p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.692C24, C22.4652+ 1+4, C22.3552- 1+4, C4⋊C4⋊7Q8, C2.10Q82, C42⋊9C4.38C2, C2.65(D4⋊3Q8), (C22×C4).602C23, (C2×C42).719C22, C22.163(C22×Q8), (C22×Q8).222C22, C2.35(C22.54C24), C23.63C23.54C2, C23.78C23.25C2, C23.65C23.83C2, C23.81C23.44C2, C2.C42.396C22, C2.44(C23.41C23), C2.110(C22.33C24), (C2×C4).87(C2×Q8), (C2×C4).473(C4○D4), (C2×C4⋊C4).502C22, C22.553(C2×C4○D4), SmallGroup(128,1524)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.692C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=abc, e2=a, f2=b, g2=ba=ab, 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: 340 in 192 conjugacy classes, 104 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×Q8, C42⋊9C4, C23.63C23, C23.65C23, C23.78C23, C23.78C23, C23.81C23, C23.81C23, C23.692C24
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.33C24, C23.41C23, D4⋊3Q8, Q82, C22.54C24, C23.692C24
►Regular action on 128 points
Generators in S128
(1 103)(2 104)(3 101)(4 102)(5 69)(6 70)(7 71)(8 72)(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 42)(38 43)(39 44)(40 41)(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)(65 128)(66 125)(67 126)(68 127)
(1 12)(2 9)(3 10)(4 11)(5 44)(6 41)(7 42)(8 43)(13 47)(14 48)(15 45)(16 46)(17 51)(18 52)(19 49)(20 50)(21 55)(22 56)(23 53)(24 54)(25 59)(26 60)(27 57)(28 58)(29 63)(30 64)(31 61)(32 62)(33 67)(34 68)(35 65)(36 66)(37 71)(38 72)(39 69)(40 70)(73 102)(74 103)(75 104)(76 101)(77 108)(78 105)(79 106)(80 107)(81 110)(82 111)(83 112)(84 109)(85 116)(86 113)(87 114)(88 115)(89 118)(90 119)(91 120)(92 117)(93 124)(94 121)(95 122)(96 123)(97 126)(98 127)(99 128)(100 125)
(1 76)(2 73)(3 74)(4 75)(5 37)(6 38)(7 39)(8 40)(9 102)(10 103)(11 104)(12 101)(13 106)(14 107)(15 108)(16 105)(17 112)(18 109)(19 110)(20 111)(21 114)(22 115)(23 116)(24 113)(25 120)(26 117)(27 118)(28 119)(29 122)(30 123)(31 124)(32 121)(33 128)(34 125)(35 126)(36 127)(41 72)(42 69)(43 70)(44 71)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)
(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 60)(2 57 104 120)(3 117 101 58)(4 59 102 118)(5 114 69 53)(6 54 70 115)(7 116 71 55)(8 56 72 113)(9 27 75 91)(10 92 76 28)(11 25 73 89)(12 90 74 26)(13 93 77 29)(14 30 78 94)(15 95 79 31)(16 32 80 96)(17 33 81 97)(18 98 82 34)(19 35 83 99)(20 100 84 36)(21 42 85 37)(22 38 86 43)(23 44 87 39)(24 40 88 41)(45 122 106 61)(46 62 107 123)(47 124 108 63)(48 64 105 121)(49 65 112 128)(50 125 109 66)(51 67 110 126)(52 127 111 68)
(1 18 12 52)(2 49 9 19)(3 20 10 50)(4 51 11 17)(5 29 44 63)(6 64 41 30)(7 31 42 61)(8 62 43 32)(13 55 47 21)(14 22 48 56)(15 53 45 23)(16 24 46 54)(25 128 59 99)(26 100 60 125)(27 126 57 97)(28 98 58 127)(33 91 67 120)(34 117 68 92)(35 89 65 118)(36 119 66 90)(37 122 71 95)(38 96 72 123)(39 124 69 93)(40 94 70 121)(73 81 102 110)(74 111 103 82)(75 83 104 112)(76 109 101 84)(77 116 108 85)(78 86 105 113)(79 114 106 87)(80 88 107 115)
