direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C23.83C23, C24.651C23, C23.294C24, (C22×C4).56Q8, (C22×C4).369D4, C23.835(C2×D4), C23.146(C2×Q8), (C23×C4).67C22, C23.372(C4○D4), C22.59(C22×Q8), (C22×C4).497C23, C22.177(C22×D4), C22.95(C22⋊Q8), C22.79(C4.4D4), C22.28(C42.C2), C22.34(C42⋊2C2), C2.C42.484C22, C22.105(C22.D4), (C2×C4).295(C2×D4), (C2×C4).121(C2×Q8), C2.9(C2×C4.4D4), C2.12(C2×C22⋊Q8), C2.6(C2×C42.C2), (C22×C4⋊C4).32C2, C2.7(C2×C42⋊2C2), (C2×C4⋊C4).837C22, C22.174(C2×C4○D4), C2.11(C2×C22.D4), (C2×C2.C42).24C2, SmallGroup(128,1126)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C23.83C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=b, f2=cb=bc, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd >
Subgroups: 516 in 302 conjugacy classes, 148 normal (16 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C4⋊C4, C22×C4, C22×C4, C24, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C23×C4, C23×C4, C2×C2.C42, C2×C2.C42, C23.83C23, C22×C4⋊C4, C2×C23.83C23
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22×D4, C22×Q8, C2×C4○D4, C23.83C23, C2×C22⋊Q8, C2×C22.D4, C2×C4.4D4, C2×C42.C2, C2×C42⋊2C2, C2×C23.83C23
►Regular action on 128 points
Generators in S128
(1 33)(2 34)(3 35)(4 36)(5 46)(6 47)(7 48)(8 45)(9 103)(10 104)(11 101)(12 102)(13 107)(14 108)(15 105)(16 106)(17 119)(18 120)(19 117)(20 118)(21 123)(22 124)(23 121)(24 122)(25 111)(26 112)(27 109)(28 110)(29 115)(30 116)(31 113)(32 114)(37 65)(38 66)(39 67)(40 68)(41 127)(42 128)(43 125)(44 126)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 69)(58 70)(59 71)(60 72)(61 81)(62 82)(63 83)(64 84)(73 96)(74 93)(75 94)(76 95)(77 100)(78 97)(79 98)(80 99)
(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 9)(2 10)(3 11)(4 12)(5 79)(6 80)(7 77)(8 78)(13 65)(14 66)(15 67)(16 68)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 69)(26 70)(27 71)(28 72)(29 81)(30 82)(31 83)(32 84)(33 103)(34 104)(35 101)(36 102)(37 107)(38 108)(39 105)(40 106)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 119)(50 120)(51 117)(52 118)(53 123)(54 124)(55 121)(56 122)(57 111)(58 112)(59 109)(60 110)(61 115)(62 116)(63 113)(64 114)(73 126)(74 127)(75 128)(76 125)
(1 105)(2 106)(3 107)(4 108)(5 96)(6 93)(7 94)(8 95)(9 39)(10 40)(11 37)(12 38)(13 35)(14 36)(15 33)(16 34)(17 55)(18 56)(19 53)(20 54)(21 51)(22 52)(23 49)(24 50)(25 63)(26 64)(27 61)(28 62)(29 59)(30 60)(31 57)(32 58)(41 80)(42 77)(43 78)(44 79)(45 76)(46 73)(47 74)(48 75)(65 101)(66 102)(67 103)(68 104)(69 113)(70 114)(71 115)(72 116)(81 109)(82 110)(83 111)(84 112)(85 121)(86 122)(87 123)(88 124)(89 117)(90 118)(91 119)(92 120)(97 125)(98 126)(99 127)(100 128)
(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 23 11 89)(2 22 12 92)(3 21 9 91)(4 24 10 90)(5 110 77 58)(6 109 78 57)(7 112 79 60)(8 111 80 59)(13 87 67 17)(14 86 68 20)(15 85 65 19)(16 88 66 18)(25 99 71 45)(26 98 72 48)(27 97 69 47)(28 100 70 46)(29 95 83 41)(30 94 84 44)(31 93 81 43)(32 96 82 42)(33 121 101 53)(34 124 102 56)(35 123 103 55)(36 122 104 54)(37 117 105 49)(38 120 106 52)(39 119 107 51)(40 118 108 50)(61 125 113 74)(62 128 114 73)(63 127 115 76)(64 126 116 75)
