direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C23.84C23, C24.652C23, C23.300C24, (C23×C4).70C22, C23.373(C4○D4), (C22×C4).500C23, C22.35(C42⋊2C2), C2.C42.487C22, C2.8(C2×C42⋊2C2), C22.180(C2×C4○D4), (C2×C2.C42).10C2, SmallGroup(128,1132)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 468 in 258 conjugacy classes, 132 normal (4 characteristic)
C1, C2 [×15], C4 [×14], C22 [×35], C2×C4 [×70], C23, C23 [×14], C22×C4 [×14], C22×C4 [×42], C24, C2.C42 [×28], C23×C4 [×7], C2×C2.C42 [×7], C23.84C23 [×8], C2×C23.84C23
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×14], C24, C42⋊2C2 [×28], C2×C4○D4 [×7], C23.84C23 [×8], C2×C42⋊2C2 [×7], C2×C23.84C23
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=bcd, f2=cb=bc, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, 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 34)(2 35)(3 36)(4 33)(5 127)(6 128)(7 125)(8 126)(9 13)(10 14)(11 15)(12 16)(17 22)(18 23)(19 24)(20 21)(25 30)(26 31)(27 32)(28 29)(37 95)(38 96)(39 93)(40 94)(41 46)(42 47)(43 48)(44 45)(49 54)(50 55)(51 56)(52 53)(57 62)(58 63)(59 64)(60 61)(65 84)(66 81)(67 82)(68 83)(69 92)(70 89)(71 90)(72 91)(73 78)(74 79)(75 80)(76 77)(85 100)(86 97)(87 98)(88 99)(101 108)(102 105)(103 106)(104 107)(109 116)(110 113)(111 114)(112 115)(117 124)(118 121)(119 122)(120 123)
(1 108)(2 105)(3 106)(4 107)(5 77)(6 78)(7 79)(8 80)(9 84)(10 81)(11 82)(12 83)(13 65)(14 66)(15 67)(16 68)(17 97)(18 98)(19 99)(20 100)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 69)(30 70)(31 71)(32 72)(33 104)(34 101)(35 102)(36 103)(37 42)(38 43)(39 44)(40 41)(45 93)(46 94)(47 95)(48 96)(49 120)(50 117)(51 118)(52 119)(53 122)(54 123)(55 124)(56 121)(57 112)(58 109)(59 110)(60 111)(61 114)(62 115)(63 116)(64 113)(73 128)(74 125)(75 126)(76 127)
(1 11)(2 12)(3 9)(4 10)(5 39)(6 40)(7 37)(8 38)(13 36)(14 33)(15 34)(16 35)(17 54)(18 55)(19 56)(20 53)(21 52)(22 49)(23 50)(24 51)(25 62)(26 63)(27 64)(28 61)(29 60)(30 57)(31 58)(32 59)(41 78)(42 79)(43 80)(44 77)(45 76)(46 73)(47 74)(48 75)(65 103)(66 104)(67 101)(68 102)(69 111)(70 112)(71 109)(72 110)(81 107)(82 108)(83 105)(84 106)(85 119)(86 120)(87 117)(88 118)(89 115)(90 116)(91 113)(92 114)(93 127)(94 128)(95 125)(96 126)(97 123)(98 124)(99 121)(100 122)
(1 84)(2 81)(3 82)(4 83)(5 42)(6 43)(7 44)(8 41)(9 108)(10 105)(11 106)(12 107)(13 101)(14 102)(15 103)(16 104)(17 121)(18 122)(19 123)(20 124)(21 117)(22 118)(23 119)(24 120)(25 113)(26 114)(27 115)(28 116)(29 109)(30 110)(31 111)(32 112)(33 68)(34 65)(35 66)(36 67)(37 77)(38 78)(39 79)(40 80)(45 125)(46 126)(47 127)(48 128)(49 88)(50 85)(51 86)(52 87)(53 98)(54 99)(55 100)(56 97)(57 72)(58 69)(59 70)(60 71)(61 90)(62 91)(63 92)(64 89)(73 96)(74 93)(75 94)(76 95)
