p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.709C24, C22.3692- (1+4), C22.4822+ (1+4), (C2×Q8).14Q8, C2.35(Q8⋊3Q8), (C22×C4).222C23, (C2×C42).728C22, C2.22(C23⋊2Q8), C22.170(C22×Q8), (C22×Q8).229C22, C23.63C23.60C2, C23.81C23.49C2, C23.67C23.62C2, C23.83C23.45C2, C2.C42.413C22, C23.65C23.87C2, C2.56(C22.57C24), C2.50(C22.49C24), C2.120(C22.33C24), (C2×C4).94(C2×Q8), (C2×C4).250(C4○D4), (C2×C4⋊C4).519C22, C22.570(C2×C4○D4), SmallGroup(128,1541)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 324 in 184 conjugacy classes, 96 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C4 [×20], C22 [×3], C22 [×4], C2×C4 [×10], C2×C4 [×40], Q8 [×4], C23, C42 [×3], C4⋊C4 [×19], C22×C4, C22×C4 [×14], C2×Q8 [×4], C2×Q8 [×2], C2.C42 [×18], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4 [×12], C22×Q8, C23.63C23 [×4], C23.65C23 [×2], C23.67C23, C23.67C23 [×2], C23.81C23 [×4], C23.83C23 [×2], C23.709C24
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C22×Q8, C2×C4○D4 [×2], 2+ (1+4) [×2], 2- (1+4) [×2], C22.33C24 [×2], C23⋊2Q8, C22.49C24, Q8⋊3Q8 [×2], C22.57C24, C23.709C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=ca=ac, e2=g2=ba=ab, f2=cb=bc, 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 >
(1 10)(2 11)(3 12)(4 9)(5 69)(6 70)(7 71)(8 72)(13 84)(14 81)(15 82)(16 83)(17 86)(18 87)(19 88)(20 85)(21 92)(22 89)(23 90)(24 91)(25 94)(26 95)(27 96)(28 93)(29 100)(30 97)(31 98)(32 99)(33 78)(34 79)(35 80)(36 77)(37 75)(38 76)(39 73)(40 74)(41 102)(42 103)(43 104)(44 101)(45 108)(46 105)(47 106)(48 107)(49 110)(50 111)(51 112)(52 109)(53 116)(54 113)(55 114)(56 115)(57 118)(58 119)(59 120)(60 117)(61 124)(62 121)(63 122)(64 123)(65 128)(66 125)(67 126)(68 127)
(1 42)(2 43)(3 44)(4 41)(5 39)(6 40)(7 37)(8 38)(9 102)(10 103)(11 104)(12 101)(13 108)(14 105)(15 106)(16 107)(17 110)(18 111)(19 112)(20 109)(21 116)(22 113)(23 114)(24 115)(25 118)(26 119)(27 120)(28 117)(29 124)(30 121)(31 122)(32 123)(33 126)(34 127)(35 128)(36 125)(45 84)(46 81)(47 82)(48 83)(49 86)(50 87)(51 88)(52 85)(53 92)(54 89)(55 90)(56 91)(57 94)(58 95)(59 96)(60 93)(61 100)(62 97)(63 98)(64 99)(65 80)(66 77)(67 78)(68 79)(69 73)(70 74)(71 75)(72 76)
(1 12)(2 9)(3 10)(4 11)(5 71)(6 72)(7 69)(8 70)(13 82)(14 83)(15 84)(16 81)(17 88)(18 85)(19 86)(20 87)(21 90)(22 91)(23 92)(24 89)(25 96)(26 93)(27 94)(28 95)(29 98)(30 99)(31 100)(32 97)(33 80)(34 77)(35 78)(36 79)(37 73)(38 74)(39 75)(40 76)(41 104)(42 101)(43 102)(44 103)(45 106)(46 107)(47 108)(48 105)(49 112)(50 109)(51 110)(52 111)(53 114)(54 115)(55 116)(56 113)(57 120)(58 117)(59 118)(60 119)(61 122)(62 123)(63 124)(64 121)(65 126)(66 127)(67 128)(68 125)
