p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.202C24, C22.262- (1+4), C22.412+ (1+4), (C4×Q8)⋊18C4, C42⋊5C4.3C2, C42.179(C2×C4), C42⋊4C4.12C2, C42⋊8C4.19C2, C22.93(C23×C4), C4.15(C42⋊C2), (C22×C4).467C23, (C2×C42).410C22, (C22×Q8).397C22, C23.63C23.1C2, C2.C42.39C22, C23.67C23.24C2, C23.65C23.29C2, C2.3(C22.36C24), C2.7(C23.32C23), C2.2(C22.35C24), C2.10(C23.33C23), (C4×C4⋊C4).34C2, (C2×C4×Q8).21C2, C4⋊C4.205(C2×C4), (C2×Q8).193(C2×C4), C22.87(C2×C4○D4), (C2×C4).645(C4○D4), (C2×C4⋊C4).176C22, (C2×C4).225(C22×C4), C2.24(C2×C42⋊C2), SmallGroup(128,1052)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 332 in 222 conjugacy classes, 140 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×20], C22 [×3], C22 [×4], C2×C4 [×18], C2×C4 [×36], Q8 [×8], C23, C42 [×12], C42 [×6], C4⋊C4 [×12], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×4], C2×Q8 [×4], C2.C42 [×2], C2.C42 [×16], C2×C42, C2×C42 [×6], C2×C4⋊C4, C2×C4⋊C4 [×8], C4×Q8 [×8], C22×Q8, C42⋊4C4, C4×C4⋊C4, C42⋊8C4 [×2], C42⋊5C4 [×2], C23.63C23 [×4], C23.65C23 [×2], C23.67C23 [×2], C2×C4×Q8, C23.202C24
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], 2+ (1+4), 2- (1+4) [×3], C2×C42⋊C2, C23.32C23, C23.33C23, C22.35C24 [×2], C22.36C24 [×2], C23.202C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=f2=a, g2=b, ab=ba, ac=ca, ede-1=gdg-1=ad=da, fef-1=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >
(1 11)(2 12)(3 9)(4 10)(5 70)(6 71)(7 72)(8 69)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 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)(65 127)(66 128)(67 125)(68 126)
(1 99)(2 100)(3 97)(4 98)(5 68)(6 65)(7 66)(8 67)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)(69 125)(70 126)(71 127)(72 128)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 119)(92 120)(93 121)(94 122)(95 123)(96 124)
(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 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 27 11 87)(2 88 12 28)(3 25 9 85)(4 86 10 26)(5 82 70 22)(6 23 71 83)(7 84 72 24)(8 21 69 81)(13 89 73 29)(14 30 74 90)(15 91 75 31)(16 32 76 92)(17 93 77 33)(18 34 78 94)(19 95 79 35)(20 36 80 96)(37 113 97 53)(38 54 98 114)(39 115 99 55)(40 56 100 116)(41 117 101 57)(42 58 102 118)(43 119 103 59)(44 60 104 120)(45 121 105 61)(46 62 106 122)(47 123 107 63)(48 64 108 124)(49 125 109 67)(50 68 110 126)(51 127 111 65)(52 66 112 128)
(1 19 11 79)(2 48 12 108)(3 17 9 77)(4 46 10 106)(5 118 70 58)(6 91 71 31)(7 120 72 60)(8 89 69 29)(13 81 73 21)(14 110 74 50)(15 83 75 23)(16 112 76 52)(18 38 78 98)(20 40 80 100)(22 42 82 102)(24 44 84 104)(25 33 85 93)(26 62 86 122)(27 35 87 95)(28 64 88 124)(30 68 90 126)(32 66 92 128)(34 114 94 54)(36 116 96 56)(37 105 97 45)(39 107 99 47)(41 109 101 49)(43 111 103 51)(53 61 113 121)(55 63 115 123)(57 67 117 125)(59 65 119 127)
(1 103 99 75)(2 44 100 16)(3 101 97 73)(4 42 98 14)(5 94 68 122)(6 35 65 63)(7 96 66 124)(8 33 67 61)(9 41 37 13)(10 102 38 74)(11 43 39 15)(12 104 40 76)(17 49 45 21)(18 110 46 82)(19 51 47 23)(20 112 48 84)(22 78 50 106)(24 80 52 108)(25 57 53 29)(26 118 54 90)(27 59 55 31)(28 120 56 92)(30 86 58 114)(32 88 60 116)(34 126 62 70)(36 128 64 72)(69 93 125 121)(71 95 127 123)(77 109 105 81)(79 111 107 83)(85 117 113 89)(87 119 115 91)
