C2xC2^3.81C2^3 - GroupNames

G = C₂×C₂³.₈₁C₂³ order 128 = 2⁷

Direct product of C₂ and C₂³.₈₁C₂³

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

Aliases: C₂×C₂³.₈₁C₂³, C₂³.₂₉₁C₂⁴, C₂⁴.₆₅₀C₂³, (C₂²×C₄).₅₅Q₈, C₂³.₈₃₃(C₂×D₄), (C₂²×C₄).₃₆₇D₄, C₂².₄₃(C₄⋊Q₈), C₂³.₁₄₅(C₂×Q₈), C₂³.₃₇₀(C₄○D₄), C₂².₅₇(C₂²×Q₈), (C₂³×C₄).₃₂₂C₂², (C₂²×C₄).₇₈₀C₂³, C₂².₁₇₄(C₂²×D₄), C₂².₉₃(C₂²⋊Q₈), C₂².₁₆₄(C₄⋊D₄), C₂².₂₇(C₄².C₂), C₂.C₄².₅₃₁C₂², C₂².₁₀₃(C₂².D₄), C₂.₆(C₂×C₄⋊Q₈), (C₂×C₄).₂₉₂(C₂×D₄), C₂.₁₁(C₂×C₄⋊D₄), (C₂×C₄).₁₂₀(C₂×Q₈), C₂.₁₀(C₂×C₂²⋊Q₈), C₂.₅(C₂×C₄².C₂), (C₂²×C₄⋊C₄).₃₁C₂, (C₂×C₄⋊C₄).₈₃₅C₂², C₂².₁₇₁(C₂×C₄○D₄), C₂.₉(C₂×C₂².D₄), (C₂×C₂.C₄²).₂₃C₂, SmallGroup(128,1123)

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

Derived series

C₁ — C₂³ — C₂×C₂³.₈₁C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂²×C₄⋊C₄ — C₂×C₂³.₈₁C₂³

Lower central

C₁ — C₂³ — C₂×C₂³.₈₁C₂³

Upper central

C₁ — C₂⁴ — C₂×C₂³.₈₁C₂³

Jennings

C₁ — C₂³ — C₂×C₂³.₈₁C₂³

Generators and relations for C₂×C₂³.₈₁C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=c, f²=g²=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, fef^-1=be=eb, bf=fb, bg=gb, cd=dc, geg^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Subgroups: 564 in 346 conjugacy classes, 164 normal (16 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂³×C₄, C₂×C₂.C₄², C₂×C₂.C₄², C₂³.₈₁C₂³, C₂²×C₄⋊C₄, C₂²×C₄⋊C₄, C₂×C₂³.₈₁C₂³
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄².C₂, C₄⋊Q₈, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂³.₈₁C₂³, C₂×C₄⋊D₄, C₂×C₂²⋊Q₈, C₂×C₂².D₄, C₂×C₄².C₂, C₂×C₄⋊Q₈, C₂×C₂³.₈₁C₂³

Smallest permutation representation of C₂×C₂³.₈₁C₂³
►Regular action on 128 points

Generators in S₁₂₈

(1 45)(2 46)(3 47)(4 48)(5 39)(6 40)(7 37)(8 38)(9 123)(10 124)(11 121)(12 122)(13 35)(14 36)(15 33)(16 34)(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)(41 74)(42 75)(43 76)(44 73)(65 104)(66 101)(67 102)(68 103)(69 111)(70 112)(71 109)(72 110)(77 99)(78 100)(79 97)(80 98)(81 119)(82 120)(83 117)(84 118)(85 107)(86 108)(87 105)(88 106)(89 127)(90 128)(91 125)(92 126)(93 115)(94 116)(95 113)(96 114)

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

(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)

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	···	2O	4A	···	4AB
order	1	2	···	2	4	···	4
size	1	1	···	1	4	···	4

44 irreducible representations

dim	1	1	1	1	2	2	2
type	+	+	+	+	+	-
image	C₁	C₂	C₂	C₂	D₄	Q₈	C₄○D₄
kernel	C₂×C₂³.₈₁C₂³	C₂×C₂.C₄²	C₂³.₈₁C₂³	C₂²×C₄⋊C₄	C₂²×C₄	C₂²×C₄	C₂³
# reps	1	3	8	4	8	8	12

Matrix representation of C₂×C₂³.₈₁C₂³ ►in GL₇(𝔽₅)

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

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

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

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

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

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

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

G:=sub<GL(7,GF(5))| [4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,2,0,0,0,0,0,4,4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,4,3],[4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,4,3] >;

C₂×C₂³.₈₁C₂³ in GAP, Magma, Sage, TeX

C_2\times C_2^3._{81}C_2^3

% in TeX

G:=Group("C2xC2^3.81C2^3");

// GroupNames label

G:=SmallGroup(128,1123);

// by ID

G=gap.SmallGroup(128,1123);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,100]);

// Polycyclic

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

// generators/relations

G = C2×C23.81C23 order 128 = 27

Direct product of C2 and C23.81C23

G = C₂×C₂³.₈₁C₂³ order 128 = 2⁷

Direct product of C₂ and C₂³.₈₁C₂³