direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C23.67C23, C23.176C24, C24.646C23, (C22×C4)⋊19Q8, (C22×Q8)⋊16C4, (Q8×C23).4C2, C22.31(C4×Q8), (C22×C4).596D4, C23.829(C2×D4), C22.41(C4⋊Q8), C23.143(C2×Q8), C22.67(C23×C4), (C22×C42).23C2, C22.74(C22×D4), C23.363(C4○D4), C22.24(C22×Q8), C23.285(C22×C4), (C23×C4).282C22, (C22×C4).450C23, C22.89(C22⋊Q8), (C2×C42).1089C22, C22.75(C4.4D4), (C22×Q8).389C22, C2.C42.511C22, C2.6(C2×C4×Q8), (C2×C4)⋊7(C2×Q8), C2.3(C2×C4⋊Q8), (C2×Q8)⋊31(C2×C4), C2.5(C2×C22⋊Q8), C4.61(C2×C22⋊C4), C2.4(C2×C4.4D4), (C2×C4).1389(C2×D4), (C22×C4⋊C4).23C2, C22.68(C2×C4○D4), (C2×C4⋊C4).791C22, (C22×C4).295(C2×C4), (C2×C4).209(C22×C4), C2.10(C22×C22⋊C4), (C2×C4).276(C22⋊C4), C22.133(C2×C22⋊C4), (C2×C2.C42).18C2, SmallGroup(128,1026)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 716 in 464 conjugacy classes, 236 normal (14 characteristic)
C1, C2 [×3], C2 [×12], C4 [×8], C4 [×20], C22 [×3], C22 [×32], C2×C4 [×44], C2×C4 [×68], Q8 [×32], C23, C23 [×14], C42 [×8], C4⋊C4 [×8], C22×C4 [×42], C22×C4 [×28], C2×Q8 [×16], C2×Q8 [×48], C24, C2.C42 [×16], C2×C42 [×4], C2×C42 [×4], C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C23×C4, C23×C4 [×6], C22×Q8 [×12], C22×Q8 [×8], C2×C2.C42 [×4], C23.67C23 [×8], C22×C42, C22×C4⋊C4, Q8×C23, C2×C23.67C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], Q8 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C2×Q8 [×12], C4○D4 [×4], C24, C2×C22⋊C4 [×12], C4×Q8 [×8], C22⋊Q8 [×8], C4.4D4 [×4], C4⋊Q8 [×4], C23×C4, C22×D4 [×2], C22×Q8 [×2], C2×C4○D4 [×2], C23.67C23 [×8], C22×C22⋊C4, C2×C4×Q8 [×2], C2×C22⋊Q8 [×2], C2×C4.4D4, C2×C4⋊Q8, C2×C23.67C23
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=bcd, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >
(1 45)(2 46)(3 47)(4 48)(5 109)(6 110)(7 111)(8 112)(9 101)(10 102)(11 103)(12 104)(13 54)(14 55)(15 56)(16 53)(17 105)(18 106)(19 107)(20 108)(21 125)(22 126)(23 127)(24 128)(25 97)(26 98)(27 99)(28 100)(29 117)(30 118)(31 119)(32 120)(33 68)(34 65)(35 66)(36 67)(37 85)(38 86)(39 87)(40 88)(41 113)(42 114)(43 115)(44 116)(49 81)(50 82)(51 83)(52 84)(57 69)(58 70)(59 71)(60 72)(61 93)(62 94)(63 95)(64 96)(73 91)(74 92)(75 89)(76 90)(77 123)(78 124)(79 121)(80 122)
(1 25)(2 26)(3 27)(4 28)(5 69)(6 70)(7 71)(8 72)(9 68)(10 65)(11 66)(12 67)(13 80)(14 77)(15 78)(16 79)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(29 81)(30 82)(31 83)(32 84)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 117)(50 118)(51 119)(52 120)(53 121)(54 122)(55 123)(56 124)(57 109)(58 110)(59 111)(60 112)(61 113)(62 114)(63 115)(64 116)(73 127)(74 128)(75 125)(76 126)
