C2^3.486C2^4 - GroupNames

G = C₂³.₄₈₆C₂⁴ order 128 = 2⁷

203^rd central stem extension by C₂³ of C₂⁴

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂³.₄₈₆C₂⁴, C₂².₂₆₈2₊¹⁺⁴, C₂.₈Q₈², C₄⋊C₄⋊₂₂Q₈, C₂.₄₄(D₄⋊₃Q₈), (C₂×C₄²).₅₈₀C₂², (C₂²×C₄).₈₄₇C₂³, C₂².₁₂₁(C₂²×Q₈), (C₂²×Q₈).₁₄₄C₂², C₂.₃₀(C₂².₃₂C₂⁴), C₂.₆₀(C₂².₄₅C₂⁴), C₂³.₆₅C₂³.₆₂C₂, C₂.C₄².₂₂₀C₂², C₂³.₆₇C₂³.₄₅C₂, C₂³.₈₁C₂³.₂₁C₂, C₂³.₆₃C₂³.₃₁C₂, C₂³.₇₈C₂³.₁₁C₂, C₂.₃₂(C₂³.₃₇C₂³), (C₄×C₄⋊C₄).₇₃C₂, (C₂×C₄).₆₃(C₂×Q₈), (C₂×C₄).₄₀₀(C₄○D₄), (C₂×C₄⋊C₄).₃₃₂C₂², C₂².₃₆₂(C₂×C₄○D₄), SmallGroup(128,1318)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂³ — C₂³.₄₈₆C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₄₈₆C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₈₆C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₈₆C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₈₆C₂⁴

Generators and relations for C₂³.₄₈₆C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=a, e²=f²=ca=ac, g²=b, ab=ba, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 356 in 206 conjugacy classes, 108 normal (22 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₄², C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²×Q₈, C₄×C₄⋊C₄, C₂³.₆₃C₂³, C₂³.₆₅C₂³, C₂³.₆₇C₂³, C₂³.₆₇C₂³, C₂³.₇₈C₂³, C₂³.₇₈C₂³, C₂³.₈₁C₂³, C₂³.₄₈₆C₂⁴
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₇C₂³, C₂².₃₂C₂⁴, C₂².₄₅C₂⁴, D₄⋊₃Q₈, Q₈², C₂³.₄₈₆C₂⁴

Smallest permutation representation of C₂³.₄₈₆C₂⁴
►Regular action on 128 points

Generators in S₁₂₈

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)(65 67)(66 68)(69 71)(70 72)(73 75)(74 76)(77 79)(78 80)(81 83)(82 84)(85 87)(86 88)(89 91)(90 92)(93 95)(94 96)(97 99)(98 100)(101 103)(102 104)(105 107)(106 108)(109 111)(110 112)(113 115)(114 116)(117 119)(118 120)(121 123)(122 124)(125 127)(126 128)

(1 9)(2 10)(3 11)(4 12)(5 71)(6 72)(7 69)(8 70)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 103)(42 104)(43 101)(44 102)(45 107)(46 108)(47 105)(48 106)(49 111)(50 112)(51 109)(52 110)(53 115)(54 116)(55 113)(56 114)(57 119)(58 120)(59 117)(60 118)(61 123)(62 124)(63 121)(64 122)(65 126)(66 127)(67 128)(68 125)

(1 101)(2 102)(3 103)(4 104)(5 100)(6 97)(7 98)(8 99)(9 43)(10 44)(11 41)(12 42)(13 47)(14 48)(15 45)(16 46)(17 51)(18 52)(19 49)(20 50)(21 55)(22 56)(23 53)(24 54)(25 59)(26 60)(27 57)(28 58)(29 63)(30 64)(31 61)(32 62)(33 68)(34 65)(35 66)(36 67)(37 72)(38 69)(39 70)(40 71)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 121)(90 122)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)

(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)

(1 119 103 85)(2 118 104 88)(3 117 101 87)(4 120 102 86)(5 114 98 84)(6 113 99 83)(7 116 100 82)(8 115 97 81)(9 57 41 25)(10 60 42 28)(11 59 43 27)(12 58 44 26)(13 61 45 29)(14 64 46 32)(15 63 47 31)(16 62 48 30)(17 66 49 33)(18 65 50 36)(19 68 51 35)(20 67 52 34)(21 70 53 37)(22 69 54 40)(23 72 55 39)(24 71 56 38)(73 123 107 89)(74 122 108 92)(75 121 105 91)(76 124 106 90)(77 127 111 93)(78 126 112 96)(79 125 109 95)(80 128 110 94)

