p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.619C24, C22.2942- 1+4, C4⋊C4.124D4, C2.70(D4⋊6D4), C42⋊8C4.42C2, C2.51(Q8⋊5D4), (C2×C42).670C22, (C22×C4).195C23, C22.428(C22×D4), C23.81C23.32C2, C2.C42.325C22, C23.65C23.72C2, C23.83C23.30C2, C23.63C23.41C2, C2.2(C22.58C24), C2.64(C23.38C23), C2.44(C22.35C24), C2.87(C22.46C24), (C2×C4).115(C2×D4), (C2×C4).435(C4○D4), (C2×C4⋊C4).432C22, C22.481(C2×C4○D4), (C2×C42.C2).27C2, SmallGroup(128,1451)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.619C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=g2=a, e2=f2=b, ab=ba, ac=ca, ede-1=ad=da, geg-1=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, gdg-1=abd, fg=gf >
Subgroups: 324 in 200 conjugacy classes, 96 normal (82 characteristic)
C1, C2, C4, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C42.C2, C42⋊8C4, C23.63C23, C23.65C23, C23.81C23, C23.83C23, C2×C42.C2, C23.619C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, C23.38C23, C22.35C24, D4⋊6D4, Q8⋊5D4, C22.46C24, C22.58C24, C23.619C24
►Regular action on 128 points
Generators in S128
(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 73)(2 74)(3 75)(4 76)(5 36)(6 33)(7 34)(8 35)(9 105)(10 106)(11 107)(12 108)(13 41)(14 42)(15 43)(16 44)(17 113)(18 114)(19 115)(20 116)(21 49)(22 50)(23 51)(24 52)(25 121)(26 122)(27 123)(28 124)(29 57)(30 58)(31 59)(32 60)(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)(69 104)(70 101)(71 102)(72 103)(81 109)(82 110)(83 111)(84 112)(89 117)(90 118)(91 119)(92 120)(97 125)(98 126)(99 127)(100 128)
(1 11)(2 12)(3 9)(4 10)(5 98)(6 99)(7 100)(8 97)(13 47)(14 48)(15 45)(16 46)(17 111)(18 112)(19 109)(20 110)(21 55)(22 56)(23 53)(24 54)(25 119)(26 120)(27 117)(28 118)(29 63)(30 64)(31 61)(32 62)(33 127)(34 128)(35 125)(36 126)(37 72)(38 69)(39 70)(40 71)(41 79)(42 80)(43 77)(44 78)(49 87)(50 88)(51 85)(52 86)(57 95)(58 96)(59 93)(60 94)(65 102)(66 103)(67 104)(68 101)(73 107)(74 108)(75 105)(76 106)(81 115)(82 116)(83 113)(84 114)(89 123)(90 124)(91 121)(92 122)
(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 89 73 117)(2 92 74 120)(3 91 75 119)(4 90 76 118)(5 114 36 18)(6 113 33 17)(7 116 34 20)(8 115 35 19)(9 121 105 25)(10 124 106 28)(11 123 107 27)(12 122 108 26)(13 57 41 29)(14 60 42 32)(15 59 43 31)(16 58 44 30)(21 66 49 37)(22 65 50 40)(23 68 51 39)(24 67 52 38)(45 93 77 61)(46 96 78 64)(47 95 79 63)(48 94 80 62)(53 101 85 70)(54 104 86 69)(55 103 87 72)(56 102 88 71)(81 125 109 97)(82 128 110 100)(83 127 111 99)(84 126 112 98)
(1 81 73 109)(2 110 74 82)(3 83 75 111)(4 112 76 84)(5 118 36 90)(6 91 33 119)(7 120 34 92)(8 89 35 117)(9 113 105 17)(10 18 106 114)(11 115 107 19)(12 20 108 116)(13 49 41 21)(14 22 42 50)(15 51 43 23)(16 24 44 52)(25 99 121 127)(26 128 122 100)(27 97 123 125)(28 126 124 98)(29 103 57 72)(30 69 58 104)(31 101 59 70)(32 71 60 102)(37 63 66 95)(38 96 67 64)(39 61 68 93)(40 94 65 62)(45 85 77 53)(46 54 78 86)(47 87 79 55)(48 56 80 88)
