p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.414C24, C22.1582- 1+4, C42⋊5C4.7C2, C42⋊8C4.28C2, (C2×C42).54C22, C4.43(C42⋊2C2), (C22×C4).527C23, (C22×Q8).122C22, C23.84C23.1C2, C23.83C23.9C2, C23.65C23.48C2, C2.C42.162C22, C23.63C23.22C2, C23.67C23.36C2, C2.47(C22.46C24), C2.57(C23.36C23), C2.19(C22.35C24), C2.27(C22.50C24), (C4×C4⋊C4).57C2, (C2×C4).135(C4○D4), (C2×C4⋊C4).862C22, C2.21(C2×C42⋊2C2), C22.291(C2×C4○D4), SmallGroup(128,1246)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.414C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=abc, e2=b, f2=ba=ab, g2=a, ac=ca, 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: 308 in 186 conjugacy classes, 96 normal (42 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×Q8, C4×C4⋊C4, C42⋊8C4, C42⋊5C4, C23.63C23, C23.65C23, C23.67C23, C23.83C23, C23.84C23, C23.414C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C42⋊2C2, C2×C4○D4, 2- 1+4, C2×C42⋊2C2, C23.36C23, C22.35C24, C22.46C24, C22.50C24, C23.414C24
►Regular action on 128 points
Generators in S128
(1 103)(2 104)(3 101)(4 102)(5 125)(6 126)(7 127)(8 128)(9 75)(10 76)(11 73)(12 74)(13 77)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 106)(38 107)(39 108)(40 105)(41 48)(42 45)(43 46)(44 47)(49 56)(50 53)(51 54)(52 55)(57 64)(58 61)(59 62)(60 63)(65 72)(66 69)(67 70)(68 71)(109 114)(110 115)(111 116)(112 113)(117 122)(118 123)(119 124)(120 121)
(1 76)(2 73)(3 74)(4 75)(5 100)(6 97)(7 98)(8 99)(9 102)(10 103)(11 104)(12 101)(13 47)(14 48)(15 45)(16 46)(17 110)(18 111)(19 112)(20 109)(21 55)(22 56)(23 53)(24 54)(25 118)(26 119)(27 120)(28 117)(29 63)(30 64)(31 61)(32 62)(33 126)(34 127)(35 128)(36 125)(37 71)(38 72)(39 69)(40 70)(41 78)(42 79)(43 80)(44 77)(49 86)(50 87)(51 88)(52 85)(57 94)(58 95)(59 96)(60 93)(65 107)(66 108)(67 105)(68 106)(81 115)(82 116)(83 113)(84 114)(89 123)(90 124)(91 121)(92 122)
(1 12)(2 9)(3 10)(4 11)(5 34)(6 35)(7 36)(8 33)(13 42)(14 43)(15 44)(16 41)(17 113)(18 114)(19 115)(20 116)(21 50)(22 51)(23 52)(24 49)(25 121)(26 122)(27 123)(28 124)(29 58)(30 59)(31 60)(32 57)(37 66)(38 67)(39 68)(40 65)(45 77)(46 78)(47 79)(48 80)(53 85)(54 86)(55 87)(56 88)(61 93)(62 94)(63 95)(64 96)(69 106)(70 107)(71 108)(72 105)(73 102)(74 103)(75 104)(76 101)(81 112)(82 109)(83 110)(84 111)(89 120)(90 117)(91 118)(92 119)(97 128)(98 125)(99 126)(100 127)
(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 90 76 124)(2 27 73 120)(3 92 74 122)(4 25 75 118)(5 114 100 84)(6 110 97 17)(7 116 98 82)(8 112 99 19)(9 123 102 89)(10 119 103 26)(11 121 104 91)(12 117 101 28)(13 63 47 29)(14 57 48 94)(15 61 45 31)(16 59 46 96)(18 127 111 34)(20 125 109 36)(21 71 55 37)(22 65 56 107)(23 69 53 39)(24 67 54 105)(30 78 64 41)(32 80 62 43)(33 81 126 115)(35 83 128 113)(38 86 72 49)(40 88 70 51)(42 95 79 58)(44 93 77 60)(50 108 87 66)(52 106 85 68)
