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
Subgroups: 740 in 454 conjugacy classes, 180 normal (8 characteristic)
C1, C2, C2 [×14], C4 [×26], C22, C22 [×34], C2×C4 [×24], C2×C4 [×82], Q8 [×32], C23, C23 [×14], C4⋊C4 [×24], C22×C4 [×38], C22×C4 [×30], C2×Q8 [×56], C24, C2.C42 [×12], C2×C4⋊C4 [×12], C2×C4⋊C4 [×12], C23×C4, C23×C4 [×6], C22×Q8 [×4], C22×Q8 [×12], C2×C2.C42 [×3], C23.78C23 [×8], C22×C4⋊C4 [×3], Q8×C23, C2×C23.78C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], Q8 [×12], C23 [×15], C2×D4 [×18], C2×Q8 [×18], C4○D4 [×2], C24, C22≀C2 [×4], C22⋊Q8 [×12], C4⋊Q8 [×12], C22×D4 [×3], C22×Q8 [×3], C2×C4○D4, C23.78C23 [×8], C2×C22≀C2, C2×C22⋊Q8 [×3], C2×C4⋊Q8 [×3], C2×C23.78C23
Generators and relations
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 >
(1 105)(2 106)(3 107)(4 108)(5 73)(6 74)(7 75)(8 76)(9 67)(10 68)(11 65)(12 66)(13 69)(14 70)(15 71)(16 72)(17 91)(18 92)(19 89)(20 90)(21 93)(22 94)(23 95)(24 96)(25 83)(26 84)(27 81)(28 82)(29 85)(30 86)(31 87)(32 88)(33 45)(34 46)(35 47)(36 48)(37 101)(38 102)(39 103)(40 104)(41 99)(42 100)(43 97)(44 98)(49 121)(50 122)(51 123)(52 124)(53 117)(54 118)(55 119)(56 120)(57 113)(58 114)(59 115)(60 116)(61 109)(62 110)(63 111)(64 112)(77 128)(78 125)(79 126)(80 127)
(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 65)(2 66)(3 67)(4 68)(5 42)(6 43)(7 44)(8 41)(9 107)(10 108)(11 105)(12 106)(13 101)(14 102)(15 103)(16 104)(17 123)(18 124)(19 121)(20 122)(21 117)(22 118)(23 119)(24 120)(25 115)(26 116)(27 113)(28 114)(29 109)(30 110)(31 111)(32 112)(33 78)(34 79)(35 80)(36 77)(37 69)(38 70)(39 71)(40 72)(45 125)(46 126)(47 127)(48 128)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(73 100)(74 97)(75 98)(76 99)
(1 71)(2 72)(3 69)(4 70)(5 46)(6 47)(7 48)(8 45)(9 101)(10 102)(11 103)(12 104)(13 107)(14 108)(15 105)(16 106)(17 117)(18 118)(19 119)(20 120)(21 123)(22 124)(23 121)(24 122)(25 109)(26 110)(27 111)(28 112)(29 115)(30 116)(31 113)(32 114)(33 76)(34 73)(35 74)(36 75)(37 67)(38 68)(39 65)(40 66)(41 125)(42 126)(43 127)(44 128)(49 95)(50 96)(51 93)(52 94)(53 91)(54 92)(55 89)(56 90)(57 87)(58 88)(59 85)(60 86)(61 83)(62 84)(63 81)(64 82)(77 98)(78 99)(79 100)(80 97)
(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 3 117)(2 118 4 120)(5 62 7 64)(6 61 8 63)(9 93 11 95)(10 96 12 94)(13 91 15 89)(14 90 16 92)(17 71 19 69)(18 70 20 72)(21 65 23 67)(22 68 24 66)(25 33 27 35)(26 36 28 34)(29 99 31 97)(30 98 32 100)(37 123 39 121)(38 122 40 124)(41 87 43 85)(42 86 44 88)(45 81 47 83)(46 84 48 82)(49 101 51 103)(50 104 52 102)(53 105 55 107)(54 108 56 106)(57 127 59 125)(58 126 60 128)(73 110 75 112)(74 109 76 111)(77 114 79 116)(78 113 80 115)