(1 43 3 41)(2 14 4 16)(5 102 7 104)(6 70 8 72)(9 79 11 77)(10 46 12 48)(13 73 15 75)(17 87 19 85)(18 54 20 56)(21 81 23 83)(22 112 24 110)(25 93 27 95)(26 64 28 62)(29 91 31 89)(30 118 32 120)(33 101 35 103)(34 69 36 71)(37 99 39 97)(38 126 40 128)(42 76 44 74)(45 105 47 107)(49 109 51 111)(50 84 52 82)(53 113 55 115)(57 119 59 117)(58 90 60 92)(61 123 63 121)(65 100 67 98)(66 127 68 125)(78 108 80 106)(86 116 88 114)(94 122 96 124)
G:=sub<Sym(128)| (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), (1,73)(2,74)(3,75)(4,76)(5,36)(6,33)(7,34)(8,35)(9,105)(10,106)(11,107)(12,108)(13,41)(14,42)(15,43)(16,44)(17,113)(18,114)(19,115)(20,116)(21,49)(22,50)(23,51)(24,52)(25,121)(26,122)(27,123)(28,124)(29,57)(30,58)(31,59)(32,60)(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)(69,104)(70,101)(71,102)(72,103)(81,109)(82,110)(83,111)(84,112)(89,117)(90,118)(91,119)(92,120)(97,125)(98,126)(99,127)(100,128), (1,11)(2,12)(3,9)(4,10)(5,98)(6,99)(7,100)(8,97)(13,47)(14,48)(15,45)(16,46)(17,111)(18,112)(19,109)(20,110)(21,55)(22,56)(23,53)(24,54)(25,119)(26,120)(27,117)(28,118)(29,63)(30,64)(31,61)(32,62)(33,127)(34,128)(35,125)(36,126)(37,72)(38,69)(39,70)(40,71)(41,79)(42,80)(43,77)(44,78)(49,87)(50,88)(51,85)(52,86)(57,95)(58,96)(59,93)(60,94)(65,102)(66,103)(67,104)(68,101)(73,107)(74,108)(75,105)(76,106)(81,115)(82,116)(83,113)(84,114)(89,123)(90,124)(91,121)(92,122), (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), (1,89,73,117)(2,92,74,120)(3,91,75,119)(4,90,76,118)(5,114,36,18)(6,113,33,17)(7,116,34,20)(8,115,35,19)(9,121,105,25)(10,124,106,28)(11,123,107,27)(12,122,108,26)(13,57,41,29)(14,60,42,32)(15,59,43,31)(16,58,44,30)(21,66,49,37)(22,65,50,40)(23,68,51,39)(24,67,52,38)(45,93,77,61)(46,96,78,64)(47,95,79,63)(48,94,80,62)(53,101,85,70)(54,104,86,69)(55,103,87,72)(56,102,88,71)(81,125,109,97)(82,128,110,100)(83,127,111,99)(84,126,112,98), (1,81,73,109)(2,110,74,82)(3,83,75,111)(4,112,76,84)(5,118,36,90)(6,91,33,119)(7,120,34,92)(8,89,35,117)(9,113,105,17)(10,18,106,114)(11,115,107,19)(12,20,108,116)(13,49,41,21)(14,22,42,50)(15,51,43,23)(16,24,44,52)(25,99,121,127)(26,128,122,100)(27,97,123,125)(28,126,124,98)(29,103,57,72)(30,69,58,104)(31,101,59,70)(32,71,60,102)(37,63,66,95)(38,96,67,64)(39,61,68,93)(40,94,65,62)(45,85,77,53)(46,54,78,86)(47,87,79,55)(48,56,80,88), (1,43,3,41)(2,14,4,16)(5,102,7,104)(6,70,8,72)(9,79,11,77)(10,46,12,48)(13,73,15,75)(17,87,19,85)(18,54,20,56)(21,81,23,83)(22,112,24,110)(25,93,27,95)(26,64,28,62)(29,91,31,89)(30,118,32,120)(33,101,35,103)(34,69,36,71)(37,99,39,97)(38,126,40,128)(42,76,44,74)(45,105,47,107)(49,109,51,111)(50,84,52,82)(53,113,55,115)(57,119,59,117)(58,90,60,92)(61,123,63,121)(65,100,67,98)(66,127,68,125)(78,108,80,106)(86,116,88,114)(94,122,96,124)>;