(1 18 10 116)(2 112 11 83)(3 20 12 114)(4 110 9 81)(5 26 36 124)(6 120 33 91)(7 28 34 122)(8 118 35 89)(13 52 44 21)(14 86 41 56)(15 50 42 23)(16 88 43 54)(17 102 115 75)(19 104 113 73)(22 48 49 78)(24 46 51 80)(25 128 123 99)(27 126 121 97)(29 39 60 66)(30 70 57 105)(31 37 58 68)(32 72 59 107)(38 96 65 62)(40 94 67 64)(45 87 79 53)(47 85 77 55)(61 71 95 106)(63 69 93 108)(74 109 101 84)(76 111 103 82)(90 100 119 125)(92 98 117 127)
(1 44 103 47)(2 48 104 41)(3 42 101 45)(4 46 102 43)(5 66 125 69)(6 70 126 67)(7 68 127 71)(8 72 128 65)(9 80 75 16)(10 13 76 77)(11 78 73 14)(12 15 74 79)(17 88 81 24)(18 21 82 85)(19 86 83 22)(20 23 84 87)(25 96 89 32)(26 29 90 93)(27 94 91 30)(28 31 92 95)(33 105 97 40)(34 37 98 106)(35 107 99 38)(36 39 100 108)(49 113 56 112)(50 109 53 114)(51 115 54 110)(52 111 55 116)(57 121 64 120)(58 117 61 122)(59 123 62 118)(60 119 63 124)
G:=sub<Sym(128)| (1,103)(2,104)(3,101)(4,102)(5,125)(6,126)(7,127)(8,128)(9,75)(10,76)(11,73)(12,74)(13,77)(14,78)(15,79)(16,80)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,106)(38,107)(39,108)(40,105)(41,48)(42,45)(43,46)(44,47)(49,56)(50,53)(51,54)(52,55)(57,64)(58,61)(59,62)(60,63)(65,72)(66,69)(67,70)(68,71)(109,114)(110,115)(111,116)(112,113)(117,122)(118,123)(119,124)(120,121), (1,76)(2,73)(3,74)(4,75)(5,100)(6,97)(7,98)(8,99)(9,102)(10,103)(11,104)(12,101)(13,47)(14,48)(15,45)(16,46)(17,110)(18,111)(19,112)(20,109)(21,55)(22,56)(23,53)(24,54)(25,118)(26,119)(27,120)(28,117)(29,63)(30,64)(31,61)(32,62)(33,126)(34,127)(35,128)(36,125)(37,71)(38,72)(39,69)(40,70)(41,78)(42,79)(43,80)(44,77)(49,86)(50,87)(51,88)(52,85)(57,94)(58,95)(59,96)(60,93)(65,107)(66,108)(67,105)(68,106)(81,115)(82,116)(83,113)(84,114)(89,123)(90,124)(91,121)(92,122), (1,12)(2,9)(3,10)(4,11)(5,34)(6,35)(7,36)(8,33)(13,42)(14,43)(15,44)(16,41)(17,113)(18,114)(19,115)(20,116)(21,50)(22,51)(23,52)(24,49)(25,121)(26,122)(27,123)(28,124)(29,58)(30,59)(31,60)(32,57)(37,66)(38,67)(39,68)(40,65)(45,77)(46,78)(47,79)(48,80)(53,85)(54,86)(55,87)(56,88)(61,93)(62,94)(63,95)(64,96)(69,106)(70,107)(71,108)(72,105)(73,102)(74,103)(75,104)(76,101)(81,112)(82,109)(83,110)(84,111)(89,120)(90,117)(91,118)(92,119)(97,128)(98,125)(99,126)(100,127), (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), (1,90,76,124)(2,27,73,120)(3,92,74,122)(4,25,75,118)(5,114,100,84)(6,110,97,17)(7,116,98,82)(8,112,99,19)(9,123,102,89)(10,119,103,26)(11,121,104,91)(12,117,101,28)(13,63,47,29)(14,57,48,94)(15,61,45,31)(16,59,46,96)(18,127,111,34)(20,125,109,36)(21,71,55,37)(22,65,56,107)(23,69,53,39)(24,67,54,105)(30,78,64,41)(32,80,62,43