(1 77 74 47)(2 48 75 78)(3 79 76 45)(4 46 73 80)(5 66 39 100)(6 97 40 67)(7 68 37 98)(8 99 38 65)(9 14 104 105)(10 106 101 15)(11 16 102 107)(12 108 103 13)(17 24 110 115)(18 116 111 21)(19 22 112 113)(20 114 109 23)(25 96 118 62)(26 63 119 93)(27 94 120 64)(28 61 117 95)(29 90 124 60)(30 57 121 91)(31 92 122 58)(32 59 123 89)(33 41 126 70)(34 71 127 42)(35 43 128 72)(36 69 125 44)(49 56 83 86)(50 87 84 53)(51 54 81 88)(52 85 82 55)
G:=sub<Sym(128)| (1,103)(2,104)(3,101)(4,102)(5,69)(6,70)(7,71)(8,72)(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)(37,42)(38,43)(39,44)(40,41)(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)(65,128)(66,125)(67,126)(68,127), (1,12)(2,9)(3,10)(4,11)(5,44)(6,41)(7,42)(8,43)(13,47)(14,48)(15,45)(16,46)(17,51)(18,52)(19,49)(20,50)(21,55)(22,56)(23,53)(24,54)(25,59)(26,60)(27,57)(28,58)(29,63)(30,64)(31,61)(32,62)(33,67)(34,68)(35,65)(36,66)(37,71)(38,72)(39,69)(40,70)(73,102)(74,103)(75,104)(76,101)(77,108)(78,105)(79,106)(80,107)(81,110)(82,111)(83,112)(84,109)(85,116)(86,113)(87,114)(88,115)(89,118)(90,119)(91,120)(92,117)(93,124)(94,121)(95,122)(96,123)(97,126)(98,127)(99,128)(100,125), (1,76)(2,73)(3,74)(4,75)(5,37)(6,38)(7,39)(8,40)(9,102)(10,103)(11,104)(12,101)(13,106)(14,107)(15,108)(16,105)(17,112)(18,109)(19,110)(20,111)(21,114)(22,115)(23,116)(24,113)(25,120)(26,117)(27,118)(28,119)(29,122)(30,123)(31,124)(32,121)(33,128)(34,125)(35,126)(36,127)(41,72)(42,69)(43,70)(44,71)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100), (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), (1,119,103,60)(2,57,104,120)(3,117,101,58)(4,59,102,118)(5,114,69,53)(6,54,70,115)(7,116,71,55)(8,56,72,113)(9,27,75,91)(10,92,76,28)(11,25,73,89)(12,90,74,26)(13,93,77,29)(14,30,78,94)(15,95,79,31)(16,32,80,96)(17,33,81,97)(18,98,82,34)(19,35,83,99)(20,100,84,36)(21,42,85,37)(22,38,86,43)(23,44,87,39)(24,40,88,41)(45,122,106,61)(46,62,107,123)(47,124,108,63)(48,64,105,121)(49,65,112,128)(50,125,109,66)(51,67,110,126)(52,127,111,68), (1,18,12,52)(2,49,9,19)(3,20,10,50)(4,51,11,17)(5,29,44,63)(6,64,41,30)(7,31,42,61)(8,62,43,32)(13,55,47,21)(14,22,48,56)(15,53,45,23)(16,24,46,54)(25,128,59,99)(26,100,60,125)(27,126,57,97)(28,98,58,127)(33,91,67,120)(34,117,68,92)(35,89,65,118)(36,119,66,90)(37,122,71,95)(38,96,72,123)(39,124,69,93)(40,94,70,121)(73,81,102,110)(74,111,103,82)(75,83,104,112)(76,109,101,84)(77,116,108,85)(78,86,105,113)(79,114,106,87)(80,88,107,115), (1,77,74,47)(2,48,75,78)(3,79,76,45)(4,46,73,80)(5,66,39,100)(6,97,40,67)(7,68,37,98)(8,99,38,65)(9,14,104,105)(10,106,101,15)(11,16,102,107)(12,108,103,13)(17,24,110,115)(18,116,111,21)(19,22,112,113)(20,114,109,23)(25,96,118,62)(26,63,119,93)(27,94,120,64)(28,61,117,95)(29,90,124,60)(30,57,121,91)(31,92,122,58)(32,59,123,89)(33,41,126,70)(34,71,127,42)(35,43,128,72)(36,69,125,44)(49,56,83,86)(50,87,84,53)(51,54,81,88)(52,85,82,55)>;