(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 95)(2 59 104 27)(3 117 101 93)(4 57 102 25)(5 114 73 90)(6 56 74 24)(7 116 75 92)(8 54 76 22)(9 118 41 94)(10 58 42 26)(11 120 43 96)(12 60 44 28)(13 61 45 29)(14 121 46 97)(15 63 47 31)(16 123 48 99)(17 126 49 78)(18 68 50 34)(19 128 51 80)(20 66 52 36)(21 71 53 37)(23 69 55 39)(30 81 62 105)(32 83 64 107)(33 86 67 110)(35 88 65 112)(38 89 72 113)(40 91 70 115)(77 85 125 109)(79 87 127 111)(82 122 106 98)(84 124 108 100)
(1 109 101 87)(2 17 102 51)(3 111 103 85)(4 19 104 49)(5 61 75 31)(6 97 76 123)(7 63 73 29)(8 99 74 121)(9 88 43 110)(10 52 44 18)(11 86 41 112)(12 50 42 20)(13 55 47 21)(14 91 48 113)(15 53 45 23)(16 89 46 115)(22 105 56 83)(24 107 54 81)(25 33 59 65)(26 127 60 77)(27 35 57 67)(28 125 58 79)(30 38 64 70)(32 40 62 72)(34 93 66 119)(36 95 68 117)(37 98 69 124)(39 100 71 122)(78 120 128 94)(80 118 126 96)(82 116 108 90)(84 114 106 92)
(1 106 103 82)(2 83 104 107)(3 108 101 84)(4 81 102 105)(5 68 73 34)(6 35 74 65)(7 66 75 36)(8 33 76 67)(9 14 41 46)(10 47 42 15)(11 16 43 48)(12 45 44 13)(17 22 49 54)(18 55 50 23)(19 24 51 56)(20 53 52 21)(25 97 57 121)(26 122 58 98)(27 99 59 123)(28 124 60 100)(29 93 61 117)(30 118 62 94)(31 95 63 119)(32 120 64 96)(37 77 71 125)(38 126 72 78)(39 79 69 127)(40 128 70 80)(85 116 109 92)(86 89 110 113)(87 114 111 90)(88 91 112 115)
G:=sub<Sym(128)| (1,10)(2,11)(3,12)(4,9)(5,69)(6,70)(7,71)(8,72)(13,84)(14,81)(15,82)(16,83)(17,86)(18,87)(19,88)(20,85)(21,92)(22,89)(23,90)(24,91)(25,94)(26,95)(27,96)(28,93)(29,100)(30,97)(31,98)(32,99)(33,78)(34,79)(35,80)(36,77)(37,75)(38,76)(39,73)(40,74)(41,102)(42,103)(43,104)(44,101)(45,108)(46,105)(47,106)(48,107)(49,110)(50,111)(51,112)(52,109)(53,116)(54,113)(55,114)(56,115)(57,118)(58,119)(59,120)(60,117)(61,124)(62,121)(63,122)(64,123)(65,128)(66,125)(67,126)(68,127), (1,42)(2,43)(3,44)(4,41)(5,39)(6,40)(7,37)(8,38)(9,102)(10,103)(11,104)(12,101)(13,108)(14,105)(15,106)(16,107)(17,110)(18,111)(19,112)(20,109)(21,116)(22,113)(23,114)(24,115)(25,118)(26,119)(27,120)(28,117)(29,124)(30,121)(31,122)(32,123)(33,126)(34,127)(35,128)(36,125)(45,84)(46,81)(47,82)(48,83)(49,86)(50,87)(51,88)(52,85)(53,92)(54,89)(55,90)(56,91)(57,94)(58,95)(59,96)(60,93)(61,100)(62,97)(63,98)(64,99)(65,80)(66,77)(67,78)(68,79)(69,73)(70,74)(71,75)(72,76), (1,12)(2,9)(3,10)(4,11)(5,71)(6,72)(7,69)(8,70)(13,82)(14,83)(15,84)(16,81)(17,88)(18,85)(19,86)(20,87)(21,90)(22,91)(23,92)(24,89)(25,96)(26,93)(27,94)(28,95)(29,98)(30,99)(31,100)(32,97)(33,80)(34,77)(35,78)(36,79)(37,73)(38,74)(39,75)(40,76)(41,104)(42,101)(43,102)(44,103)(45,106)(46,107)(47,108)(48,105)(49,112)(50,109)(51,110)(52,111)(53,114)(54,115)(55,116)(56,113)(57,120)(58,117)(59,118)(60,119)(61,122)(62,123)(63,124)(64,121)(65,126)(66,127)(67,128)(68,125), (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), (1,119,103,95)(2,59,104,27)(3,117,101,93)(4,57,102,25)(5,114,73,90)(6,56,74,24)(7,116,75,92)(8,54,76,22)(9,118,41,94)(10,58,42,26)(11