G:=Group( (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), (1,73)(2,74)(3,75)(4,76)(5,36)(6,33)(7,34)(8,35)(9,105)(10,106)(11,107)(12,108)(13,41)(14,42)(15,43)(16,44)(17,113)(18,114)(19,115)(20,116)(21,49)(22,50)(23,51)(24,52)(25,121)(26,122)(27,123)(28,124)(29,57)(30,58)(31,59)(32,60)(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)(69,104)(70,101)(71,102)(72,103)(81,109)(82,110)(83,111)(84,112)(89,117)(90,118)(91,119)(92,120)(97,125)(98,126)(99,127)(100,128), (1,11)(2,12)(3,9)(4,10)(5,98)(6,99)(7,100)(8,97)(13,47)(14,48)(15,45)(16,46)(17,111)(18,112)(19,109)(20,110)(21,55)(22,56)(23,53)(24,54)(25,119)(26,120)(27,117)(28,118)(29,63)(30,64)(31,61)(32,62)(33,127)(34,128)(35,125)(36,126)(37,72)(38,69)(39,70)(40,71)(41,79)(42,80)(43,77)(44,78)(49,87)(50,88)(51,85)(52,86)(57,95)(58,96)(59,93)(60,94)(65,102)(66,103)(67,104)(68,101)(73,107)(74,108)(75,105)(76,106)(81,115)(82,116)(83,113)(84,114)(89,123)(90,124)(91,121)(92,122), (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), (1,89,73,117)(2,92,74,120)(3,91,75,119)(4,90,76,118)(5,114,36,18)(6,113,33,17)(7,116,34,20)(8,115,35,19)(9,121,105,25)(10,124,106,28)(11,123,107,27)(12,122,108,26)(13,57,41,29)(14,60,42,32)(15,59,43,31)(16,58,44,30)(21,66,49,37)(22,65,50,40)(23,68,51,39)(24,67,52,38)(45,93,77,61)(46,96,78,64)(47,95,79,63)(48,94,80,62)(53,101,85,70)(54,104,86,69)(55,103,87,72)(56,102,88,71)(81,125,109,97)(82,128,110,100)(83,127,111,99)(84,126,112,98), (1,81,73,109)(2,110,74,82)(3,83,75,111)(4,112,76,84)(5,118,36,90)(6,91,33,119)(7,120,34,92)(8,89,35,117)(9,113,105,17)(10,18,106,114)(11,115,107,19)(12,20,108,116)(13,49,41,21)(14,22,42,50)(15,51,43,23)(16,24,44,52)(25,99,121,127)(26,128,122,100)(27,97,123,125)(28,126,124,98)(29,103,57,72)(30,69,58,104)(31,101,59,70)(32,71,60,102)(37,63,66,95)(38,96,67,64)(39,61,68,93)(40,94,65,62)(45,85,77,53)(46,54,78,86)(47,87,79,55)(48,56,80,88), (1,43,3,41)(2,14,4,16)(5,102,7,104)(6,70,8,72)(9,79,11,77)(10,46,12,48)(13,73,15,75)(17,87,19,85)(18,54,20,56)(21,81,23,83)(22,112,24,110)(25,93,27,95)(26,64,28,62)(29,91,31,89)(30,118,32,120)(33,101,35,103)(34,69,36,71)(37,99,39,97)(38,126,40,128)(42,76,44,74)(45,105,47,107)(49,109,51,111)(50,84,52,82)(53,113,55,115)(57,119,59,117)(58,90,60,92)(61,123,63,121)(65,100,67,98)(66,127,68,125)(78,108,80,106)(86,116,88,114)(94,122,96,124) );
G=PermutationGroup([[(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)], [(1,73),(2,74),(3,75),(4,76),(5,36),(6,33),(7,34),(8,35),(9,105),(10,106),(11,107),(12,108),(13,41),(14,42),(15,43),(16,44),(17,113),(18,114),(19,115),(20,116),(21,49),(22,50),(23,51),(24,52),(25,121),(26,122),(27,123),(28,124),(29,57),(30,58),(31,59),(32,60),(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),(69,104),(70,101),(71,102),(72,103),(81,109),(82,110),(83,111),(84,112),(89,117),(90,118),(91,119),(92,120),(97,125),(98,126),(99,127),(100,128)], [(1,11),(2,12),(3,9),(4,10),(5,98),(6,99),(7,100),(8,97),(13,47),(14,48),(15,45),(16,46),(17,111),(18,112),(19,109),(20,110),(21,55),(22,56),(23,53),(24,54),(25,119),(26,120),(27,117),(28,118),(29,63),(30,64),(31,61),(32,62