(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 85 82 52)(2 22 83 120)(3 87 84 50)(4 24 81 118)(5 109 44 31)(6 59 41 72)(7 111 42 29)(8 57 43 70)(9 117 106 23)(10 51 107 88)(11 119 108 21)(12 49 105 86)(13 124 103 18)(14 56 104 99)(15 122 101 20)(16 54 102 97)(17 68 123 35)(19 66 121 33)(25 75 115 96)(26 127 116 45)(27 73 113 94)(28 125 114 47)(30 80 112 38)(32 78 110 40)(34 100 67 53)(36 98 65 55)(37 69 79 60)(39 71 77 58)(46 91 128 64)(48 89 126 62)(61 95 92 74)(63 93 90 76)
(1 111 108 60)(2 70 105 30)(3 109 106 58)(4 72 107 32)(5 21 77 85)(6 49 78 120)(7 23 79 87)(8 51 80 118)(9 71 84 31)(10 110 81 59)(11 69 82 29)(12 112 83 57)(13 90 65 26)(14 113 66 64)(15 92 67 28)(16 115 68 62)(17 46 97 94)(18 74 98 125)(19 48 99 96)(20 76 100 127)(22 41 86 40)(24 43 88 38)(25 35 89 102)(27 33 91 104)(34 114 101 61)(36 116 103 63)(37 50 42 117)(39 52 44 119)(45 122 93 53)(47 124 95 55)(54 73 123 128)(56 75 121 126)
G:=sub<Sym(128)| (1,34)(2,35)(3,36)(4,33)(5,127)(6,128)(7,125)(8,126)(9,13)(10,14)(11,15)(12,16)(17,22)(18,23)(19,24)(20,21)(25,30)(26,31)(27,32)(28,29)(37,95)(38,96)(39,93)(40,94)(41,46)(42,47)(43,48)(44,45)(49,54)(50,55)(51,56)(52,53)(57,62)(58,63)(59,64)(60,61)(65,84)(66,81)(67,82)(68,83)(69,92)(70,89)(71,90)(72,91)(73,78)(74,79)(75,80)(76,77)(85,100)(86,97)(87,98)(88,99)(101,108)(102,105)(103,106)(104,107)(109,116)(110,113)(111,114)(112,115)(117,124)(118,121)(119,122)(120,123), (1,108)(2,105)(3,106)(4,107)(5,77)(6,78)(7,79)(8,80)(9,84)(10,81)(11,82)(12,83)(13,65)(14,66)(15,67)(16,68)(17,97)(18,98)(19,99)(20,100)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,69)(30,70)(31,71)(32,72)(33,104)(34,101)(35,102)(36,103)(37,42)(38,43)(39,44)(40,41)(45,93)(46,94)(47,95)(48,96)(49,120)(50,117)(51,118)(52,119)(53,122)(54,123)(55,124)(56,121)(57,112)(58,109)(59,110)(60,111)(61,114)(62,115)(63,116)(64,113)(73,128)(74,125)(75,126)(76,127), (1,11)(2,12)(3,9)(4,10)(5,39)(6,40)(7,37)(8,38)(13,36)(14,33)(15,34)(16,35)(17,54)(18,55)(19,56)(20,53)(21,52)(22,49)(23,50)(24,51)(25,62)(26,63)(27,64)(28,61)(29,60)(30,57)(31,58)(32,59)(41,78)(42,79)(43,80)(44,77)(45,76)(46,73)(47,74)(48,75)(65,103)(66,104)(67,101)(68,102)(69,111)(70,112)(71,109)(72,110)(81,107)(82,108)(83,105)(84,106)(85,119)(86,120)(87,117)(88,118)(89,115)(90,116)(91,113)(92,114)(93,127)(94,128)(95,125)(96,126)(97,123)(98,124)(99,121)(100,122), (1,84)(2,81)(3,82)(4,83)(5,42)(6,43)(7,44)(8,41)(9,108)(10,105)(11,106)(12,107)(13,101)(14,102)(15,103)(16,104)(17,121)(18,122)(19,123)(20,124)(21,117)(22,118)(23,119)(24,120)(25,113)(26,114)(27,115)(28,116)(29,109)(30,110)(31,111)(32,112)(33,68)(34,65)(35,66)(36,67)(37,77)(38,78)(39,79)(40,80)(45,