(1 111 3 109)(2 32 4 30)(5 50 7 52)(6 91 8 89)(9 85 11 87)(10 62 12 64)(13 83 15 81)(14 60 16 58)(17 76 19 74)(18 100 20 98)(21 78 23 80)(22 34 24 36)(25 71 27 69)(26 40 28 38)(29 65 31 67)(33 119 35 117)(37 115 39 113)(41 49 43 51)(42 90 44 92)(45 55 47 53)(46 96 48 94)(54 126 56 128)(57 101 59 103)(61 105 63 107)(66 112 68 110)(70 116 72 114)(73 122 75 124)(77 118 79 120)(82 102 84 104)(86 106 88 108)(93 125 95 127)(97 123 99 121)
G:=sub<Sym(128)| (1,105)(2,106)(3,107)(4,108)(5,73)(6,74)(7,75)(8,76)(9,67)(10,68)(11,65)(12,66)(13,69)(14,70)(15,71)(16,72)(17,91)(18,92)(19,89)(20,90)(21,93)(22,94)(23,95)(24,96)(25,83)(26,84)(27,81)(28,82)(29,85)(30,86)(31,87)(32,88)(33,45)(34,46)(35,47)(36,48)(37,101)(38,102)(39,103)(40,104)(41,99)(42,100)(43,97)(44,98)(49,121)(50,122)(51,123)(52,124)(53,117)(54,118)(55,119)(56,120)(57,113)(58,114)(59,115)(60,116)(61,109)(62,110)(63,111)(64,112)(77,128)(78,125)(79,126)(80,127), (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), (1,65)(2,66)(3,67)(4,68)(5,42)(6,43)(7,44)(8,41)(9,107)(10,108)(11,105)(12,106)(13,101)(14,102)(15,103)(16,104)(17,123)(18,124)(19,121)(20,122)(21,117)(22,118)(23,119)(24,120)(25,115)(26,116)(27,113)(28,114)(29,109)(30,110)(31,111)(32,112)(33,78)(34,79)(35,80)(36,77)(37,69)(38,70)(39,71)(40,72)(45,125)(46,126)(47,127)(48,128)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(73,100)(74,97)(75,98)(76,99), (1,71)(2,72)(3,69)(4,70)(5,46)(6,47)(7,48)(8,45)(9,101)(10,102)(11,103)(12,104)(13,107)(14,108)(15,105)(16,106)(17,117)(18,118)(19,119)(20,120)(21,123)(22,124)(23,121)(24,122)(25,109)(26,110)(27,111)(28,112)(29,115)(30,116)(31,113)(32,114)(33,76)(34,73)(35,74)(36,75)(37,67)(38,68)(39,65)(40,66)(41,125)(42,126)(43,127)(44,128)(49,95)(50,96)(51,93)(52,94)(53,91)(54,92)(55,89)(56,90)(57,87)(58,88)(59,85)(60,86)(61,83)(62,84)(63,81)(64,82)(77,98)(78,99)(79,100)(80,97), (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), (1,119,3,117)(2,118,4,120)(5,62,7,64)(6,61,8,63)(9,93,11,95)(10,96,12,94)(13,91,15,89)(14,90,16,92)(17,71,19,69)(18,70,20,72)(21,65,23,67)(22,68,24,66)(25,33,27,35)(26,36,28,34)(29,99,31,97)(30,98,32,100)(37,123,39,121)(38,122,40,124)(41,87,43,85)(42,86,44,88)(45,81,47,83)(46,84,48,82)(49,101,51,103)(50,104,52,102)(53,105,55,107)(54,108,56,106)(57,127,59,125)(58,126,60,128)(73,110,75,112)(74,109,76,111)(77,114,79,116)(78,113,80,115), (1,111,3,109)(2,32,4,30)(5,50,7,52)(6,91,8,89)(9,85,11,87)(10,62,12,64)(13,83,15,81)(14,60,16,58)(17,76,19,74)(18,100,20,98)(21,78,23,80)(22,34,24,36)(25,71,27,69)(26,40,28,38)(29,65,31,67)(33,119,35,117)(37,115,39,113)(41,49,43,51)(42,90,44,92)(45,55,47,53)(46,96,48,94)(54,126,56,128)(57,101,59,103)(61,105,63,107)(66,112,68,110)(70,116,72,114)(73,122,75,124)(77,118,79,120)(82,102,84,104)(86,106,88,108)(93,125,95,127)(97,123,99,121)>;