(1 111 103 77)(2 50 104 18)(3 109 101 79)(4 52 102 20)(5 32 98 64)(6 89 99 123)(7 30 100 62)(8 91 97 121)(9 49 41 17)(10 112 42 78)(11 51 43 19)(12 110 44 80)(13 53 45 21)(14 116 46 82)(15 55 47 23)(16 114 48 84)(22 74 54 108)(24 76 56 106)(25 35 57 68)(26 96 58 126)(27 33 59 66)(28 94 60 128)(29 39 61 72)(31 37 63 70)(34 118 67 88)(36 120 65 86)(38 122 71 92)(40 124 69 90)(73 115 107 81)(75 113 105 83)(85 95 119 125)(87 93 117 127)

(1 13 9 73)(2 16 10 76)(3 15 11 75)(4 14 12 74)(5 126 71 65)(6 125 72 68)(7 128 69 67)(8 127 70 66)(17 81 77 21)(18 84 78 24)(19 83 79 23)(20 82 80 22)(25 89 85 29)(26 92 86 32)(27 91 87 31)(28 90 88 30)(33 97 93 37)(34 100 94 40)(35 99 95 39)(36 98 96 38)(41 107 103 45)(42 106 104 48)(43 105 101 47)(44 108 102 46)(49 115 111 53)(50 114 112 56)(51 113 109 55)(52 116 110 54)(57 123 119 61)(58 122 120 64)(59 121 117 63)(60 124 118 62)

G:=sub<Sym(128)| (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84)(85,87)(86,88)(89,91)(90,92)(93,95)(94,96)(97,99)(98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128), (1,9)(2,10)(3,11)(4,12)(5,71)(6,72)(7,69)(8,70)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,103)(42,104)(43,101)(44,102)(45,107)(46,108)(47,105)(48,106)(49,111)(50,112)(51,109)(52,110)(53,115)(54,116)(55,113)(56,114)(57,119)(58,120)(59,117)(60,118)(61,123)(62,124)(63,121)(64,122)(65,126)(66,127)(67,128)(68,125), (1,101)(2,102)(3,103)(4,104)(5,100)(6,97)(7,98)(8,99)(9,43)(10,44)(11,41)(12,42)(13,47)(14,48)(15,45)(16,46)(17,51)(18,52)(19,49)(20,50)(21,55)(22,56)(23,53)(24,54)(25,59)(26,60)(27,57)(28,58)(29,63)(30,64)(31,61)(32,62)(33,68)(34,65)(35,66)(36,67)(37,72)(38,69)(39,70)(40,71)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128), (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), (1,119,103,85)(2,118,104,88)(3,117,101,87)(4,120,102,86)(5,114,98,84)(6,113,99,83)(7,116,100,82)(8,115,97,81)(9,57,41,25)(10,60,42,28)(11,59,43,27)(12,58,44,26)(13,61,45,29)(14,64,46,32)(15,63,47,31)(16,62,48,30)(17,66,49,33)(18,65,50,36)(19,68,51,35)(20,67,52,34)(21,70,53,37)(22,69,54,40)(23,72,55,39)(24,71,56,38)(73,123,107,89)(74,122,108,92)(75,121,105,91)(76,124,106,90)(77,127,111,93)(78,126,112,96)(79,125,109,95)(80,128,110,94), (1,111,103,77)(2,50,104,18)(3,109,101,79)(4,52,102,20)(5,32,98,64)(6,89,99,123)(7,30,100,62)(8,91,97,121)(9,49,41,17)(10,112,42,78)(11,51,43,19)(12,110,44,80)(13,53,45,21)(14,116,46,82)(15,55,47,23)(16,114,48,84)(22,74,54,108)(24,76,56,106)(25,35,57,68)(26,96,58,126)(27,33,59,66)(28,94,60,128)(29,39,61,72)(31,37,63,70)(34,118,67,88)(36,120,65,86)(38,122,71,92)(40,124,69,90)(73,115,107,81)(75,113,105,83)(85,95,119,125)(87,93,117,127), (1,13,9,73)(2,16,10,76)(3,15,11,75)(4,14,12,74)(5,126,71,65)(6,125,72,68)(7,128,69,67)(8,127,70,66)(17,81,77,21)(18,84,78,24)(19,83,79,23)(20,82,80,22)(25,89,85,29)(26,92,86,32)(27,91,87,31)(28,90,88,30)(33,97,93,37)(34,100,94,40)(35,99,95,39)(36,98,96,38)(41,107,103,45)(42,106,104,48)(43,105,101,47)(44,108,102,46)(49,115,111,53)(50,114,112,56)(51,113,109,55)(52,116,110,54)(57,123,119,61)(58,122,120,64)(59,121,117,63)(60,124,118,62)>;