(1 83 3 81)(2 32 4 30)(5 24 7 22)(6 89 8 91)(9 31 11 29)(10 84 12 82)(13 27 15 25)(14 72 16 70)(17 74 19 76)(18 128 20 126)(21 78 23 80)(26 66 28 68)(33 63 35 61)(34 114 36 116)(37 59 39 57)(38 110 40 112)(41 51 43 49)(42 118 44 120)(45 55 47 53)(46 122 48 124)(50 94 52 96)(54 98 56 100)(58 108 60 106)(62 104 64 102)(65 71 67 69)(73 86 75 88)(77 90 79 92)(85 127 87 125)(93 117 95 119)(97 121 99 123)(101 115 103 113)(105 111 107 109)
G:=sub<Sym(128)| (1,33)(2,34)(3,35)(4,36)(5,46)(6,47)(7,48)(8,45)(9,103)(10,104)(11,101)(12,102)(13,107)(14,108)(15,105)(16,106)(17,119)(18,120)(19,117)(20,118)(21,123)(22,124)(23,121)(24,122)(25,111)(26,112)(27,109)(28,110)(29,115)(30,116)(31,113)(32,114)(37,65)(38,66)(39,67)(40,68)(41,127)(42,128)(43,125)(44,126)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,69)(58,70)(59,71)(60,72)(61,81)(62,82)(63,83)(64,84)(73,96)(74,93)(75,94)(76,95)(77,100)(78,97)(79,98)(80,99), (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), (1,9)(2,10)(3,11)(4,12)(5,79)(6,80)(7,77)(8,78)(13,65)(14,66)(15,67)(16,68)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,69)(26,70)(27,71)(28,72)(29,81)(30,82)(31,83)(32,84)(33,103)(34,104)(35,101)(36,102)(37,107)(38,108)(39,105)(40,106)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,119)(50,120)(51,117)(52,118)(53,123)(54,124)(55,121)(56,122)(57,111)(58,112)(59,109)(60,110)(61,115)(62,116)(63,113)(64,114)(73,126)(74,127)(75,128)(76,125), (1,105)(2,106)(3,107)(4,108)(5,96)(6,93)(7,94)(8,95)(9,39)(10,40)(11,37)(12,38)(13,35)(14,36)(15,33)(16,34)(17,55)(18,56)(19,53)(20,54)(21,51)(22,52)(23,49)(24,50)(25,63)(26,64)(27,61)(28,62)(29,59)(30,60)(31,57)(32,58)(41,80)(42,77)(43,78)(44,79)(45,76)(46,73)(47,74)(48,75)(65,101)(66,102)(67,103)(68,104)(69,113)(70,114)(71,115)(72,116)(81,109)(82,110)(83,111)(84,112)(85,121)(86,122)(87,123)(88,124)(89,117)(90,118)(91,119)(92,120)(97,125)(98,126)(99,127)(100,128), (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), (1,23,11,89)(2,22,12,92)(3,21,9,91)(4,24,10,90)(5,110,77,58)(6,109,78,57)(7,112,79,60)(8,111,80,59)(13,87,67,17)(14,86,68,20)(15,85,65,19)(16,88,66,18)(25,99,71,45)(26,98,72,48)(27,97,69,47)(28,100,70,46)(29,95,83,41)(30,94,84,44)(31,93,81,43)(32,96,82,42)(33,121,101,53)(34,124,102,56)(35,123,103,55)(36,122,104,54)(37,117,105,49)(38,120,106,52)(39,119,107,51)(40,118,108,50)(61,125,113,74)(62,128,114,73)(63,127,115,76)(64,126,116,75), (1,83,3,81)(2,32,4,30)(5,24,7,22)(6,89,8,91)(9,31,11,29)(10,84,12,82)(13,27,15,25)(14,72,16,70)(17,74,19,76)(18,128,20,126)(21,78,23,80)(26,66,28,68)(33,63,35,61)(34,114,36,116)(37,59,39,57)(38,110,40,112)(41,51,43,49)(42,118,44,120)(45,55,47,53)(46,122,48,124)(50,94,52,96)(54,98,56,100)(58,108,60,106)(62,104,64,102)(65,71,67,69)(73,86,75,88)(77,90,79,92)(85,127,87,125)(93,117,95,119)(97,121,99,123)(101,115,103,113)(105,111,107,109)>;