G:=Group( (1,105)(2,106)(3,107)(4,108)(5,73)(6,74)(7,75)(8,76)(9,67)(10,68)(11,65)(12,66)(13,69)(14,70)(15,71)(16,72)(17,91)(18,92)(19,89)(20,90)(21,93)(22,94)(23,95)(24,96)(25,83)(26,84)(27,81)(28,82)(29,85)(30,86)(31,87)(32,88)(33,45)(34,46)(35,47)(36,48)(37,101)(38,102)(39,103)(40,104)(41,99)(42,100)(43,97)(44,98)(49,121)(50,122)(51,123)(52,124)(53,117)(54,118)(55,119)(56,120)(57,113)(58,114)(59,115)(60,116)(61,109)(62,110)(63,111)(64,112)(77,128)(78,125)(79,126)(80,127), (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), (1,65)(2,66)(3,67)(4,68)(5,42)(6,43)(7,44)(8,41)(9,107)(10,108)(11,105)(12,106)(13,101)(14,102)(15,103)(16,104)(17,123)(18,124)(19,121)(20,122)(21,117)(22,118)(23,119)(24,120)(25,115)(26,116)(27,113)(28,114)(29,109)(30,110)(31,111)(32,112)(33,78)(34,79)(35,80)(36,77)(37,69)(38,70)(39,71)(40,72)(45,125)(46,126)(47,127)(48,128)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(73,100)(74,97)(75,98)(76,99), (1,71)(2,72)(3,69)(4,70)(5,46)(6,47)(7,48)(8,45)(9,101)(10,102)(11,103)(12,104)(13,107)(14,108)(15,105)(16,106)(17,117)(18,118)(19,119)(20,120)(21,123)(22,124)(23,121)(24,122)(25,109)(26,110)(27,111)(28,112)(29,115)(30,116)(31,113)(32,114)(33,76)(34,73)(35,74)(36,75)(37,67)(38,68)(39,65)(40,66)(41,125)(42,126)(43,127)(44,128)(49,95)(50,96)(51,93)(52,94)(53,91)(54,92)(55,89)(56,90)(57,87)(58,88)(59,85)(60,86)(61,83)(62,84)(63,81)(64,82)(77,98)(78,99)(79,100)(80,97), (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), (1,119,3,117)(2,118,4,120)(5,62,7,64)(6,61,8,63)(9,93,11,95)(10,96,12,94)(13,91,15,89)(14,90,16,92)(17,71,19,69)(18,70,20,72)(21,65,23,67)(22,68,24,66)(25,33,27,35)(26,36,28,34)(29,99,31,97)(30,98,32,100)(37,123,39,121)(38,122,40,124)(41,87,43,85)(42,86,44,88)(45,81,47,83)(46,84,48,82)(49,101,51,103)(50,104,52,102)(53,105,55,107)(54,108,56,106)(57,127,59,125)(58,126,60,128)(73,110,75,112)(74,109,76,111)(77,114,79,116)(78,113,80,115), (1,111,3,109)(2,32,4,30)(5,50,7,52)(6,91,8,89)(9,85,11,87)(10,62,12,64)(13,83,15,81)(14,60,16,58)(17,76,19,74)(18,100,20,98)(21,78,23,80)(22,34,24,36)(25,71,27,69)(26,40,28,38)(29,65,31,67)(33,119,35,117)(37,115,39,113)(41,49,43,51)(42,90,44,92)(45,55,47,53)(46,96,48,94)(54,126,56,128)(57,101,59,103)(61,105,63,107)(66,112,68,110)(70,116,72,114)(73,122,75,124)(77,118,79,120)(82,102,84,104)(86,106,88,108)(93,125,95,127)(97,123,99,121) );