G:=Group( (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84)(85,87)(86,88)(89,91)(90,92)(93,95)(94,96)(97,99)(98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128), (1,9)(2,10)(3,11)(4,12)(5,71)(6,72)(7,69)(8,70)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,103)(42,104)(43,101)(44,102)(45,107)(46,108)(47,105)(48,106)(49,111)(50,112)(51,109)(52,110)(53,115)(54,116)(55,113)(56,114)(57,119)(58,120)(59,117)(60,118)(61,123)(62,124)(63,121)(64,122)(65,126)(66,127)(67,128)(68,125), (1,101)(2,102)(3,103)(4,104)(5,100)(6,97)(7,98)(8,99)(9,43)(10,44)(11,41)(12,42)(13,47)(14,48)(15,45)(16,46)(17,51)(18,52)(19,49)(20,50)(21,55)(22,56)(23,53)(24,54)(25,59)(26,60)(27,57)(28,58)(29,63)(30,64)(31,61)(32,62)(33,68)(34,65)(35,66)(36,67)(37,72)(38,69)(39,70)(40,71)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128), (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), (1,119,103,85)(2,118,104,88)(3,117,101,87)(4,120,102,86)(5,114,98,84)(6,113,99,83)(7,116,100,82)(8,115,97,81)(9,57,41,25)(10,60,42,28)(11,59,43,27)(12,58,44,26)(13,61,45,29)(14,64,46,32)(15,63,47,31)(16,62,48,30)(17,66,49,33)(18,65,50,36)(19,68,51,35)(20,67,52,34)(21,70,53,37)(22,69,54,40)(23,72,55,39)(24,71,56,38)(73,123,107,89)(74,122,108,92)(75,121,105,91)(76,124,106,90)(77,127,111,93)(78,126,112,96)(79,125,109,95)(80,128,110,94), (1,111,103,77)(2,50,104,18)(3,109,101,79)(4,52,102,20)(5,32,98,64)(6,89,99,123)(7,30,100,62)(8,91,97,121)(9,49,41,17)(10,112,42,78)(11,51,43,19)(12,110,44,80)(13,53,45,21)(14,116,46,82)(15,55,47,23)(16,114,48,84)(22,74,54,108)(24,76,56,106)(25,35,57,68)(26,96,58,126)(27,33,59,66)(28,94,60,128)(29,39,61,72)(31,37,63,70)(34,118,67,88)(36,120,65,86)(38,122,71,92)(40,124,69,90)(73,115,107,81)(75,113,105,83)(85,95,119,125)(87,93,117,127), (1,13,9,73)(2,16,10,76)(3,15,11,75)(4,14,12,74)(5,126,71,65)(6,125,72,68)(7,128,69,67)(8,127,70,66)(17,81,77,21)(18,84,78,24)(19,83,79,23)(20,82,80,22)(25,89,85,29)(26,92,86,32)(27,91,87,31)(28,90,88,30)(33,97,93,37)(34,100,94,40)(35,99,95,39)(36,98,96,38)(41,107,103,45)(42,106,104,48)(43,105,101,47)(44,108,102,46)(49,115,111,53)(50,114,112,56)(51,113,109,55)(52,116,110,54)(57,123,119,61)(58,122,120,64)(59,121,117,63)(60,124,118,62) );