)(33,81,126,115)(35,83,128,113)(38,86,72,49)(40,88,70,51)(42,95,79,58)(44,93,77,60)(50,108,87,66)(52,106,85,68), (1,18,10,116)(2,112,11,83)(3,20,12,114)(4,110,9,81)(5,26,36,124)(6,120,33,91)(7,28,34,122)(8,118,35,89)(13,52,44,21)(14,86,41,56)(15,50,42,23)(16,88,43,54)(17,102,115,75)(19,104,113,73)(22,48,49,78)(24,46,51,80)(25,128,123,99)(27,126,121,97)(29,39,60,66)(30,70,57,105)(31,37,58,68)(32,72,59,107)(38,96,65,62)(40,94,67,64)(45,87,79,53)(47,85,77,55)(61,71,95,106)(63,69,93,108)(74,109,101,84)(76,111,103,82)(90,100,119,125)(92,98,117,127), (1,44,103,47)(2,48,104,41)(3,42,101,45)(4,46,102,43)(5,66,125,69)(6,70,126,67)(7,68,127,71)(8,72,128,65)(9,80,75,16)(10,13,76,77)(11,78,73,14)(12,15,74,79)(17,88,81,24)(18,21,82,85)(19,86,83,22)(20,23,84,87)(25,96,89,32)(26,29,90,93)(27,94,91,30)(28,31,92,95)(33,105,97,40)(34,37,98,106)(35,107,99,38)(36,39,100,108)(49,113,56,112)(50,109,53,114)(51,115,54,110)(52,111,55,116)(57,121,64,120)(58,117,61,122)(59,123,62,118)(60,119,63,124)>;
G:=Group( (1,103)(2,104)(3,101)(4,102)(5,125)(6,126)(7,127)(8,128)(9,75)(10,76)(11,73)(12,74)(13,77)(14,78)(15,79)(16,80)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,106)(38,107)(39,108)(40,105)(41,48)(42,45)(43,46)(44,47)(49,56)(50,53)(51,54)(52,55)(57,64)(58,61)(59,62)(60,63)(65,72)(66,69)(67,70)(68,71)(109,114)(110,115)(111,116)(112,113)(117,122)(118,123)(119,124)(120,121), (1,76)(2,73)(3,74)(4,75)(5,100)(6,97)(7,98)(8,99)(9,102)(10,103)(11,104)(12,101)(13,47)(14,48)(15,45)(16,46)(17,110)(18,111)(19,112)(20,109)(21,55)(22,56)(23,53)(24,54)(25,118)(26,119)(27,120)(28,117)(29,63)(30,64)(31,61)(32,62)(33,126)(34,127)(35,128)(36,125)(37,71)(38,72)(39,69)(40,70)(41,78)(42,79)(43,80)(44,77)(49,86)(50,87)(51,88)(52,85)(57,94)(58,95)(59,96)(60,93)(65,107)(66,108)(67,105)(68,106)(81,115)(82,116)(83,113)(84,114)(89,123)(90,124)(91,121)(92,122), (1,12)(2,9)(3,10)(4,11)(5,34)(6,35)(7,36)(8,33)(13,42)(14,43)(15,44)(16,41)(17,113)(18,114)(19,115)(20,116)(21,50)(22,51)(23,52)(24,49)(25,121)(26,122)(27,123)(28,124)(29,58)(30,59)(31,60)(32,57)(37,66)(38,67)(39,68)(40,65)(45,77)(46,78)(47,79)(48,80)(53,85)(54,86)(55,87)(56,88)(61,93)(62,94)(63,95)(64,96)(69,106)(70,107)(71,108)(72,105)(73,102)(74,103)(75,104)(76,101)(81,112)(82,109)(83,110)(84,111)(89,120)(90,117)(91,118)(92,119)(97,128)(98,125)(99,126)(100,127), (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), (1,90,76,124)(2,27,73,120)(3,92,74,122)(4,25,75,118)(5,114,100,84)(6,110,97,17)(7,116,98,82)(8,112,99,19)(9,123,102,89)(10,119,103,26)(11,121,104,91)(12,117,101,28)(13,63,47,29)(14,57,48,94)(15,61,45,31)(16,59,46,96)(18,127,111,34)(20,125,109,36)(21,71,55,37)(22,65,56,107)(23,69,53,39)(24,67,54,105)(30,78,64,41)(32,80,62,43)(33,81