p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.351C24, C22.1592+ 1+4, C22.1172- 1+4, C2.3Q82, C4⋊C4⋊13Q8, (C2×Q8)⋊8Q8, C2.14(D4×Q8), C4⋊C4.325D4, C2.7(Q8⋊3Q8), C4.53(C22⋊Q8), C2.40(D4⋊5D4), C22.77(C22×Q8), (C2×C42).494C22, (C22×C4).806C23, C22.231(C22×D4), (C22×Q8).426C22, C23.78C23.3C2, C23.81C23.4C2, C2.C42.108C22, C23.63C23.14C2, C23.67C23.33C2, C23.65C23.40C2, C2.16(C23.37C23), C2.18(C22.36C24), (C4×C4⋊C4).49C2, (C2×C4×Q8).30C2, (C2×C4⋊Q8).30C2, (C2×C4).32(C2×Q8), (C2×C4).331(C2×D4), C2.24(C2×C22⋊Q8), (C2×C4).106(C4○D4), (C2×C4⋊C4).233C22, C22.228(C2×C4○D4), SmallGroup(128,1183)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.351C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=f2=b, g2=a, ab=ba, 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: 404 in 242 conjugacy classes, 120 normal (42 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4⋊Q8, C22×Q8, C22×Q8, C4×C4⋊C4, C23.63C23, C23.65C23, C23.67C23, C23.67C23, C23.78C23, C23.81C23, C2×C4×Q8, C2×C4⋊Q8, C23.351C24
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C22⋊Q8, C23.37C23, C22.36C24, D4⋊5D4, D4×Q8, Q8⋊3Q8, Q82, C23.351C24
►Regular action on 128 points
Generators in S128
(1 71)(2 72)(3 69)(4 70)(5 33)(6 34)(7 35)(8 36)(9 73)(10 74)(11 75)(12 76)(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)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 126)(66 127)(67 128)(68 125)
(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 99)(2 100)(3 97)(4 98)(5 68)(6 65)(7 66)(8 67)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 69)(38 70)(39 71)(40 72)(41 73)(42 74)(43 75)(44 76)(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)
(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 21 3 23)(2 86 4 88)(5 81 7 83)(6 18 8 20)(9 27 11 25)(10 92 12 90)(13 31 15 29)(14 96 16 94)(17 35 19 33)(22 70 24 72)(26 74 28 76)(30 78 32 80)(34 82 36 84)(37 55 39 53)(38 116 40 114)(41 59 43 57)(42 120 44 118)(45 63 47 61)(46 124 48 122)(49 66 51 68)(50 128 52 126)(54 98 56 100)(58 102 60 104)(62 106 64 108)(65 110 67 112)(69 87 71 85)(73 91 75 89)(77 95 79 93)(97 115 99 113)(101 119 103 117)(105 123 107 121)(109 127 111 125)
(1 13 3 15)(2 16 4 14)(5 57 7 59)(6 60 8 58)(9 19 11 17)(10 18 12 20)(21 123 23 121)(22 122 24 124)(25 127 27 125)(26 126 28 128)(29 113 31 115)(30 116 32 114)(33 117 35 119)(34 120 36 118)(37 47 39 45)(38 46 40 48)(41 51 43 49)(42 50 44 52)(53 95 55 93)(54 94 56 96)(61 85 63 87)(62 88 64 86)(65 92 67 90)(66 91 68 89)(69 79 71 77)(70 78 72 80)(73 83 75 81)(74 82 76 84)(97 107 99 105)(98 106 100 108)(101 111 103 109)(102 110 104 112)
(1 75 71 11)(2 12 72 76)(3 73 69 9)(4 10 70 74)(5 93 33 29)(6 30 34 94)(7 95 35 31)(8 32 36 96)(13 81 77 17)(14 18 78 82)(15 83 79 19)(16 20 80 84)(21 89 85 25)(22 26 86 90)(23 91 87 27)(24 28 88 92)(37 101 97 41)(38 42 98 102)(39 103 99 43)(40 44 100 104)(45 109 105 49)(46 50 106 110)(47 111 107 51)(48 52 108 112)(53 117 113 57)(54 58 114 118)(55 119 115 59)(56 60 116 120)(61 125 121 68)(62 65 122 126)(63 127 123 66)(64 67 124 128)