G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64),(65,67),(66,68),(69,71),(70,72),(73,75),(74,76),(77,79),(78,80),(81,83),(82,84),(85,87),(86,88),(89,91),(90,92),(93,95),(94,96),(97,99),(98,100),(101,103),(102,104),(105,107),(106,108),(109,111),(110,112),(113,115),(114,116),(117,119),(118,120),(121,123),(122,124),(125,127),(126,128)], [(1,9),(2,10),(3,11),(4,12),(5,71),(6,72),(7,69),(8,70),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,103),(42,104),(43,101),(44,102),(45,107),(46,108),(47,105),(48,106),(49,111),(50,112),(51,109),(52,110),(53,115),(54,116),(55,113),(56,114),(57,119),(58,120),(59,117),(60,118),(61,123),(62,124),(63,121),(64,122),(65,126),(66,127),(67,128),(68,125)], [(1,101),(2,102),(3,103),(4,104),(5,100),(6,97),(7,98),(8,99),(9,43),(10,44),(11,41),(12,42),(13,47),(14,48),(15,45),(16,46),(17,51),(18,52),(19,49),(20,50),(21,55),(22,56),(23,53),(24,54),(25,59),(26,60),(27,57),(28,58),(29,63),(30,64),(31,61),(32,62),(33,68),(34,65),(35,66),(36,67),(37,72),(38,69),(39,70),(40,71),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112),(81,113),(82,114),(83,115),(84,116),(85,117),(86,118),(87,119),(88,120),(89,121),(90,122),(91,123),(92,124),(93,125),(94,126),(95,127),(96,128)], [(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)], [(1,119,103,85),(2,118,104,88),(3,117,101,87),(4,120,102,86),(5,114,98,84),(6,113,99,83),(7,116,100,82),(8,115,97,81),(9,57,41,25),(10,60,42,28),(11,59,43,27),(12,58,44,26),(13,61,45,29),(14,64,46,32),(15,63,47,31),(16,62,48,30),(17,66,49,33),(18,65,50,36),(19,68,51,35),(20,67,52,34),(21,70,53,37),(22,69,54,40),(23,72,55,39),(24,71,56,38),(73,123,107,89),(74,122,108,92),(75,121,105,91),(76,124,106,90),(77,127,111,93),(78,126,112,96),(79,125,109,95),(80,128,110,94)], [(1,111,103,77),(2,50,104,18),(3,109,101,79),(4,52,102,20),(5,32,98,64),(6,89,99,123),(7,30,100,62),(8,91,97,121),(9,49,41,17),(10,112,42,78),(11,51,43,19),(12,110,44,80),(13,53,45,21),(14,116,46,82),(15,55,47,23),(16,114,48,84),(22,74,54,108),(24,76,56,106),(25,35,57,68),(26,96,58,126),(27,33,59,66),(28,94,60,128),(29,39,61,72),(31,37,63,70),(34,118,67,88),(36,120,65,86),(38,122,71,92),(40,124,69,90),(73,115,107,81),(75,113,105,83),(85,95,119,125),(87,93,117,127)], [(1,13,9,73),(2,16,10,76),(3,15,11,75),(4,14,12,74),(5,126,71,65),(6,125,72,68),(7,128,69,67),(8,127,70,66),(17,81,77,21),(18,84,78,24),(19,83,79,23),(20,82,80,22),(25,89,85,29),(26,92,86,32),(27,91,87,31),(28,90,88,30),(33,97,93,37),(34,100,94,40),(35,99,95,39),(36,98,96,38),(41,107,103,45),(42,106,104,48),(43,105,101,47),(44,108,102,46),(49,115,111,53),(50,114,112,56),(51,113,109,55),(52,116,110,54),(57,123,119,61),(58,122,120,64),(59,121,117,63),(60,124,118,62)]])

38 conjugacy classes

class	1	2A	···	2G	4A	···	4H	4I	···	4Z	4AA	4AB	4AC	4AD
order	1	2	···	2	4	···	4	4	···	4	4	4	4	4
size	1	1	···	1	2	···	2	4	···	4	8	8	8	8

38 irreducible representations

dim	1	1	1	1	1	1	1	2	2	4
type	+	+	+	+	+	+	+	-		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	C₄○D₄	2₊¹⁺⁴
kernel	C₂³.₄₈₆C₂⁴	C₄×C₄⋊C₄	C₂³.₆₃C₂³	C₂³.₆₅C₂³	C₂³.₆₇C₂³	C₂³.₇₈C₂³	C₂³.₈₁C₂³	C₄⋊C₄	C₂×C₄	C₂²
# reps	1	2	4	2	3	3	1	8	12	2

Matrix representation of C₂³.₄₈₆C₂⁴ ►in GL₆(𝔽₅)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	1	0
0	0	0	0	0	1

4	0	0	0	0	0
0	4	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

4	0	0	0	0	0
0	4	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	4

4	2	0	0	0	0
0	1	0	0	0	0
0	0	2	0	0	0
0	0	1	3	0	0
0	0	0	0	1	0
0	0	0	0	0	1

2	1	0	0	0	0
0	3	0	0	0	0
0	0	1	1	0	0
0	0	3	4	0	0
0	0	0	0	2	0
0	0	0	0	3	3

1	3	0	0	0	0
1	4	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	1	2
0	0	0	0	4	4

3	0	0	0	0	0
0	3	0	0	0	0
0	0	2	2	0	0
0	0	1	3	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,2,1,0,0,0,0,0,0,2,1,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,1,3,0,0,0,0,0,0,1,3,0,0,0,0,1,4,0,0,0,0,0,0,2,3,0,0,0,0,0,3],[1,1,0,0,0,0,3,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,4,0,0,0,0,2,4],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,1,0,0,0,0,2,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂³.₄₈₆C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{486}C_2^4

% in TeX

G:=Group("C2^3.486C2^4");

// GroupNames label

G:=SmallGroup(128,1318);

// by ID

G=gap.SmallGroup(128,1318);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,568,758,723,436,675,136]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=a,e^2=f^2=c*a=a*c,g^2=b,a*b=b*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;

// generators/relations

G = C23.486C24 order 128 = 27

203rd central stem extension by C23 of C24

G = C₂³.₄₈₆C₂⁴ order 128 = 2⁷

203^rd central stem extension by C₂³ of C₂⁴