p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.420C24, C22.2132+ 1+4, C22.1622- 1+4, C4⋊C4⋊16Q8, C42⋊8C4.29C2, C2.14(Q8⋊3Q8), C2.27(D4⋊3Q8), C22.94(C22×Q8), (C22×C4).530C23, (C2×C42).535C22, (C22×Q8).123C22, C23.78C23.5C2, C2.36(C22.45C24), C23.83C23.10C2, C2.C42.168C22, C23.81C23.10C2, C23.67C23.37C2, C23.63C23.23C2, C2.63(C23.36C23), C2.51(C22.46C24), C2.37(C22.36C24), C2.23(C23.37C23), (C4×C4⋊C4).58C2, (C2×C4).47(C2×Q8), (C2×C4).379(C4○D4), (C2×C4⋊C4).283C22, C22.297(C2×C4○D4), SmallGroup(128,1252)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.420C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=ca=ac, e2=f2=a, g2=b, ab=ba, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >
Subgroups: 324 in 194 conjugacy classes, 100 normal (82 characteristic)
C1, C2, C4, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, C4×C4⋊C4, C42⋊8C4, C23.63C23, C23.67C23, C23.78C23, C23.81C23, C23.83C23, C23.420C24
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C23.36C23, C23.37C23, C22.36C24, C22.45C24, C22.46C24, D4⋊3Q8, Q8⋊3Q8, C23.420C24
►Regular action on 128 points
Generators in S128
(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 26 10 95)(2 96 11 27)(3 28 12 93)(4 94 9 25)(5 114 69 55)(6 56 70 115)(7 116 71 53)(8 54 72 113)(13 100 84 29)(14 30 81 97)(15 98 82 31)(16 32 83 99)(17 33 86 78)(18 79 87 34)(19 35 88 80)(20 77 85 36)(21 75 92 37)(22 38 89 76)(23 73 90 39)(24 40 91 74)(41 57 102 118)(42 119 103 58)(43 59 104 120)(44 117 101 60)(45 124 108 61)(46 62 105 121)(47 122 106 63)(48 64 107 123)(49 67 110 126)(50 127 111 68)(51 65 112 128)(52 125 109 66)
(1 18 10 87)(2 112 11 51)(3 20 12 85)(4 110 9 49)(5 61 69 124)(6 97 70 30)(7 63 71 122)(8 99 72 32)(13 92 84 21)(14 54 81 113)(15 90 82 23)(16 56 83 115)(17 102 86 41)(19 104 88 43)(22 105 89 46)(24 107 91 48)(25 128 94 65)(26 36 95 77)(27 126 96 67)(28 34 93 79)(29 39 100 73)(31 37 98 75)(33 59 78 120)(35 57 80 118)(38 64 76 123)(40 62 74 121)(42 111 103 50)(44 109 101 52)(45 116 108 53)(47 114 106 55)(58 66 119 125)(60 68 117 127)
(1 47 42 82)(2 107 43 16)(3 45 44 84)(4 105 41 14)(5 127 39 34)(6 65 40 80)(7 125 37 36)(8 67 38 78)(9 46 102 81)(10 106 103 15)(11 48 104 83)(12 108 101 13)(17 54 110 89)(18 114 111 23)(19 56 112 91)(20 116 109 21)(22 86 113 49)(24 88 115 51)(25 62 118 97)(26 122 119 31)(27 64 120 99)(28 124 117 29)(30 94 121 57)(32 96 123 59)(33 72 126 76)(35 70 128 74)(50 90 87 55)(52 92 85 53)(58 98 95 63)(60 100 93 61)(66 75 77 71)(68 73 79 69)
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), (1,26,10,95)(2,96,11,27)(3,28,12,93)(4,94,9,25)(5,114,69,55)(6,56,70,115)(7,116,71,53)(8,54,72,113)(13,100,84,29)(14,30,81,97)(15,98,82,31)(16,32,83,99)(17,33,86,78)(18,79,87,34)(19,35,88,80)(20,77,85,36)(21,75,92,37)(22,38,89,76)(23,73,90,39)(24,40,91,74)(41,57,102,118)(42,119,103,58)(43,59,104,120)(44,117,101,60)(45,124,108,61)(46,62,105,121)(47,122,106,63)(48,64,107,123)(49,67,110,126)(50,127,111,68)(51,65,112,128)(52,125,109,66), (1,18,10,87)(2,112,11,51)(3,20,12,85)(4,110,9,49)(5,61,69,124)(6,97,70,30)(7,63,71,122)(8,99,72,32)(13,92,84,21)(14,54,81,113)(15,90,82,23)(16,56,83,115)(17,102,86,41)(19,104,88,43)(22,105,89,46)(24,107,91,48)(25,128,94,65)(26,36,95,77)(27,126,96,67)(28,34,93,79)(29,39,100,73)(31,37,98,75)(33,59,78,120)(35,57,80,118)(38,64,76,123)(40,62,74,121)(42,111,103,50)(44,109,101,52)(45,116,108,53)(47,114,106,55)(58,66,119,125)(60,68,117,127), (1,47,42,82)(2,107,43,16)(3,45,44,84)(4,105,41,14)(5,127,39,34)(6,65,40,80)(7,125,37,36)(8,67,38,78)(9,46,102,81)(10,106,103,15)(11,48,104,83)(12,108,101,13)(17,54,110,89)(18,114,111,23)(19,56,112,91)(20,116,109,21)(22,86,113,49)(24,88,115,51)(25,62,118,97)(26,122,119,31)(27,64,120,99)(28,124,117,29)(30,94,121,57)(32,96,123,59)(33,72,126,76)(35,70,128,74)(50,90,87,55)(52,92,85,53)(58,98,95,63)(60,100,93,61)(66,75,77,71)(68,73,79,69)>;