,126,115)(35,83,128,113)(38,86,72,49)(40,88,70,51)(42,95,79,58)(44,93,77,60)(50,108,87,66)(52,106,85,68), (1,18,10,116)(2,112,11,83)(3,20,12,114)(4,110,9,81)(5,26,36,124)(6,120,33,91)(7,28,34,122)(8,118,35,89)(13,52,44,21)(14,86,41,56)(15,50,42,23)(16,88,43,54)(17,102,115,75)(19,104,113,73)(22,48,49,78)(24,46,51,80)(25,128,123,99)(27,126,121,97)(29,39,60,66)(30,70,57,105)(31,37,58,68)(32,72,59,107)(38,96,65,62)(40,94,67,64)(45,87,79,53)(47,85,77,55)(61,71,95,106)(63,69,93,108)(74,109,101,84)(76,111,103,82)(90,100,119,125)(92,98,117,127), (1,44,103,47)(2,48,104,41)(3,42,101,45)(4,46,102,43)(5,66,125,69)(6,70,126,67)(7,68,127,71)(8,72,128,65)(9,80,75,16)(10,13,76,77)(11,78,73,14)(12,15,74,79)(17,88,81,24)(18,21,82,85)(19,86,83,22)(20,23,84,87)(25,96,89,32)(26,29,90,93)(27,94,91,30)(28,31,92,95)(33,105,97,40)(34,37,98,106)(35,107,99,38)(36,39,100,108)(49,113,56,112)(50,109,53,114)(51,115,54,110)(52,111,55,116)(57,121,64,120)(58,117,61,122)(59,123,62,118)(60,119,63,124) );
G=PermutationGroup([[(1,103),(2,104),(3,101),(4,102),(5,125),(6,126),(7,127),(8,128),(9,75),(10,76),(11,73),(12,74),(13,77),(14,78),(15,79),(16,80),(17,81),(18,82),(19,83),(20,84),(21,85),(22,86),(23,87),(24,88),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(33,97),(34,98),(35,99),(36,100),(37,106),(38,107),(39,108),(40,105),(41,48),(42,45),(43,46),(44,47),(49,56),(50,53),(51,54),(52,55),(57,64),(58,61),(59,62),(60,63),(65,72),(66,69),(67,70),(68,71),(109,114),(110,115),(111,116),(112,113),(117,122),(118,123),(119,124),(120,121)], [(1,76),(2,73),(3,74),(4,75),(5,100),(6,97),(7,98),(8,99),(9,102),(10,103),(11,104),(12,101),(13,47),(14,48),(15,45),(16,46),(17,110),(18,111),(19,112),(20,109),(21,55),(22,56),(23,53),(24,54),(25,118),(26,119),(27,120),(28,117),(29,63),(30,64),(31,61),(32,62),(33,126),(34,127),(35,128),(36,125),(37,71),(38,72),(39,69),(40,70),(41,78),(42,79),(43,80),(44,77),(49,86),(50,87),(51,88),(52,85),(57,94),(58,95),(59,96),(60,93),(65,107),(66,108),(67,105),(68,106),(81,115),(82,116),(83,113),(84,114),(89,123),(90,124),(91,121),(92,122)], [(1,12),(2,9),(3,10),(4,11),(5,34),(6,35),(7,36),(8,33),(13,42),(14,43),(15,44),(16,41),(17,113),(18,114),(19,115),(20,116),(21,50),(22,51),(23,52),(24,49),(25,121),(26,122),(27,123),(28,124),(29,58),(30,59),(31,60),(32,57),(37,66),(38,67),(39,68),(40,65),(45,77),(46,78),(47,79),(48,80),(53,85),(54,86),(55,87),(56,88),(61,93),(62,94),(63,95),(64,96),(69,106),(70,107),(71,108),(72,105),(73,102),(74,103),(75,104),(76,101),(81,112),(82,109),(83,110),(84,111),(89,120),(90,117),(91,118),(92,119),(97,128),(98,125),(99,126),(100,127)], [(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)], [(1,90,76,124),(2,27,73,120),(3,92,74,122),(4,25,75