),(33,127),(34,128),(35,125),(36,126),(37,72),(38,69),(39,70),(40,71),(41,79),(42,80),(43,77),(44,78),(49,87),(50,88),(51,85),(52,86),(57,95),(58,96),(59,93),(60,94),(65,102),(66,103),(67,104),(68,101),(73,107),(74,108),(75,105),(76,106),(81,115),(82,116),(83,113),(84,114),(89,123),(90,124),(91,121),(92,122)], [(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)], [(1,89,73,117),(2,92,74,120),(3,91,75,119),(4,90,76,118),(5,114,36,18),(6,113,33,17),(7,116,34,20),(8,115,35,19),(9,121,105,25),(10,124,106,28),(11,123,107,27),(12,122,108,26),(13,57,41,29),(14,60,42,32),(15,59,43,31),(16,58,44,30),(21,66,49,37),(22,65,50,40),(23,68,51,39),(24,67,52,38),(45,93,77,61),(46,96,78,64),(47,95,79,63),(48,94,80,62),(53,101,85,70),(54,104,86,69),(55,103,87,72),(56,102,88,71),(81,125,109,97),(82,128,110,100),(83,127,111,99),(84,126,112,98)], [(1,81,73,109),(2,110,74,82),(3,83,75,111),(4,112,76,84),(5,118,36,90),(6,91,33,119),(7,120,34,92),(8,89,35,117),(9,113,105,17),(10,18,106,114),(11,115,107,19),(12,20,108,116),(13,49,41,21),(14,22,42,50),(15,51,43,23),(16,24,44,52),(25,99,121,127),(26,128,122,100),(27,97,123,125),(28,126,124,98),(29,103,57,72),(30,69,58,104),(31,101,59,70),(32,71,60,102),(37,63,66,95),(38,96,67,64),(39,61,68,93),(40,94,65,62),(45,85,77,53),(46,54,78,86),(47,87,79,55),(48,56,80,88)], [(1,43,3,41),(2,14,4,16),(5,102,7,104),(6,70,8,72),(9,79,11,77),(10,46,12,48),(13,73,15,75),(17,87,19,85),(18,54,20,56),(21,81,23,83),(22,112,24,110),(25,93,27,95),(26,64,28,62),(29,91,31,89),(30,118,32,120),(33,101,35,103),(34,69,36,71),(37,99,39,97),(38,126,40,128),(42,76,44,74),(45,105,47,107),(49,109,51,111),(50,84,52,82),(53,113,55,115),(57,119,59,117),(58,90,60,92),(61,123,63,121),(65,100,67,98),(66,127,68,125),(78,108,80,106),(86,116,88,114),(94,122,96,124)]])
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 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2- 1+4 |
| kernel | C23.619C24 | C42⋊8C4 | C23.63C23 | C23.65C23 | C23.81C23 | C23.83C23 | C2×C42.C2 | C4⋊C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 3 | 3 | 2 | 4 | 8 | 4 |
Matrix representation of C23.619C24 ►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 | 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 | 2 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,4,0,0,0,0,2,4,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,2,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[3,2,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C23.619C24 in GAP, Magma, Sage, TeX
C_2^3._{619}C_2^4 % in TeX
G:=Group("C2^3.619C2^4"); // GroupNames label
G:=SmallGroup(128,1451);
// by ID
G=gap.SmallGroup(128,1451);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,184,1571,346,80]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=g^2=a,e^2=f^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g^-1=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,g*d*g^-1=a*b*d,f*g=g*f>;
// generators/relations