G:=sub<Sym(128)| (1,71)(2,72)(3,69)(4,70)(5,33)(6,34)(7,35)(8,36)(9,73)(10,74)(11,75)(12,76)(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)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,126)(66,127)(67,128)(68,125), (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), (1,99)(2,100)(3,97)(4,98)(5,68)(6,65)(7,66)(8,67)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,69)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,76)(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), (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), (1,21,3,23)(2,86,4,88)(5,81,7,83)(6,18,8,20)(9,27,11,25)(10,92,12,90)(13,31,15,29)(14,96,16,94)(17,35,19,33)(22,70,24,72)(26,74,28,76)(30,78,32,80)(34,82,36,84)(37,55,39,53)(38,116,40,114)(41,59,43,57)(42,120,44,118)(45,63,47,61)(46,124,48,122)(49,66,51,68)(50,128,52,126)(54,98,56,100)(58,102,60,104)(62,106,64,108)(65,110,67,112)(69,87,71,85)(73,91,75,89)(77,95,79,93)(97,115,99,113)(101,119,103,117)(105,123,107,121)(109,127,111,125), (1,13,3,15)(2,16,4,14)(5,57,7,59)(6,60,8,58)(9,19,11,17)(10,18,12,20)(21,123,23,121)(22,122,24,124)(25,127,27,125)(26,126,28,128)(29,113,31,115)(30,116,32,114)(33,117,35,119)(34,120,36,118)(37,47,39,45)(38,46,40,48)(41,51,43,49)(42,50,44,52)(53,95,55,93)(54,94,56,96)(61,85,63,87)(62,88,64,86)(65,92,67,90)(66,91,68,89)(69,79,71,77)(70,78,72,80)(73,83,75,81)(74,82,76,84)(97,107,99,105)(98,106,100,108)(101,111,103,109)(102,110,104,112), (1,75,71,11)(2,12,72,76)(3,73,69,9)(4,10,70,74)(5,93,33,29)(6,30,34,94)(7,95,35,31)(8,32,36,96)(13,81,77,17)(14,18,78,82)(15,83,79,19)(16,20,80,84)(21,89,85,25)(22,26,86,90)(23,91,87,27)(24,28,88,92)(37,101,97,41)(38,42,98,102)(39,103,99,43)(40,44,100,104)(45,109,105,49)(46,50,106,110)(47,111,107,51)(48,52,108,112)(53,117,113,57)(54,58,114,118)(55,119,115,59)(56,60,116,120)(61,125,121,68)(62,65,122,126)(63,127,123,66)(64,67,124,128)>;
G:=Group( (1,71)(2,72)(3,69)(4,70)(5,33)(6,34)(7,35)(8,36)(9,73)(10,74)(11,75)(12,76)(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)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,126)(66,127)(67,128)(68,125), (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), (1,99)(2,100)(3,97)(4,98)(5,68)(6,65)(7,66)(8,67)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,69)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,76)(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), (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), (1,21,3,23)(2,86,4,88)(5,81,7,83)(6,18,8,20)(9,27,11,25)(10,92,12,90)(13,31,15,29)(14,96,16,94)(17,35,19,33)(22,70,24,72)(26,74,28,76)(30,78,32,80)(34,82,36,84)(37,55,39,53)(38,116,40,114)(41,59,43,57)(42,120,44,118)(45,63,47,61)(46,124,48,122)(49,66,51,68)(50,128,52,126)(54,98,56,100)(58,102,60,104)(62,106,64,108)(65,110,67,112)(69,87,71,85)(73,91,75,89)(77,95,79,93)(97,115,99,113)(101,119,103,117)(105,123,107,121)(109,127,111,125), (1,13,3,15)(2,16,4,14)(5,57,7,59)(6,60,8,58)(9,19,11,17)(10,18,12,20)(21,123,23,121)(22,122,24,124)(25,127,27,125)(26,126,28,128)(29,113,31,115)(30,116,32,114)(33,117,35,119)(34,120,36,118)(37,47,39,45)(38,46,40,48)(41,51,43,49)(42,50,44,52)(53,95,55,93)(54,94,56,96)(61,85,63,87)(62,88,64,86)(65,92,67,90)(66,91,68,89)(69,79,71,77)(70,78,72,80)(73,83,75,81)(74,82,76,84)(97,107,99,105)(98,106,100,108)(101,111,103,109)(102,110,104,112), (1,75,71,11)(2,12,72,76)(3,73,69,9)(4,10,70,74)(5,93,33,29)(6,30,34,94)(7,95,35,31)(8,32,36,96)(13,81,77,17)(14,18,78,82)(15,83,79,19)(16,20,80,84)(21,89,85,25)(22,26,86,90)(23,91,87,27)(24,28,88,92)(37,101,97,41)(38,42,98,102)(39,103,99,43)(40,44,100,104)(45,109,105,49)(46,50,106,110)(47,111,107,51)(48,52,108,112)(53,117,113,57)(54,58,114,118)(55,119,115,59)(56,60,116,120)(61,125,121,68)(62,65,122,126)(63,127,123,66)(64,67,124,128) );