G:=Group( (1,33)(2,34)(3,35)(4,36)(5,46)(6,47)(7,48)(8,45)(9,103)(10,104)(11,101)(12,102)(13,107)(14,108)(15,105)(16,106)(17,119)(18,120)(19,117)(20,118)(21,123)(22,124)(23,121)(24,122)(25,111)(26,112)(27,109)(28,110)(29,115)(30,116)(31,113)(32,114)(37,65)(38,66)(39,67)(40,68)(41,127)(42,128)(43,125)(44,126)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,69)(58,70)(59,71)(60,72)(61,81)(62,82)(63,83)(64,84)(73,96)(74,93)(75,94)(76,95)(77,100)(78,97)(79,98)(80,99), (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), (1,9)(2,10)(3,11)(4,12)(5,79)(6,80)(7,77)(8,78)(13,65)(14,66)(15,67)(16,68)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,69)(26,70)(27,71)(28,72)(29,81)(30,82)(31,83)(32,84)(33,103)(34,104)(35,101)(36,102)(37,107)(38,108)(39,105)(40,106)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,119)(50,120)(51,117)(52,118)(53,123)(54,124)(55,121)(56,122)(57,111)(58,112)(59,109)(60,110)(61,115)(62,116)(63,113)(64,114)(73,126)(74,127)(75,128)(76,125), (1,105)(2,106)(3,107)(4,108)(5,96)(6,93)(7,94)(8,95)(9,39)(10,40)(11,37)(12,38)(13,35)(14,36)(15,33)(16,34)(17,55)(18,56)(19,53)(20,54)(21,51)(22,52)(23,49)(24,50)(25,63)(26,64)(27,61)(28,62)(29,59)(30,60)(31,57)(32,58)(41,80)(42,77)(43,78)(44,79)(45,76)(46,73)(47,74)(48,75)(65,101)(66,102)(67,103)(68,104)(69,113)(70,114)(71,115)(72,116)(81,109)(82,110)(83,111)(84,112)(85,121)(86,122)(87,123)(88,124)(89,117)(90,118)(91,119)(92,120)(97,125)(98,126)(99,127)(100,128), (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), (1,23,11,89)(2,22,12,92)(3,21,9,91)(4,24,10,90)(5,110,77,58)(6,109,78,57)(7,112,79,60)(8,111,80,59)(13,87,67,17)(14,86,68,20)(15,85,65,19)(16,88,66,18)(25,99,71,45)(26,98,72,48)(27,97,69,47)(28,100,70,46)(29,95,83,41)(30,94,84,44)(31,93,81,43)(32,96,82,42)(33,121,101,53)(34,124,102,56)(35,123,103,55)(36,122,104,54)(37,117,105,49)(38,120,106,52)(39,119,107,51)(40,118,108,50)(61,125,113,74)(62,128,114,73)(63,127,115,76)(64,126,116,75), (1,83,3,81)(2,32,4,30)(5,24,7,22)(6,89,8,91)(9,31,11,29)(10,84,12,82)(13,27,15,25)(14,72,16,70)(17,74,19,76)(18,128,20,126)(21,78,23,80)(26,66,28,68)(33,63,35,61)(34,114,36,116)(37,59,39,57)(38,110,40,112)(41,51,43,49)(42,118,44,120)(45,55,47,53)(46,122,48,124)(50,94,52,96)(54,98,56,100)(58,108,60,106)(62,104,64,102)(65,71,67,69)(73,86,75,88)(77,90,79,92)(85,127,87,125)(93,117,95,119)(97,121,99,123)(101,115,103,113)(105,111,107,109) );
G=PermutationGroup([[(1,33),(2,34),(3,35),(4,36),(5,46),(6,47),(7,48),(8,45),(9,103),(10,104),(11,101),(12,102),(13,107),(14,108),(15,105),(16,106),(17,119),(18,120),(19,117),(20,118),(21,123),(22,124),(23,121),(24,122),(25,111),(26,112),(27,109),(28,110),(29,115),(30,116),(31,113),(32,114),(37,65),(38,66),(39,67),(40,68),(41,127),(42,128),(43,125),(44,126),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,91),(56,92),(57,69),(58,70),(59,71),(60,72),(61,81),(62,82),(63,83),(64,84),(73,96),(74,93),(75,94),(76