C2xC2^3.78C2^3 - GroupNames

G = C₂×C₂³.₇₈C₂³ order 128 = 2⁷

Direct product of C₂ and C₂³.₇₈C₂³

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

Aliases: C₂×C₂³.₇₈C₂³, C₂⁴.₆₄₈C₂³, C₂³.₂₈₇C₂⁴, (C₂²×C₄)⋊₆Q₈, (Q₈×C₂³).₇C₂, (C₂²×C₄).₃₆₄D₄, C₂³.₈₃₁(C₂×D₄), C₂².₄₂(C₄⋊Q₈), C₂³.₁₄₄(C₂×Q₈), (C₂³×C₄).₆₅C₂², C₂².₁₁₁C₂²≀C₂, C₂³.₃₆₇(C₄○D₄), C₂².₅₅(C₂²×Q₈), (C₂²×C₄).₇₇₇C₂³, C₂².₁₇₀(C₂²×D₄), C₂².₉₁(C₂²⋊Q₈), (C₂²×Q₈).₄₀₈C₂², C₂.C₄².₅₂₉C₂², (C₂×C₄)⋊₅(C₂×Q₈), C₂.₅(C₂×C₄⋊Q₈), C₂.₈(C₂×C₂²⋊Q₈), C₂.₈(C₂×C₂²≀C₂), (C₂×C₄).₂₈₈(C₂×D₄), (C₂²×C₄⋊C₄).₃₀C₂, (C₂×C₄⋊C₄).₈₃₃C₂², C₂².₁₆₇(C₂×C₄○D₄), (C₂×C₂.C₄²).₂₂C₂, SmallGroup(128,1119)

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

Derived series

C₁ — C₂³ — C₂×C₂³.₇₈C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂²×C₄⋊C₄ — C₂×C₂³.₇₈C₂³

Lower central

C₁ — C₂³ — C₂×C₂³.₇₈C₂³

Upper central

C₁ — C₂⁴ — C₂×C₂³.₇₈C₂³

Jennings

C₁ — C₂³ — C₂×C₂³.₇₈C₂³

Generators and relations for C₂×C₂³.₇₈C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=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: 740 in 454 conjugacy classes, 180 normal (8 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂³×C₄, C₂²×Q₈, C₂²×Q₈, C₂×C₂.C₄², C₂³.₇₈C₂³, C₂²×C₄⋊C₄, Q₈×C₂³, C₂×C₂³.₇₈C₂³
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²≀C₂, C₂²⋊Q₈, C₄⋊Q₈, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂³.₇₈C₂³, C₂×C₂²≀C₂, C₂×C₂²⋊Q₈, C₂×C₄⋊Q₈, C₂×C₂³.₇₈C₂³

Smallest permutation representation of C₂×C₂³.₇₈C₂³
►Regular action on 128 points

Generators in S₁₂₈

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

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

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

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

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

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

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

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

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	1	2	2	2
type	+	+	+	+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	D₄	Q₈	C₄○D₄
kernel	C₂×C₂³.₇₈C₂³	C₂×C₂.C₄²	C₂³.₇₈C₂³	C₂²×C₄⋊C₄	Q₈×C₂³	C₂²×C₄	C₂²×C₄	C₂³
# reps	1	3	8	3	1	12	12	4

Matrix representation of C₂×C₂³.₇₈C₂³ ►in GL₇(𝔽₅)

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

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	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	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

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

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

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

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

C₂×C₂³.₇₈C₂³ in GAP, Magma, Sage, TeX

C_2\times C_2^3._{78}C_2^3

% in TeX

G:=Group("C2xC2^3.78C2^3");

// GroupNames label

G:=SmallGroup(128,1119);

// by ID

G=gap.SmallGroup(128,1119);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,100]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=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×C23.78C23 order 128 = 27

Direct product of C2 and C23.78C23

G = C₂×C₂³.₇₈C₂³ order 128 = 2⁷

Direct product of C₂ and C₂³.₇₈C₂³