G=PermutationGroup([[(1,71),(2,72),(3,69),(4,70),(5,33),(6,34),(7,35),(8,36),(9,73),(10,74),(11,75),(12,76),(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),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120),(61,121),(62,122),(63,123),(64,124),(65,126),(66,127),(67,128),(68,125)], [(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)], [(1,99),(2,100),(3,97),(4,98),(5,68),(6,65),(7,66),(8,67),(9,101),(10,102),(11,103),(12,104),(13,105),(14,106),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,113),(22,114),(23,115),(24,116),(25,117),(26,118),(27,119),(28,120),(29,121),(30,122),(31,123),(32,124),(33,125),(34,126),(35,127),(36,128),(37,69),(38,70),(39,71),(40,72),(41,73),(42,74),(43,75),(44,76),(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)], [(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)], [(1,21,3,23),(2,86,4,88),(5,81,7,83),(6,18,8,20),(9,27,11,25),(10,92,12,90),(13,31,15,29),(14,96,16,94),(17,35,19,33),(22,70,24,72),(26,74,28,76),(30,78,32,80),(34,82,36,84),(37,55,39,53),(38,116,40,114),(41,59,43,57),(42,120,44,118),(45,63,47,61),(46,124,48,122),(49,66,51,68),(50,128,52,126),(54,98,56,100),(58,102,60,104),(62,106,64,108),(65,110,67,112),(69,87,71,85),(73,91,75,89),(77,95,79,93),(97,115,99,113),(101,119,103,117),(105,123,107,121),(109,127,111,125)], [(1,13,3,15),(2,16,4,14),(5,57,7,59),(6,60,8,58),(9,19,11,17),(10,18,12,20),(21,123,23,121),(22,122,24,124),(25,127,27,125),(26,126,28,128),(29,113,31,115),(30,116,32,114),(33,117,35,119),(34,120,36,118),(37,47,39,45),(38,46,40,48),(41,51,43,49),(42,50,44,52),(53,95,55,93),(54,94,56,96),(61,85,63,87),(62,88,64,86),(65,92,67,90),(66,91,68,89),(69,79,71,77),(70,78,72,80),(73,83,75,81),(74,82,76,84),(97,107,99,105),(98,106,100,108),(101,111,103,109),(102,110,104,112)], [(1,75,71,11),(2,12,72,76),(3,73,69,9),(4,10,70,74),(5,93,33,29),(6,30,34,94),(7,95,35,31),(8,32,36,96),(13,81,77,17),(14,18,78,82),(15,83,79,19),(16,20,80,84),(21,89,85,25),(22,26,86,90),(23,91,87,27),(24,28,88,92),(37,101,97,41),(38,42,98,102),(39,103,99,43),(40,44,100,104),(45,109,105,49),(46,50,106,110),(47,111,107,51),(48,52,108,112),(53,117,113,57),(54,58,114,118),(55,119,115,59),(56,60,116,120),(61,125,121,68),(62,65,122,126),(63,127,123,66),(64,67,124,128)]])
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 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | Q8 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C23.351C24 | C4×C4⋊C4 | C23.63C23 | C23.65C23 | C23.67C23 | C23.78C23 | C23.81C23 | C2×C4×Q8 | C2×C4⋊Q8 | C4⋊C4 | C4⋊C4 | C2×Q8 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 4 | 4 | 4 | 8 | 1 | 1 |
Matrix representation of C23.351C24 ►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 |
| 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 |
| 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 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 | 3 | 2 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 1 | 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 | 4 | 2 |
| 0 | 0 | 0 | 0 | 4 | 1 |
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],[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],[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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,3,3,0,0,0,0,0,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,4,0,0,0,0,2,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,1,1,0,0,0,0,3,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,4,0,0,0,0,2,1] >;
C23.351C24 in GAP, Magma, Sage, TeX
C_2^3._{351}C_2^4 % in TeX
G:=Group("C2^3.351C2^4"); // GroupNames label
G:=SmallGroup(128,1183);
// by ID
G=gap.SmallGroup(128,1183);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,232,758,723,184,675,80]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=f^2=b,g^2=a,a*b=b*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