G:=sub<Sym(128)| (1,11)(2,12)(3,9)(4,10)(5,70)(6,71)(7,72)(8,69)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(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)(65,127)(66,128)(67,125)(68,126), (1,99)(2,100)(3,97)(4,98)(5,68)(6,65)(7,66)(8,67)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(69,125)(70,126)(71,127)(72,128)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124), (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), (1,2,3,4)(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), (1,27,11,87)(2,88,12,28)(3,25,9,85)(4,86,10,26)(5,82,70,22)(6,23,71,83)(7,84,72,24)(8,21,69,81)(13,89,73,29)(14,30,74,90)(15,91,75,31)(16,32,76,92)(17,93,77,33)(18,34,78,94)(19,95,79,35)(20,36,80,96)(37,113,97,53)(38,54,98,114)(39,115,99,55)(40,56,100,116)(41,117,101,57)(42,58,102,118)(43,119,103,59)(44,60,104,120)(45,121,105,61)(46,62,106,122)(47,123,107,63)(48,64,108,124)(49,125,109,67)(50,68,110,126)(51,127,111,65)(52,66,112,128), (1,19,11,79)(2,48,12,108)(3,17,9,77)(4,46,10,106)(5,118,70,58)(6,91,71,31)(7,120,72,60)(8,89,69,29)(13,81,73,21)(14,110,74,50)(15,83,75,23)(16,112,76,52)(18,38,78,98)(20,40,80,100)(22,42,82,102)(24,44,84,104)(25,33,85,93)(26,62,86,122)(27,35,87,95)(28,64,88,124)(30,68,90,126)(32,66,92,128)(34,114,94,54)(36,116,96,56)(37,105,97,45)(39,107,99,47)(41,109,101,49)(43,111,103,51)(53,61,113,121)(55,63,115,123)(57,67,117,125)(59,65,119,127), (1,103,99,75)(2,44,100,16)(3,101,97,73)(4,42,98,14)(5,94,68,122)(6,35,65,63)(7,96,66,124)(8,33,67,61)(9,41,37,13)(10,102,38,74)(11,43,39,15)(12,104,40,76)(17,49,45,21)(18,110,46,82)(19,51,47,23)(20,112,48,84)(22,78,50,106)(24,80,52,108)(25,57,53,29)(26,118,54,90)(27,59,55,31)(28,120,56,92)(30,86,58,114)(32,88,60,116)(34,126,62,70)(36,128,64,72)(69,93,125,121)(71,95,127,123)(77,109,105,81)(79,111,107,83)(85,117,113,89)(87,119,115,91)>;
G:=Group( (1,11)(2,12)(3,9)(4,10)(5,70)(6,71)(7,72)(8,69)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(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)(65,127)(66,128)(67,125)(68,126), (1,99)(2,100)(3,97)(4,98)(5,68)(6,65)(7,66)(8,67)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(69,125)(70,126)(71,127)(72,128)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124), (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), (1,2,3,4)(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), (1,27,11,87)(2,88,12,28)(3,25,9,85)(4,86,10,26)(5,82,70,22)(6,23,71,83)(7,84,72,24)(8,21,69,81)(13,89,73,29)(14,30,74,90)(15,91,75,31)(16,32,76,92)(17,93,77,33)(18,34,78,94)(19,95,79,35)(20,36,80,96)(37,113,97,53)(38,54,98,114)(39,115,99,55)(40,56,100,116)(41,117,101,57)(42,58,102,118)(43,119,103,59)(44,60,104,120)(45,121,105,61)(46,62,106,122)(47,123,107,63)(48,64,108,124)(49,125,109,67)(50,68,110,126)(51,127,111,65)(52,66,112,128), (1,19,11,79)(2,48,12,108)(3,17,9,77)(4,46,10,106)(5,118,70,58)(6,91,71,31)(7,120,72,60)(8,89,69,29)(13,81,73,21)(14,110,74,50)(15,83,75,23)(16,112,76,52)(18,38,78,98)(20,40,80,100)(22,42,82,102)(24,44,84,104)(25,33,85,93)(26,62,86,122)(27,35,87,95)(28,64,88,124)(30,68,90,126)(32,66,92,128)(34,114,94,54)(36,116,96,56)(37,105,97,45)(39,107,99,47)(41,109,101,49)(43,111,103,51)(53,61,113,121)(55,63,115,123)(57,67,117,125)(59,65,119,127), (1,103,99,75)(2,44,100,16)(3,101,97,73)(4,42,98,14)(5,94,68,122)(6,35,65,63)(7,96,66,124)(8,33,67,61)(9,41,37,13)(10,102,38,74)(11,43,39,15)(12,104,40,76)(17,49,45,21)(18,110,46,82)(19,51,47,23)(20,112,48,84)(22,78,50,106)(24,80,52,108)(25,57,53,29)(26,118,54,90)(27,59,55,31)(28,120,56,92)(30,86,58,114)(32,88,60,116)(34,126,62,70)(36,128,64,72)(69,93,125,121)(71,95,127,123)(77,109,105,81)(79,111,107,83)(85,117,113,89)(87,119,115,91) );