G=PermutationGroup([(1,105),(2,106),(3,107),(4,108),(5,73),(6,74),(7,75),(8,76),(9,67),(10,68),(11,65),(12,66),(13,69),(14,70),(15,71),(16,72),(17,91),(18,92),(19,89),(20,90),(21,93),(22,94),(23,95),(24,96),(25,83),(26,84),(27,81),(28,82),(29,85),(30,86),(31,87),(32,88),(33,45),(34,46),(35,47),(36,48),(37,101),(38,102),(39,103),(40,104),(41,99),(42,100),(43,97),(44,98),(49,121),(50,122),(51,123),(52,124),(53,117),(54,118),(55,119),(56,120),(57,113),(58,114),(59,115),(60,116),(61,109),(62,110),(63,111),(64,112),(77,128),(78,125),(79,126),(80,127)], [(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)], [(1,65),(2,66),(3,67),(4,68),(5,42),(6,43),(7,44),(8,41),(9,107),(10,108),(11,105),(12,106),(13,101),(14,102),(15,103),(16,104),(17,123),(18,124),(19,121),(20,122),(21,117),(22,118),(23,119),(24,120),(25,115),(26,116),(27,113),(28,114),(29,109),(30,110),(31,111),(32,112),(33,78),(34,79),(35,80),(36,77),(37,69),(38,70),(39,71),(40,72),(45,125),(46,126),(47,127),(48,128),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(56,96),(57,81),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,88),(73,100),(74,97),(75,98),(76,99)], [(1,71),(2,72),(3,69),(4,70),(5,46),(6,47),(7,48),(8,45),(9,101),(10,102),(11,103),(12,104),(13,107),(14,108),(15,105),(16,106),(17,117),(18,118),(19,119),(20,120),(21,123),(22,124),(23,121),(24,122),(25,109),(26,110),(27,111),(28,112),(29,115),(30,116),(31,113),(32,114),(33,76),(34,73),(35,74),(36,75),(37,67),(38,68),(39,65),(40,66),(41,125),(42,126),(43,127),(44,128),(49,95),(50,96),(51,93),(52,94),(53,91),(54,92),(55,89),(56,90),(57,87),(58,88),(59,85),(60,86),(61,83),(62,84),(63,81),(64,82),(77,98),(78,99),(79,100),(80,97)], [(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)], [(1,119,3,117),(2,118,4,120),(5,62,7,64),(6,61,8,63),(9,93,11,95),(10,96,12,94),(13,91,15,89),(14,90,16,92),(17,71,19,69),(18,70,20,72),(21,65,23,67),(22,68,24,66),(25,33,27,35),(26,36,28,34),(29,99,31,97),(30,98,32,100),(37,123,39,121),(38,122,40,124),(41,87,43,85),(42,86,44,88),(45,81,47,83),(46,84,48,82),(49,101,51,103),(50,104,52,102),(53,105,55,107),(54,108,56,106),(57,127,59,125),(58,126,60,128),(73,110,75,112),(74,109,76,111),(77,114,79,116),(78,113,80,115)], [(1,111,3,109),(2,32,4,30),(5,50,7,52),(6,91,8,89),(9,85,11,87),(10,62,12,64),(13,83,15,81),(14,60,16,58),(17,76,19,74),(18,100,20,98),(21,78,23,80),(22,34,24,36),(25,71,27,69),(26,40,28,38),(29,65,31,67),(33,119,35,117),(37,115,39,113),(41,49,43,51),(42,90,44,92),(45,55,47,53),(46,96,48,94),(54,126,56,128),(57,101,59,103),(61,105,63,107),(66,112,68,110),(70,116,72,114),(73,122,75,124),(77,118,79,120),(82,102,84,104),(86,106,88,108),(93,125,95,127),(97,123,99,121)])
Matrix representation ►G ⊆ 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] >;
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 |
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