G:=Group( (1,103)(2,104)(3,101)(4,102)(5,69)(6,70)(7,71)(8,72)(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)(37,42)(38,43)(39,44)(40,41)(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)(65,128)(66,125)(67,126)(68,127), (1,12)(2,9)(3,10)(4,11)(5,44)(6,41)(7,42)(8,43)(13,47)(14,48)(15,45)(16,46)(17,51)(18,52)(19,49)(20,50)(21,55)(22,56)(23,53)(24,54)(25,59)(26,60)(27,57)(28,58)(29,63)(30,64)(31,61)(32,62)(33,67)(34,68)(35,65)(36,66)(37,71)(38,72)(39,69)(40,70)(73,102)(74,103)(75,104)(76,101)(77,108)(78,105)(79,106)(80,107)(81,110)(82,111)(83,112)(84,109)(85,116)(86,113)(87,114)(88,115)(89,118)(90,119)(91,120)(92,117)(93,124)(94,121)(95,122)(96,123)(97,126)(98,127)(99,128)(100,125), (1,76)(2,73)(3,74)(4,75)(5,37)(6,38)(7,39)(8,40)(9,102)(10,103)(11,104)(12,101)(13,106)(14,107)(15,108)(16,105)(17,112)(18,109)(19,110)(20,111)(21,114)(22,115)(23,116)(24,113)(25,120)(26,117)(27,118)(28,119)(29,122)(30,123)(31,124)(32,121)(33,128)(34,125)(35,126)(36,127)(41,72)(42,69)(43,70)(44,71)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100), (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), (1,119,103,60)(2,57,104,120)(3,117,101,58)(4,59,102,118)(5,114,69,53)(6,54,70,115)(7,116,71,55)(8,56,72,113)(9,27,75,91)(10,92,76,28)(11,25,73,89)(12,90,74,26)(13,93,77,29)(14,30,78,94)(15,95,79,31)(16,32,80,96)(17,33,81,97)(18,98,82,34)(19,35,83,99)(20,100,84,36)(21,42,85,37)(22,38,86,43)(23,44,87,39)(24,40,88,41)(45,122,106,61)(46,62,107,123)(47,124,108,63)(48,64,105,121)(49,65,112,128)(50,125,109,66)(51,67,110,126)(52,127,111,68), (1,18,12,52)(2,49,9,19)(3,20,10,50)(4,51,11,17)(5,29,44,63)(6,64,41,30)(7,31,42,61)(8,62,43,32)(13,55,47,21)(14,22,48,56)(15,53,45,23)(16,24,46,54)(25,128,59,99)(26,100,60,125)(27,126,57,97)(28,98,58,127)(33,91,67,120)(34,117,68,92)(35,89,65,118)(36,119,66,90)(37,122,71,95)(38,96,72,123)(39,124,69,93)(40,94,70,121)(73,81,102,110)(74,111,103,82)(75,83,104,112)(76,109,101,84)(77,116,108,85)(78,86,105,113)(79,114,106,87)(80,88,107,115), (1,77,74,47)(2,48,75,78)(3,79,76,45)(4,46,73,80)(5,66,39,100)(6,97,40,67)(7,68,37,98)(8,99,38,65)(9,14,104,105)(10,106,101,15)(11,16,102,107)(12,108,103,13)(17,24,110,115)(18,116,111,21)(19,22,112,113)(20,114,109,23)(25,96,118,62)(26,63,119,93)(27,94,120,64)(28,61,117,95)(29,90,124,60)(30,57,121,91)(31,92,122,58)(32,59,123,89)(33,41,126,70)(34,71,127,42)(35,43,128,72)(36,69,125,44)(49,56,83,86)(50,87,84,53)(51,54,81,88)(52,85,82,55) );
G=PermutationGroup([[(1,103),(2,104),(3,101),(4,102),(5,69),(6,70),(7,71),(8,72),(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),(37,42),(38,43),(39,44),(40,41),(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),(65,128),(66,125),(67,126),(68,127)], [(1,12),(2,9),(3,10),(4,11),(5,44),(6,41),(7,42),(8,43),(13,47),(14,48),(15,45),(16,46),(17,51),(18,52),(19,49),(20,50),(21,55),(22,56),(23,53),(24,54),(25,59),(26,60),(27,57),(28,58),(29,63),(30,64),(31,61),(32,62),(33,67),(34,68),(35,65),(36,66),(37,71),(38,72),(39,69),(40,70),(73,102),(74,103),(75,104),(76,101),(77,108),(78,105),(79,106),(80,107),(81,110),(82,111),(83,112),(84,109),(85,116),