,118),(5,114,100,84),(6,110,97,17),(7,116,98,82),(8,112,99,19),(9,123,102,89),(10,119,103,26),(11,121,104,91),(12,117,101,28),(13,63,47,29),(14,57,48,94),(15,61,45,31),(16,59,46,96),(18,127,111,34),(20,125,109,36),(21,71,55,37),(22,65,56,107),(23,69,53,39),(24,67,54,105),(30,78,64,41),(32,80,62,43),(33,81,126,115),(35,83,128,113),(38,86,72,49),(40,88,70,51),(42,95,79,58),(44,93,77,60),(50,108,87,66),(52,106,85,68)], [(1,18,10,116),(2,112,11,83),(3,20,12,114),(4,110,9,81),(5,26,36,124),(6,120,33,91),(7,28,34,122),(8,118,35,89),(13,52,44,21),(14,86,41,56),(15,50,42,23),(16,88,43,54),(17,102,115,75),(19,104,113,73),(22,48,49,78),(24,46,51,80),(25,128,123,99),(27,126,121,97),(29,39,60,66),(30,70,57,105),(31,37,58,68),(32,72,59,107),(38,96,65,62),(40,94,67,64),(45,87,79,53),(47,85,77,55),(61,71,95,106),(63,69,93,108),(74,109,101,84),(76,111,103,82),(90,100,119,125),(92,98,117,127)], [(1,44,103,47),(2,48,104,41),(3,42,101,45),(4,46,102,43),(5,66,125,69),(6,70,126,67),(7,68,127,71),(8,72,128,65),(9,80,75,16),(10,13,76,77),(11,78,73,14),(12,15,74,79),(17,88,81,24),(18,21,82,85),(19,86,83,22),(20,23,84,87),(25,96,89,32),(26,29,90,93),(27,94,91,30),(28,31,92,95),(33,105,97,40),(34,37,98,106),(35,107,99,38),(36,39,100,108),(49,113,56,112),(50,109,53,114),(51,115,54,110),(52,111,55,116),(57,121,64,120),(58,117,61,122),(59,123,62,118),(60,119,63,124)]])
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 | 1 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | 2- 1+4 |
| kernel | C23.414C24 | C4×C4⋊C4 | C42⋊8C4 | C42⋊5C4 | C23.63C23 | C23.65C23 | C23.67C23 | C23.83C23 | C23.84C23 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 2 | 2 | 1 | 3 | 2 | 2 | 20 | 2 |
Matrix representation of C23.414C24 ►in 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 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 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 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 1 | 1 | 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 | 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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[2,0,0,0,0,0,4,3,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[2,3,0,0,0,0,4,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[4,1,0,0,0,0,3,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
C23.414C24 in GAP, Magma, Sage, TeX
C_2^3._{414}C_2^4 % in TeX
G:=Group("C2^3.414C2^4"); // GroupNames label
G:=SmallGroup(128,1246);
// by ID
G=gap.SmallGroup(128,1246);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,232,758,723,268,675,80]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=a*b*c,e^2=b,f^2=b*a=a*b,g^2=a,a*c=c*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