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), (1,26,10,95)(2,96,11,27)(3,28,12,93)(4,94,9,25)(5,114,69,55)(6,56,70,115)(7,116,71,53)(8,54,72,113)(13,100,84,29)(14,30,81,97)(15,98,82,31)(16,32,83,99)(17,33,86,78)(18,79,87,34)(19,35,88,80)(20,77,85,36)(21,75,92,37)(22,38,89,76)(23,73,90,39)(24,40,91,74)(41,57,102,118)(42,119,103,58)(43,59,104,120)(44,117,101,60)(45,124,108,61)(46,62,105,121)(47,122,106,63)(48,64,107,123)(49,67,110,126)(50,127,111,68)(51,65,112,128)(52,125,109,66), (1,18,10,87)(2,112,11,51)(3,20,12,85)(4,110,9,49)(5,61,69,124)(6,97,70,30)(7,63,71,122)(8,99,72,32)(13,92,84,21)(14,54,81,113)(15,90,82,23)(16,56,83,115)(17,102,86,41)(19,104,88,43)(22,105,89,46)(24,107,91,48)(25,128,94,65)(26,36,95,77)(27,126,96,67)(28,34,93,79)(29,39,100,73)(31,37,98,75)(33,59,78,120)(35,57,80,118)(38,64,76,123)(40,62,74,121)(42,111,103,50)(44,109,101,52)(45,116,108,53)(47,114,106,55)(58,66,119,125)(60,68,117,127), (1,47,42,82)(2,107,43,16)(3,45,44,84)(4,105,41,14)(5,127,39,34)(6,65,40,80)(7,125,37,36)(8,67,38,78)(9,46,102,81)(10,106,103,15)(11,48,104,83)(12,108,101,13)(17,54,110,89)(18,114,111,23)(19,56,112,91)(20,116,109,21)(22,86,113,49)(24,88,115,51)(25,62,118,97)(26,122,119,31)(27,64,120,99)(28,124,117,29)(30,94,121,57)(32,96,123,59)(33,72,126,76)(35,70,128,74)(50,90,87,55)(52,92,85,53)(58,98,95,63)(60,100,93,61)(66,75,77,71)(68,73,79,69) );
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)], [(1,26,10,95),(2,96,11,27),(3,28,12,93),(4,94,9,25),(5,114,69,55),(6,56,70,115),(7,116,71,53),(8,54,72,113),(13,100,84,29),(14,30,81,97),(15,98,82,31),(16,32,83,99),(17,33,86,78),(18,79,87,34),(19,35,88,80),(20,77,85,36),(21,75,92,37),(22,38,89,76),(23,73,90,39),(24,40,91,74),(41,57,102,118),(42,119,103,58),(43,59,104,120),(44,117,101,60),(45,124,108,61),(46,62,105,121),(47,122,106,63),(48,64,107,123),(49,67,110,126),(50,127,111,68),(51,65,112,128),(52,125,109,66)], [(1,18,10,87),(2,112,11,51),(3,20,12,85),(4,110,9,49),(5,61,69,124),(6,97,70,30),(7,63,71,122),(8,99,72,32),(13,92,84,21),(14,54,81,113),(15,90,82,23),(16,56,83,115),(17,102,86,41),(19,104,88,43),(22,105,89,46),(24,107,91,48),(25,128,94,65),(26,36,95,77),(27,126,96,67),(28,34,93,79),(29,39,100,73),(31,37,98,75),(33,59,78,120),(35,57,80,118),(38,64,76,123),(40,62,74,121),(42,111,103,50),(44,109,101,52),(45,116,108,53),(47,114,106,55),(58,66,119,125),(60,68,117,127)], [(1,47,42,82),(2,107,43,16),(3,45,44,84),(4,105,41,14),(5,127,39,34),(6,65,40,80),(7,125,37,36),(8,67,38,78),(9,46,102,81),(10,106,103,15),(11,48,104,83),(12,108,101,13),(17,54,110,89),(18,114,111,23),(19,56,112,91),(20,116,109,21),(22,86,113,49),(24,88,115,51),(25,62,118,97),(26,122,119,31),(27,64,120,99),(28,124,117,29),(30,94,121,57),(32,96,123,59),(33,72,126,76),(35,70,128,74),(50,90,87,55),(52,92,85,53),(58,98,95,63),(60,100,93,61),(66,75,77,71),(68,73,79,69)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4Z | 4AA | 4AB | 4AC | 4AD |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C23.420C24 | C4×C4⋊C4 | C42⋊8C4 | C23.63C23 | C23.67C23 | C23.78C23 | C23.81C23 | C23.83C23 | C4⋊C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 6 | 2 | 1 | 2 | 1 | 4 | 16 | 1 | 1 |
Matrix representation of C23.420C24 ►in GL6(𝔽5)
| 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 |
| 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 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 | 2 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [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],[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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C23.420C24 in GAP, Magma, Sage, TeX
C_2^3._{420}C_2^4 % in TeX
G:=Group("C2^3.420C2^4"); // GroupNames label
G:=SmallGroup(128,1252);
// by ID
G=gap.SmallGroup(128,1252);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,100,675,136]);
// 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=f^2=a,g^2=b,a*b=b*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,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,e*g=g*e,f*g=g*f>;
// generators/relations