C2^3.706C2^4 - GroupNames

G = C₂³.₇₀₆C₂⁴ order 128 = 2⁷

423^rd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₇₀₆C₂⁴, C₂².₃₆₆2_-¹⁺⁴, C₂².₄₇₉2₊¹⁺⁴, C₄⋊C₄⋊₉Q₈, C₂.₁₁Q₈², (C₂×Q₈)⋊₅Q₈, C₂.₃₄(Q₈⋊₃Q₈), C₂.₆₉(D₄⋊₃Q₈), (C₂²×C₄).₆₁₂C₂³, (C₂×C₄²).₇₂₆C₂², C₂.₂₁(C₂³⋊₂Q₈), C₂².₁₆₈(C₂²×Q₈), (C₂²×Q₈).₂₂₇C₂², C₂.₁₁₁(C₂².₃₂C₂⁴), C₂³.₆₃C₂³.₅₉C₂, C₂³.₈₁C₂³.₄₈C₂, C₂³.₆₇C₂³.₆₁C₂, C₂³.₆₅C₂³.₈₆C₂, C₂.C₄².₄₁₀C₂², C₂³.₇₈C₂³.₂₈C₂, C₂.₄₇(C₂³.₄₁C₂³), C₂.₄₆(C₂².₅₆C₂⁴), (C₂×C₄).₉₂(C₂×Q₈), (C₂×C₄).₂₄₇(C₄○D₄), (C₂×C₄⋊C₄).₅₁₆C₂², C₂².₅₆₇(C₂×C₄○D₄), SmallGroup(128,1538)

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

Derived series

C₁ — C₂³ — C₂³.₇₀₆C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₅C₂³ — C₂³.₇₀₆C₂⁴

Lower central

C₁ — C₂³ — C₂³.₇₀₆C₂⁴

Upper central

C₁ — C₂³ — C₂³.₇₀₆C₂⁴

Jennings

C₁ — C₂³ — C₂³.₇₀₆C₂⁴

Generators and relations for C₂³.₇₀₆C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=abc, e²=a, f²=cb=bc, g²=ba=ab, ac=ca, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Subgroups: 356 in 196 conjugacy classes, 104 normal (30 characteristic)
C₁, C₂, C₄, C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₄², C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²×Q₈, C₂³.₆₃C₂³, C₂³.₆₅C₂³, C₂³.₆₅C₂³, C₂³.₆₇C₂³, C₂³.₆₇C₂³, C₂³.₇₈C₂³, C₂³.₈₁C₂³, C₂³.₇₀₆C₂⁴
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂³⋊₂Q₈, C₂³.₄₁C₂³, D₄⋊₃Q₈, Q₈⋊₃Q₈, Q₈², C₂².₅₆C₂⁴, C₂³.₇₀₆C₂⁴

Smallest permutation representation of C₂³.₇₀₆C₂⁴
►Regular action on 128 points

Generators in S₁₂₈

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

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

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

(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)

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	4A	···	4R	4S	···	4X
order	1	2	···	2	4	···	4	4	···	4
size	1	1	···	1	4	···	4	8	···	8

32 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4	4
type	+	+	+	+	+	+	-	-		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	Q₈	Q₈	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂³.₇₀₆C₂⁴	C₂³.₆₃C₂³	C₂³.₆₅C₂³	C₂³.₆₇C₂³	C₂³.₇₈C₂³	C₂³.₈₁C₂³	C₄⋊C₄	C₂×Q₈	C₂×C₄	C₂²	C₂²
# reps	1	2	3	4	2	4	4	4	4	3	1

Matrix representation of C₂³.₇₀₆C₂⁴ ►in GL₆(𝔽₅)

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
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

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

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

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

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

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

C₂³.₇₀₆C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{706}C_2^4

% in TeX

G:=Group("C2^3.706C2^4");

// GroupNames label

G:=SmallGroup(128,1538);

// by ID

G=gap.SmallGroup(128,1538);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,120,758,723,520,1571,346,192]);

// Polycyclic

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

// generators/relations

G = C23.706C24 order 128 = 27

423rd central stem extension by C23 of C24

G = C₂³.₇₀₆C₂⁴ order 128 = 2⁷

423^rd central stem extension by C₂³ of C₂⁴