125)(46,126)(47,127)(48,128)(49,88)(50,85)(51,86)(52,87)(53,98)(54,99)(55,100)(56,97)(57,72)(58,69)(59,70)(60,71)(61,90)(62,91)(63,92)(64,89)(73,96)(74,93)(75,94)(76,95), (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), (1,85,82,52)(2,22,83,120)(3,87,84,50)(4,24,81,118)(5,109,44,31)(6,59,41,72)(7,111,42,29)(8,57,43,70)(9,117,106,23)(10,51,107,88)(11,119,108,21)(12,49,105,86)(13,124,103,18)(14,56,104,99)(15,122,101,20)(16,54,102,97)(17,68,123,35)(19,66,121,33)(25,75,115,96)(26,127,116,45)(27,73,113,94)(28,125,114,47)(30,80,112,38)(32,78,110,40)(34,100,67,53)(36,98,65,55)(37,69,79,60)(39,71,77,58)(46,91,128,64)(48,89,126,62)(61,95,92,74)(63,93,90,76), (1,111,108,60)(2,70,105,30)(3,109,106,58)(4,72,107,32)(5,21,77,85)(6,49,78,120)(7,23,79,87)(8,51,80,118)(9,71,84,31)(10,110,81,59)(11,69,82,29)(12,112,83,57)(13,90,65,26)(14,113,66,64)(15,92,67,28)(16,115,68,62)(17,46,97,94)(18,74,98,125)(19,48,99,96)(20,76,100,127)(22,41,86,40)(24,43,88,38)(25,35,89,102)(27,33,91,104)(34,114,101,61)(36,116,103,63)(37,50,42,117)(39,52,44,119)(45,122,93,53)(47,124,95,55)(54,73,123,128)(56,75,121,126)>;
G:=Group( (1,34)(2,35)(3,36)(4,33)(5,127)(6,128)(7,125)(8,126)(9,13)(10,14)(11,15)(12,16)(17,22)(18,23)(19,24)(20,21)(25,30)(26,31)(27,32)(28,29)(37,95)(38,96)(39,93)(40,94)(41,46)(42,47)(43,48)(44,45)(49,54)(50,55)(51,56)(52,53)(57,62)(58,63)(59,64)(60,61)(65,84)(66,81)(67,82)(68,83)(69,92)(70,89)(71,90)(72,91)(73,78)(74,79)(75,80)(76,77)(85,100)(86,97)(87,98)(88,99)(101,108)(102,105)(103,106)(104,107)(109,116)(110,113)(111,114)(112,115)(117,124)(118,121)(119,122)(120,123), (1,108)(2,105)(3,106)(4,107)(5,77)(6,78)(7,79)(8,80)(9,84)(10,81)(11,82)(12,83)(13,65)(14,66)(15,67)(16,68)(17,97)(18,98)(19,99)(20,100)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,69)(30,70)(31,71)(32,72)(33,104)(34,101)(35,102)(36,103)(37,42)(38,43)(39,44)(40,41)(45,93)(46,94)(47,95)(48,96)(49,120)(50,117)(51,118)(52,119)(53,122)(54,123)(55,124)(56,121)(57,112)(58,109)(59,110)(60,111)(61,114)(62,115)(63,116)(64,113)(73,128)(74,125)(75,126)(76,127), (1,11)(2,12)(3,9)(4,10)(5,39)(6,40)(7,37)(8,38)(13,36)(14,33)(15,34)(16,35)(17,54)(18,55)(19,56)(20,53)(21,52)(22,49)(23,50)(24,51)(25,62)(26,63)(27,64)(28,61)(29,60)(30,57)(31,58)(32,59)(41,78)(42,79)(43,80)(44,77)(45,76)(46,73)(47,74)(48,75)(65,103)(66,104)(67,101)(68,102)(69,111)(70,112)(71,109)(72,110)(81,107)(82,108)(83,105)(84,106)(85,119)(86,120)(87,117)(88,118)(89,115)(90,116)(91,113)(92,114)(93,127)(94,128)(95,125)(96,126)(97,123)(98,124)(99,121)(100,122), (1,84)(2,81)(3,82)(4,83)(5,42)(6,43)(7,44)(8,41)(9,108)(10,105)(11,106)(12,107)(13,101)(14,102)(15,103)(16,104)(17,121)(18,122)(19,123)(20,124)(21,117)(22,118)(23