,120,43,96)(12,60,44,28)(13,61,45,29)(14,121,46,97)(15,63,47,31)(16,123,48,99)(17,126,49,78)(18,68,50,34)(19,128,51,80)(20,66,52,36)(21,71,53,37)(23,69,55,39)(30,81,62,105)(32,83,64,107)(33,86,67,110)(35,88,65,112)(38,89,72,113)(40,91,70,115)(77,85,125,109)(79,87,127,111)(82,122,106,98)(84,124,108,100), (1,109,101,87)(2,17,102,51)(3,111,103,85)(4,19,104,49)(5,61,75,31)(6,97,76,123)(7,63,73,29)(8,99,74,121)(9,88,43,110)(10,52,44,18)(11,86,41,112)(12,50,42,20)(13,55,47,21)(14,91,48,113)(15,53,45,23)(16,89,46,115)(22,105,56,83)(24,107,54,81)(25,33,59,65)(26,127,60,77)(27,35,57,67)(28,125,58,79)(30,38,64,70)(32,40,62,72)(34,93,66,119)(36,95,68,117)(37,98,69,124)(39,100,71,122)(78,120,128,94)(80,118,126,96)(82,116,108,90)(84,114,106,92), (1,106,103,82)(2,83,104,107)(3,108,101,84)(4,81,102,105)(5,68,73,34)(6,35,74,65)(7,66,75,36)(8,33,76,67)(9,14,41,46)(10,47,42,15)(11,16,43,48)(12,45,44,13)(17,22,49,54)(18,55,50,23)(19,24,51,56)(20,53,52,21)(25,97,57,121)(26,122,58,98)(27,99,59,123)(28,124,60,100)(29,93,61,117)(30,118,62,94)(31,95,63,119)(32,120,64,96)(37,77,71,125)(38,126,72,78)(39,79,69,127)(40,128,70,80)(85,116,109,92)(86,89,110,113)(87,114,111,90)(88,91,112,115)>;
G:=Group( (1,10)(2,11)(3,12)(4,9)(5,69)(6,70)(7,71)(8,72)(13,84)(14,81)(15,82)(16,83)(17,86)(18,87)(19,88)(20,85)(21,92)(22,89)(23,90)(24,91)(25,94)(26,95)(27,96)(28,93)(29,100)(30,97)(31,98)(32,99)(33,78)(34,79)(35,80)(36,77)(37,75)(38,76)(39,73)(40,74)(41,102)(42,103)(43,104)(44,101)(45,108)(46,105)(47,106)(48,107)(49,110)(50,111)(51,112)(52,109)(53,116)(54,113)(55,114)(56,115)(57,118)(58,119)(59,120)(60,117)(61,124)(62,121)(63,122)(64,123)(65,128)(66,125)(67,126)(68,127), (1,42)(2,43)(3,44)(4,41)(5,39)(6,40)(7,37)(8,38)(9,102)(10,103)(11,104)(12,101)(13,108)(14,105)(15,106)(16,107)(17,110)(18,111)(19,112)(20,109)(21,116)(22,113)(23,114)(24,115)(25,118)(26,119)(27,120)(28,117)(29,124)(30,121)(31,122)(32,123)(33,126)(34,127)(35,128)(36,125)(45,84)(46,81)(47,82)(48,83)(49,86)(50,87)(51,88)(52,85)(53,92)(54,89)(55,90)(56,91)(57,94)(58,95)(59,96)(60,93)(61,100)(62,97)(63,98)(64,99)(65,80)(66,77)(67,78)(68,79)(69,73)(70,74)(71,75)(72,76), (1,12)(2,9)(3,10)(4,11)(5,71)(6,72)(7,69)(8,70)(13,82)(14,83)(15,84)(16,81)(17,88)(18,85)(19,86)(20,87)(21,90)(22,91)(23,92)(24,89)(25,96)(26,93)(27,94)(28,95)(29,98)(30,99)(31,100)(32,97)(33,80)(34,77)(35,78)(36,79)(37,73)(38,74)(39,75)(40,76)(41,104)(42,101)(43,102)(44,103)(45,106)(46,107)(47,108)(48,105)(49,112)(50,109)(51,110)(52,111)(53,114)(54,115)(55,116)(56,113)(57,120)(58,117)(59,118)(60,119)(61,122)(62,123)(63,124)(64,121)(65,126)(66,127)(67,128)(68,125), (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), (1,119,103,95)(2,59,104,27)(3,117,101,93)(4,57,102,25)(5,114,73,90)(6,56,74,24)(7,116,75,92)(8,54,76,22)(9,118,41,94)(10,58,42,26)(11,120,43,96)(12,60