G=PermutationGroup([(1,11),(2,12),(3,9),(4,10),(5,70),(6,71),(7,72),(8,69),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(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),(65,127),(66,128),(67,125),(68,126)], [(1,99),(2,100),(3,97),(4,98),(5,68),(6,65),(7,66),(8,67),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,61),(34,62),(35,63),(36,64),(69,125),(70,126),(71,127),(72,128),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112),(85,113),(86,114),(87,115),(88,116),(89,117),(90,118),(91,119),(92,120),(93,121),(94,122),(95,123),(96,124)], [(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)], [(1,2,3,4),(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)], [(1,27,11,87),(2,88,12,28),(3,25,9,85),(4,86,10,26),(5,82,70,22),(6,23,71,83),(7,84,72,24),(8,21,69,81),(13,89,73,29),(14,30,74,90),(15,91,75,31),(16,32,76,92),(17,93,77,33),(18,34,78,94),(19,95,79,35),(20,36,80,96),(37,113,97,53),(38,54,98,114),(39,115,99,55),(40,56,100,116),(41,117,101,57),(42,58,102,118),(43,119,103,59),(44,60,104,120),(45,121,105,61),(46,62,106,122),(47,123,107,63),(48,64,108,124),(49,125,109,67),(50,68,110,126),(51,127,111,65),(52,66,112,128)], [(1,19,11,79),(2,48,12,108),(3,17,9,77),(4,46,10,106),(5,118,70,58),(6,91,71,31),(7,120,72,60),(8,89,69,29),(13,81,73,21),(14,110,74,50),(15,83,75,23),(16,112,76,52),(18,38,78,98),(20,40,80,100),(22,42,82,102),(24,44,84,104),(25,33,85,93),(26,62,86,122),(27,35,87,95),(28,64,88,124),(30,68,90,126),(32,66,92,128),(34,114,94,54),(36,116,96,56),(37,105,97,45),(39,107,99,47),(41,109,101,49),(43,111,103,51),(53,61,113,121),(55,63,115,123),(57,67,117,125),(59,65,119,127)], [(1,103,99,75),(2,44,100,16),(3,101,97,73),(4,42,98,14),(5,94,68,122),(6,35,65,63),(7,96,66,124),(8,33,67,61),(9,41,37,13),(10,102,38,74),(11,43,39,15),(12,104,40,76),(17,49,45,21),(18,110,46,82),(19,51,47,23),(20,112,48,84),(22,78,50,106),(24,80,52,108),(25,57,53,29),(26,118,54,90),(27,59,55,31),(28,120,56,92),(30,86,58,114),(32,88,60,116),(34,126,62,70),(36,128,64,72),(69,93,125,121),(71,95,127,123),(77,109,105,81),(79,111,107,83),(85,117,113,89),(87,119,115,91)])
Matrix representation ►G ⊆ 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 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 | 3 | 4 |
| 0 | 0 | 0 | 0 | 4 | 2 | 4 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 3 | 2 |
| 0 | 0 | 0 | 0 | 3 | 0 | 3 | 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 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 4 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 3 | 1 | 2 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 4 | 3 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 | 1 | 1 |
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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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],[1,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,1,4,0,3,0,0,0,0,4,2,1,0,0,0,0,0,3,4,3,3,0,0,0,0,4,1,2,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,4,0,0,0,0,0,4,3,0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,2,0],[4,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,2,0,2,0,0,0,0,1,4,0,4,0,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,2,4,0,0,0,0,1,0,2,4,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,1] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C23.202C24 | C42⋊4C4 | C4×C4⋊C4 | C42⋊8C4 | C42⋊5C4 | C23.63C23 | C23.65C23 | C23.67C23 | C2×C4×Q8 | C4×Q8 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 16 | 8 | 1 | 3 |
In GAP, Magma, Sage, TeX
C_2^3._{202}C_2^4 % in TeX
G:=Group("C2^3.202C2^4"); // GroupNames label
G:=SmallGroup(128,1052);
// by ID
G=gap.SmallGroup(128,1052);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,219,100,675,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=f^2=a,g^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,f*e*f^-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,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
// generators/relations