,119)(24,120)(25,113)(26,114)(27,115)(28,116)(29,109)(30,110)(31,111)(32,112)(33,68)(34,65)(35,66)(36,67)(37,77)(38,78)(39,79)(40,80)(45,125)(46,126)(47,127)(48,128)(49,88)(50,85)(51,86)(52,87)(53,98)(54,99)(55,100)(56,97)(57,72)(58,69)(59,70)(60,71)(61,90)(62,91)(63,92)(64,89)(73,96)(74,93)(75,94)(76,95), (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), (1,85,82,52)(2,22,83,120)(3,87,84,50)(4,24,81,118)(5,109,44,31)(6,59,41,72)(7,111,42,29)(8,57,43,70)(9,117,106,23)(10,51,107,88)(11,119,108,21)(12,49,105,86)(13,124,103,18)(14,56,104,99)(15,122,101,20)(16,54,102,97)(17,68,123,35)(19,66,121,33)(25,75,115,96)(26,127,116,45)(27,73,113,94)(28,125,114,47)(30,80,112,38)(32,78,110,40)(34,100,67,53)(36,98,65,55)(37,69,79,60)(39,71,77,58)(46,91,128,64)(48,89,126,62)(61,95,92,74)(63,93,90,76), (1,111,108,60)(2,70,105,30)(3,109,106,58)(4,72,107,32)(5,21,77,85)(6,49,78,120)(7,23,79,87)(8,51,80,118)(9,71,84,31)(10,110,81,59)(11,69,82,29)(12,112,83,57)(13,90,65,26)(14,113,66,64)(15,92,67,28)(16,115,68,62)(17,46,97,94)(18,74,98,125)(19,48,99,96)(20,76,100,127)(22,41,86,40)(24,43,88,38)(25,35,89,102)(27,33,91,104)(34,114,101,61)(36,116,103,63)(37,50,42,117)(39,52,44,119)(45,122,93,53)(47,124,95,55)(54,73,123,128)(56,75,121,126) );
G=PermutationGroup([(1,34),(2,35),(3,36),(4,33),(5,127),(6,128),(7,125),(8,126),(9,13),(10,14),(11,15),(12,16),(17,22),(18,23),(19,24),(20,21),(25,30),(26,31),(27,32),(28,29),(37,95),(38,96),(39,93),(40,94),(41,46),(42,47),(43,48),(44,45),(49,54),(50,55),(51,56),(52,53),(57,62),(58,63),(59,64),(60,61),(65,84),(66,81),(67,82),(68,83),(69,92),(70,89),(71,90),(72,91),(73,78),(74,79),(75,80),(76,77),(85,100),(86,97),(87,98),(88,99),(101,108),(102,105),(103,106),(104,107),(109,116),(110,113),(111,114),(112,115),(117,124),(118,121),(119,122),(120,123)], [(1,108),(2,105),(3,106),(4,107),(5,77),(6,78),(7,79),(8,80),(9,84),(10,81),(11,82),(12,83),(13,65),(14,66),(15,67),(16,68),(17,97),(18,98),(19,99),(20,100),(21,85),(22,86),(23,87),(24,88),(25,89),(26,90),(27,91),(28,92),(29,69),(30,70),(31,71),(32,72),(33,104),(34,101),(35,102),(36,103),(37,42),(38,43),(39,44),(40,41),(45,93),(46,94),(47,95),(48,96),(49,120),(50,117),(51,118),(52,119),(53,122),(54,123),(55,124),(56,121),(57,112),(58,109),(59,110),(60,111),(61,114),(62,115),(63,116),(64,113),(73,128),(74,125),(75,126),(76,127)], [(1,11),(2,12),(3,9),(4,10),(5,39),(6,40),(7,37),(8,38),(13,36),(14,33),(15,34),(16,35),(17,54),(18,55),(19,56),(20,53),(21,52),(22,49),(23,50),(24,51),(25,62),(26,63),(27,64),(28,61),(29,60),(30,57),(31,58),(32,59),(41,78),(42,79),(43,80),(44,77),(45,76),(46,73),(47,74),(48,75),(65,103),(66,104),(67,101),(68,102),(69,111),(70,112),(71,109),(72,110