(86,113),(87,114),(88,115),(89,118),(90,119),(91,120),(92,117),(93,124),(94,121),(95,122),(96,123),(97,126),(98,127),(99,128),(100,125)], [(1,76),(2,73),(3,74),(4,75),(5,37),(6,38),(7,39),(8,40),(9,102),(10,103),(11,104),(12,101),(13,106),(14,107),(15,108),(16,105),(17,112),(18,109),(19,110),(20,111),(21,114),(22,115),(23,116),(24,113),(25,120),(26,117),(27,118),(28,119),(29,122),(30,123),(31,124),(32,121),(33,128),(34,125),(35,126),(36,127),(41,72),(42,69),(43,70),(44,71),(45,77),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,85),(54,86),(55,87),(56,88),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,99),(68,100)], [(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)], [(1,119,103,60),(2,57,104,120),(3,117,101,58),(4,59,102,118),(5,114,69,53),(6,54,70,115),(7,116,71,55),(8,56,72,113),(9,27,75,91),(10,92,76,28),(11,25,73,89),(12,90,74,26),(13,93,77,29),(14,30,78,94),(15,95,79,31),(16,32,80,96),(17,33,81,97),(18,98,82,34),(19,35,83,99),(20,100,84,36),(21,42,85,37),(22,38,86,43),(23,44,87,39),(24,40,88,41),(45,122,106,61),(46,62,107,123),(47,124,108,63),(48,64,105,121),(49,65,112,128),(50,125,109,66),(51,67,110,126),(52,127,111,68)], [(1,18,12,52),(2,49,9,19),(3,20,10,50),(4,51,11,17),(5,29,44,63),(6,64,41,30),(7,31,42,61),(8,62,43,32),(13,55,47,21),(14,22,48,56),(15,53,45,23),(16,24,46,54),(25,128,59,99),(26,100,60,125),(27,126,57,97),(28,98,58,127),(33,91,67,120),(34,117,68,92),(35,89,65,118),(36,119,66,90),(37,122,71,95),(38,96,72,123),(39,124,69,93),(40,94,70,121),(73,81,102,110),(74,111,103,82),(75,83,104,112),(76,109,101,84),(77,116,108,85),(78,86,105,113),(79,114,106,87),(80,88,107,115)], [(1,77,74,47),(2,48,75,78),(3,79,76,45),(4,46,73,80),(5,66,39,100),(6,97,40,67),(7,68,37,98),(8,99,38,65),(9,14,104,105),(10,106,101,15),(11,16,102,107),(12,108,103,13),(17,24,110,115),(18,116,111,21),(19,22,112,113),(20,114,109,23),(25,96,118,62),(26,63,119,93),(27,94,120,64),(28,61,117,95),(29,90,124,60),(30,57,121,91),(31,92,122,58),(32,59,123,89),(33,41,126,70),(34,71,127,42),(35,43,128,72),(36,69,125,44),(49,56,83,86),(50,87,84,53),(51,54,81,88),(52,85,82,55)]])
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 | C23.692C24 | C42⋊9C4 | C23.63C23 | C23.65C23 | C23.78C23 | C23.81C23 | C4⋊C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 4 | 4 | 3 | 3 | 8 | 4 | 3 | 1 |
Matrix representation of C23.692C24 ►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 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 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],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C23.692C24 in GAP, Magma, Sage, TeX
C_2^3._{692}C_2^4 % in TeX
G:=Group("C2^3.692C2^4"); // GroupNames label
G:=SmallGroup(128,1524);
// by ID
G=gap.SmallGroup(128,1524);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,120,758,723,184,1571,346,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=a*b*c,e^2=a,f^2=b,g^2=b*a=a*b,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