,95),(77,100),(78,97),(79,98),(80,99)], [(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)], [(1,9),(2,10),(3,11),(4,12),(5,79),(6,80),(7,77),(8,78),(13,65),(14,66),(15,67),(16,68),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,69),(26,70),(27,71),(28,72),(29,81),(30,82),(31,83),(32,84),(33,103),(34,104),(35,101),(36,102),(37,107),(38,108),(39,105),(40,106),(41,93),(42,94),(43,95),(44,96),(45,97),(46,98),(47,99),(48,100),(49,119),(50,120),(51,117),(52,118),(53,123),(54,124),(55,121),(56,122),(57,111),(58,112),(59,109),(60,110),(61,115),(62,116),(63,113),(64,114),(73,126),(74,127),(75,128),(76,125)], [(1,105),(2,106),(3,107),(4,108),(5,96),(6,93),(7,94),(8,95),(9,39),(10,40),(11,37),(12,38),(13,35),(14,36),(15,33),(16,34),(17,55),(18,56),(19,53),(20,54),(21,51),(22,52),(23,49),(24,50),(25,63),(26,64),(27,61),(28,62),(29,59),(30,60),(31,57),(32,58),(41,80),(42,77),(43,78),(44,79),(45,76),(46,73),(47,74),(48,75),(65,101),(66,102),(67,103),(68,104),(69,113),(70,114),(71,115),(72,116),(81,109),(82,110),(83,111),(84,112),(85,121),(86,122),(87,123),(88,124),(89,117),(90,118),(91,119),(92,120),(97,125),(98,126),(99,127),(100,128)], [(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)], [(1,23,11,89),(2,22,12,92),(3,21,9,91),(4,24,10,90),(5,110,77,58),(6,109,78,57),(7,112,79,60),(8,111,80,59),(13,87,67,17),(14,86,68,20),(15,85,65,19),(16,88,66,18),(25,99,71,45),(26,98,72,48),(27,97,69,47),(28,100,70,46),(29,95,83,41),(30,94,84,44),(31,93,81,43),(32,96,82,42),(33,121,101,53),(34,124,102,56),(35,123,103,55),(36,122,104,54),(37,117,105,49),(38,120,106,52),(39,119,107,51),(40,118,108,50),(61,125,113,74),(62,128,114,73),(63,127,115,76),(64,126,116,75)], [(1,83,3,81),(2,32,4,30),(5,24,7,22),(6,89,8,91),(9,31,11,29),(10,84,12,82),(13,27,15,25),(14,72,16,70),(17,74,19,76),(18,128,20,126),(21,78,23,80),(26,66,28,68),(33,63,35,61),(34,114,36,116),(37,59,39,57),(38,110,40,112),(41,51,43,49),(42,118,44,120),(45,55,47,53),(46,122,48,124),(50,94,52,96),(54,98,56,100),(58,108,60,106),(62,104,64,102),(65,71,67,69),(73,86,75,88),(77,90,79,92),(85,127,87,125),(93,117,95,119),(97,121,99,123),(101,115,103,113),(105,111,107,109)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4AB |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | D4 | Q8 | C4○D4 |
| kernel | C2×C23.83C23 | C2×C2.C42 | C23.83C23 | C22×C4⋊C4 | C22×C4 | C22×C4 | C23 |
| # reps | 1 | 5 | 8 | 2 | 4 | 4 | 20 |
Matrix representation of C2×C23.83C23 ►in GL8(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2],[0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0] >;
C2×C23.83C23 in GAP, Magma, Sage, TeX
C_2\times C_2^3._{83}C_2^3 % in TeX
G:=Group("C2xC2^3.83C2^3"); // GroupNames label
G:=SmallGroup(128,1126);
// by ID
G=gap.SmallGroup(128,1126);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,100]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=b,f^2=c*b=b*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d>;
// generators/relations