,42),(5,48,8,43),(11,56,16,51),(12,57,17,52),(13,58,18,53),(14,59,19,54),(15,60,20,55),(21,66,26,61),(22,67,27,62),(23,68,28,63),(24,69,29,64),(25,70,30,65),(31,76,36,71),(32,77,37,72),(33,78,38,73),(34,79,39,74),(35,80,40,75),(81,146,86,141),(82,147,87,142),(83,148,88,143),(84,149,89,144),(85,150,90,145),(91,156,96,151),(92,157,97,152),(93,158,98,153),(94,159,99,154),(95,160,100,155),(101,121,106,126),(102,122,107,127),(103,123,108,128),(104,124,109,129),(105,125,110,130),(111,131,116,136),(112,132,117,137),(113,133,118,138),(114,134,119,139),(115,135,120,140)], [(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)], [(1,95,14,85),(2,94,15,84),(3,93,11,83),(4,92,12,82),(5,91,13,81),(6,98,16,88),(7,97,17,87),(8,96,18,86),(9,100,19,90),(10,99,20,89),(21,113,31,103),(22,112,32,102),(23,111,33,101),(24,115,34,105),(25,114,35,104),(26,118,36,108),(27,117,37,107),(28,116,38,106),(29,120,39,110),(30,119,40,109),(41,133,51,123),(42,132,52,122),(43,131,53,121),(44,135,54,125),(45,134,55,124),(46,138,56,128),(47,137,57,127),(48,136,58,126),(49,140,59,130),(50,139,60,129),(61,153,71,143),(62,152,72,142),(63,151,73,141),(64,155,74,145),(65,154,75,144),(66,158,76,148),(67,157,77,147),(68,156,78,146),(69,160,79,150),(70,159,80,149)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | SD16 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8.C22 | D4⋊2D5 | Q8⋊D5 | D4.9D10 |
| kernel | C22⋊Q8.D5 | C20.Q8 | C20.55D4 | Q8⋊Dic5 | C2×C4⋊Dic5 | C5×C22⋊Q8 | C2×C20 | C22×C10 | C22⋊Q8 | C20 | C2×C10 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C4 | C23 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 4 | 4 | 4 |
Matrix representation of C22⋊Q8.D5 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 8 |
| 0 | 0 | 0 | 0 | 23 | 26 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 9 | 0 | 0 |
| 0 | 0 | 40 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 30 |
| 0 | 0 | 0 | 0 | 34 | 2 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 33 | 0 | 0 | 0 | 0 |
| 3 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 19 |
| 0 | 0 | 0 | 0 | 1 | 8 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,23,0,0,0,0,8,26],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,40,0,0,0,0,9,25,0,0,0,0,0,0,39,34,0,0,0,0,30,2],[0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,3,0,0,0,0,33,5,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,33,1,0,0,0,0,19,8] >;
C22⋊Q8.D5 in GAP, Magma, Sage, TeX
C_2^2\rtimes Q_8.D_5
% in TeX
G:=Group("C2^2:Q8.D5"); // GroupNames label
G:=SmallGroup(320,670);
// by ID
G=gap.SmallGroup(320,670);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,254,219,184,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=e^5=1,d^2=c^2,f^2=b,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=f*c*f^-1=c^-1,c*e=e*c,d*e=e*d,f*d*f^-1=c^-1*d,f*e*f^-1=e^-1>;
// generators/relations