,44,28)(13,61,45,29)(14,121,46,97)(15,63,47,31)(16,123,48,99)(17,126,49,78)(18,68,50,34)(19,128,51,80)(20,66,52,36)(21,71,53,37)(23,69,55,39)(30,81,62,105)(32,83,64,107)(33,86,67,110)(35,88,65,112)(38,89,72,113)(40,91,70,115)(77,85,125,109)(79,87,127,111)(82,122,106,98)(84,124,108,100), (1,109,101,87)(2,17,102,51)(3,111,103,85)(4,19,104,49)(5,61,75,31)(6,97,76,123)(7,63,73,29)(8,99,74,121)(9,88,43,110)(10,52,44,18)(11,86,41,112)(12,50,42,20)(13,55,47,21)(14,91,48,113)(15,53,45,23)(16,89,46,115)(22,105,56,83)(24,107,54,81)(25,33,59,65)(26,127,60,77)(27,35,57,67)(28,125,58,79)(30,38,64,70)(32,40,62,72)(34,93,66,119)(36,95,68,117)(37,98,69,124)(39,100,71,122)(78,120,128,94)(80,118,126,96)(82,116,108,90)(84,114,106,92), (1,106,103,82)(2,83,104,107)(3,108,101,84)(4,81,102,105)(5,68,73,34)(6,35,74,65)(7,66,75,36)(8,33,76,67)(9,14,41,46)(10,47,42,15)(11,16,43,48)(12,45,44,13)(17,22,49,54)(18,55,50,23)(19,24,51,56)(20,53,52,21)(25,97,57,121)(26,122,58,98)(27,99,59,123)(28,124,60,100)(29,93,61,117)(30,118,62,94)(31,95,63,119)(32,120,64,96)(37,77,71,125)(38,126,72,78)(39,79,69,127)(40,128,70,80)(85,116,109,92)(86,89,110,113)(87,114,111,90)(88,91,112,115) );
G=PermutationGroup([(1,10),(2,11),(3,12),(4,9),(5,69),(6,70),(7,71),(8,72),(13,84),(14,81),(15,82),(16,83),(17,86),(18,87),(19,88),(20,85),(21,92),(22,89),(23,90),(24,91),(25,94),(26,95),(27,96),(28,93),(29,100),(30,97),(31,98),(32,99),(33,78),(34,79),(35,80),(36,77),(37,75),(38,76),(39,73),(40,74),(41,102),(42,103),(43,104),(44,101),(45,108),(46,105),(47,106),(48,107),(49,110),(50,111),(51,112),(52,109),(53,116),(54,113),(55,114),(56,115),(57,118),(58,119),(59,120),(60,117),(61,124),(62,121),(63,122),(64,123),(65,128),(66,125),(67,126),(68,127)], [(1,42),(2,43),(3,44),(4,41),(5,39),(6,40),(7,37),(8,38),(9,102),(10,103),(11,104),(12,101),(13,108),(14,105),(15,106),(16,107),(17,110),(18,111),(19,112),(20,109),(21,116),(22,113),(23,114),(24,115),(25,118),(26,119),(27,120),(28,117),(29,124),(30,121),(31,122),(32,123),(33,126),(34,127),(35,128),(36,125),(45,84),(46,81),(47,82),(48,83),(49,86),(50,87),(51,88),(52,85),(53,92),(54,89),(55,90),(56,91),(57,94),(58,95),(59,96),(60,93),(61,100),(62,97),(63,98),(64,99),(65,80),(66,77),(67,78),(68,79),(69,73),(70,74),(71,75),(72,76)], [(1,12),(2,9),(3,10),(4,11),(5,71),(6,72),(7,69),(8,70),(13,82),(14,83),(15,84),(16,81),(17,88),(18,85),(19,86),(20,87),(21,90),(22,91),(23,92),(24,89),(25,96),(26,93),(27,94),(28,95),(29,98),(30,99),(31,100),(32,97),(33,80),(34,77),(35,78),(36,79),(37,73),(38,74),(39,75),(40,76),(41,104),(42,101),(43,102),(44,103),(45,106),(46,107),(47,108),(48,105),(49,112),(50,109),(51,110),(52,111),(53,114),(54,115),(55,116),(56,113),(57,120),(58,117),(59,118),(60,119),(61,122),(62,123),(63,124),(64,121),(65,126),(66,127),(67,128),(68,125)], [(