),(81,107),(82,108),(83,105),(84,106),(85,119),(86,120),(87,117),(88,118),(89,115),(90,116),(91,113),(92,114),(93,127),(94,128),(95,125),(96,126),(97,123),(98,124),(99,121),(100,122)], [(1,84),(2,81),(3,82),(4,83),(5,42),(6,43),(7,44),(8,41),(9,108),(10,105),(11,106),(12,107),(13,101),(14,102),(15,103),(16,104),(17,121),(18,122),(19,123),(20,124),(21,117),(22,118),(23,119),(24,120),(25,113),(26,114),(27,115),(28,116),(29,109),(30,110),(31,111),(32,112),(33,68),(34,65),(35,66),(36,67),(37,77),(38,78),(39,79),(40,80),(45,125),(46,126),(47,127),(48,128),(49,88),(50,85),(51,86),(52,87),(53,98),(54,99),(55,100),(56,97),(57,72),(58,69),(59,70),(60,71),(61,90),(62,91),(63,92),(64,89),(73,96),(74,93),(75,94),(76,95)], [(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)], [(1,85,82,52),(2,22,83,120),(3,87,84,50),(4,24,81,118),(5,109,44,31),(6,59,41,72),(7,111,42,29),(8,57,43,70),(9,117,106,23),(10,51,107,88),(11,119,108,21),(12,49,105,86),(13,124,103,18),(14,56,104,99),(15,122,101,20),(16,54,102,97),(17,68,123,35),(19,66,121,33),(25,75,115,96),(26,127,116,45),(27,73,113,94),(28,125,114,47),(30,80,112,38),(32,78,110,40),(34,100,67,53),(36,98,65,55),(37,69,79,60),(39,71,77,58),(46,91,128,64),(48,89,126,62),(61,95,92,74),(63,93,90,76)], [(1,111,108,60),(2,70,105,30),(3,109,106,58),(4,72,107,32),(5,21,77,85),(6,49,78,120),(7,23,79,87),(8,51,80,118),(9,71,84,31),(10,110,81,59),(11,69,82,29),(12,112,83,57),(13,90,65,26),(14,113,66,64),(15,92,67,28),(16,115,68,62),(17,46,97,94),(18,74,98,125),(19,48,99,96),(20,76,100,127),(22,41,86,40),(24,43,88,38),(25,35,89,102),(27,33,91,104),(34,114,101,61),(36,116,103,63),(37,50,42,117),(39,52,44,119),(45,122,93,53),(47,124,95,55),(54,73,123,128),(56,75,121,126)])
Matrix representation ►G ⊆ GL8(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 3 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 1 |
G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,4],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,3,3],[4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,1,1] >;
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 | 2 |
| type | + | + | + | |
| image | C1 | C2 | C2 | C4○D4 |
| kernel | C2×C23.84C23 | C2×C2.C42 | C23.84C23 | C23 |
| # reps | 1 | 7 | 8 | 28 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3._{84}C_2^3 % in TeX
G:=Group("C2xC2^3.84C2^3"); // GroupNames label
G:=SmallGroup(128,1132);
// by ID
G=gap.SmallGroup(128,1132);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,344,758,723,100]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=b*c*d,f^2=c*b=b*c,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*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