C2^3.424C2^4 - GroupNames

G = C₂³.₄₂₄C₂⁴ order 128 = 2⁷

141^st central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₄₂₄C₂⁴, C₂².₁₆₅2_-¹⁺⁴, C₄²⋊₅C₄.₈C₂, C₄²⋊₈C₄.₃₀C₂, (C₂²×C₄).₈₆C₂³, (C₂×C₄²).₅₃₉C₂², C₂³.₈₁C₂³.₁₁C₂, C₂³.₆₃C₂³.₂₄C₂, C₂³.₈₃C₂³.₁₁C₂, C₂.C₄².₁₇₂C₂², C₂.₂₁(C₂².₃₅C₂⁴), C₂.₅₂(C₂².₄₆C₂⁴), C₂.₆₇(C₂³.₃₆C₂³), C₂.₃₀(C₂².₅₀C₂⁴), (C₄×C₄⋊C₄).₅₉C₂, (C₂×C₄).₁₄₃(C₄○D₄), (C₂×C₄⋊C₄).₈₆₄C₂², C₂².₃₀₁(C₂×C₄○D₄), SmallGroup(128,1256)

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

Derived series

C₁ — C₂³ — C₂³.₄₂₄C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₄₂₄C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₂₄C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₂₄C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₂₄C₂⁴

Generators and relations for C₂³.₄₂₄C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=c, e²=f²=a, g²=ba=ab, ac=ca, 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: 292 in 182 conjugacy classes, 92 normal (82 characteristic)
C₁, C₂, C₄, C₂², C₂×C₄, C₂×C₄, C₂³, C₄², C₄⋊C₄, C₂²×C₄, C₂.C₄², C₂×C₄², C₂×C₄⋊C₄, C₄×C₄⋊C₄, C₄²⋊₈C₄, C₄²⋊₅C₄, C₂³.₆₃C₂³, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂³.₄₂₄C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2_-¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₅C₂⁴, C₂².₄₆C₂⁴, C₂².₅₀C₂⁴, C₂³.₄₂₄C₂⁴

Smallest permutation representation of C₂³.₄₂₄C₂⁴
►Regular action on 128 points

Generators in S₁₂₈

(1 11)(2 12)(3 9)(4 10)(5 70)(6 71)(7 72)(8 69)(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)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 127)(66 128)(67 125)(68 126)

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

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

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

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

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

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

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

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	4
type	+	+	+	+	+	+	+		-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₄○D₄	2_-¹⁺⁴
kernel	C₂³.₄₂₄C₂⁴	C₄×C₄⋊C₄	C₄²⋊₈C₄	C₄²⋊₅C₄	C₂³.₆₃C₂³	C₂³.₈₁C₂³	C₂³.₈₃C₂³	C₂×C₄	C₂²
# reps	1	2	1	2	6	2	2	20	2

Matrix representation of C₂³.₄₂₄C₂⁴ ►in GL₆(𝔽₅)

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	4	0	0	0
0	0	0	4	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	4	0
0	0	0	0	0	4

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

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

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

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

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

C₂³.₄₂₄C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{424}C_2^4

% in TeX

G:=Group("C2^3.424C2^4");

// GroupNames label

G:=SmallGroup(128,1256);

// by ID

G=gap.SmallGroup(128,1256);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,100,675,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=f^2=a,g^2=b*a=a*b,a*c=c*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 = C23.424C24 order 128 = 27

141st central stem extension by C23 of C24

G = C₂³.₄₂₄C₂⁴ order 128 = 2⁷

141^st central stem extension by C₂³ of C₂⁴