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)], [(1,119,103,95),(2,59,104,27),(3,117,101,93),(4,57,102,25),(5,114,73,90),(6,56,74,24),(7,116,75,92),(8,54,76,22),(9,118,41,94),(10,58,42,26),(11,120,43,96),(12,60,44,28),(13,61,45,29),(14,121,46,97),(15,63,47,31),(16,123,48,99),(17,126,49,78),(18,68,50,34),(19,128,51,80),(20,66,52,36),(21,71,53,37),(23,69,55,39),(30,81,62,105),(32,83,64,107),(33,86,67,110),(35,88,65,112),(38,89,72,113),(40,91,70,115),(77,85,125,109),(79,87,127,111),(82,122,106,98),(84,124,108,100)], [(1,109,101,87),(2,17,102,51),(3,111,103,85),(4,19,104,49),(5,61,75,31),(6,97,76,123),(7,63,73,29),(8,99,74,121),(9,88,43,110),(10,52,44,18),(11,86,41,112),(12,50,42,20),(13,55,47,21),(14,91,48,113),(15,53,45,23),(16,89,46,115),(22,105,56,83),(24,107,54,81),(25,33,59,65),(26,127,60,77),(27,35,57,67),(28,125,58,79),(30,38,64,70),(32,40,62,72),(34,93,66,119),(36,95,68,117),(37,98,69,124),(39,100,71,122),(78,120,128,94),(80,118,126,96),(82,116,108,90),(84,114,106,92)], [(1,106,103,82),(2,83,104,107),(3,108,101,84),(4,81,102,105),(5,68,73,34),(6,35,74,65),(7,66,75,36),(8,33,76,67),(9,14,41,46),(10,47,42,15),(11,16,43,48),(12,45,44,13),(17,22,49,54),(18,55,50,23),(19,24,51,56),(20,53,52,21),(25,97,57,121),(26,122,58,98),(27,99,59,123),(28,124,60,100),(29,93,61,117),(30,118,62,94),(31,95,63,119),(32,120,64,96),(37,77,71,125),(38,126,72,78),(39,79,69,127),(40,128,70,80),(85,116,109,92),(86,89,110,113),(87,114,111,90),(88,91,112,115)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 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 |
| 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 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 3 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 4 | 4 |
G:=sub<GL(6,GF(5))| [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],[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],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,2,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,4,0,0,0,0,3,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,1,0,0,0,0,0,2,0,0,0,0,0,0,1,4,0,0,0,0,2,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,4,0,0,0,0,2,4] >;
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 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C23.709C24 | C23.63C23 | C23.65C23 | C23.67C23 | C23.81C23 | C23.83C23 | C2×Q8 | C2×C4 | C22 | C22 |
| # reps | 1 | 4 | 2 | 3 | 4 | 2 | 4 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2^3._{709}C_2^4 % in TeX
G:=Group("C2^3.709C2^4"); // GroupNames label
G:=SmallGroup(128,1541);
// by ID
G=gap.SmallGroup(128,1541);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,344,758,723,520,1571,346,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*a=a*c,e^2=g^2=b*a=a*b,f^2=c*b=b*c,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