(1 103)(2 104)(3 101)(4 102)(5 39)(6 40)(7 37)(8 38)(9 47)(10 48)(11 45)(12 46)(13 128)(14 125)(15 126)(16 127)(17 59)(18 60)(19 57)(20 58)(21 55)(22 56)(23 53)(24 54)(25 35)(26 36)(27 33)(28 34)(29 63)(30 64)(31 61)(32 62)(41 51)(42 52)(43 49)(44 50)(65 100)(66 97)(67 98)(68 99)(69 107)(70 108)(71 105)(72 106)(73 79)(74 80)(75 77)(76 78)(81 115)(82 116)(83 113)(84 114)(85 111)(86 112)(87 109)(88 110)(89 123)(90 124)(91 121)(92 122)(93 119)(94 120)(95 117)(96 118)
(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 61 33 81)(2 114 34 30)(3 63 35 83)(4 116 36 32)(5 125 105 79)(6 76 106 13)(7 127 107 77)(8 74 108 15)(9 117 97 41)(10 50 98 94)(11 119 99 43)(12 52 100 96)(14 71 73 39)(16 69 75 37)(17 121 109 21)(18 54 110 90)(19 123 111 23)(20 56 112 92)(22 86 122 58)(24 88 124 60)(25 113 101 29)(26 62 102 82)(27 115 103 31)(28 64 104 84)(38 80 70 126)(40 78 72 128)(42 65 118 46)(44 67 120 48)(45 93 68 49)(47 95 66 51)(53 57 89 85)(55 59 91 87)
(1 107 103 69)(2 70 104 108)(3 105 101 71)(4 72 102 106)(5 25 39 35)(6 36 40 26)(7 27 37 33)(8 34 38 28)(9 59 47 17)(10 18 48 60)(11 57 45 19)(12 20 46 58)(13 116 128 82)(14 83 125 113)(15 114 126 84)(16 81 127 115)(21 41 55 51)(22 52 56 42)(23 43 53 49)(24 50 54 44)(29 73 63 79)(30 80 64 74)(31 75 61 77)(32 78 62 76)(65 86 100 112)(66 109 97 87)(67 88 98 110)(68 111 99 85)(89 93 123 119)(90 120 124 94)(91 95 121 117)(92 118 122 96)
G:=sub<Sym(128)| (1,45)(2,46)(3,47)(4,48)(5,109)(6,110)(7,111)(8,112)(9,101)(10,102)(11,103)(12,104)(13,54)(14,55)(15,56)(16,53)(17,105)(18,106)(19,107)(20,108)(21,125)(22,126)(23,127)(24,128)(25,97)(26,98)(27,99)(28,100)(29,117)(30,118)(31,119)(32,120)(33,68)(34,65)(35,66)(36,67)(37,85)(38,86)(39,87)(40,88)(41,113)(42,114)(43,115)(44,116)(49,81)(50,82)(51,83)(52,84)(57,69)(58,70)(59,71)(60,72)(61,93)(62,94)(63,95)(64,96)(73,91)(74,92)(75,89)(76,90)(77,123)(78,124)(79,121)(80,122), (1,25)(2,26)(3,27)(4,28)(5,69)(6,70)(7,71)(8,72)(9,68)(10,65)(11,66)(12,67)(13,80)(14,77)(15,78)(16,79)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(29,81)(30,82)(31,83)(32,84)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,117)(50,118)(51,119)(52,120)(53,121)(54,122)(55,123)(56,124)(57,109)(58,110)(59,111)(60,112)(61,113)(62,114)(63,115)(64,116)(73,127)(74,128)(75,125)(76,126), (1,103)(2,104)(3,101)(4,102)(5,39)(6,40)(7,37)(8,38)(9,47)(10,48)(11,45)(12,46)(13,128)(14,125)(15,126)(16,127)(17,59)(18,60)(19,57)(20,58)(21,55)(22,56)(23,53)(24,54)(25,35)(26,36)(27,33)(28,34)(29,63)(30,64)(31,61)(32,62)(41,51)(42,52)(43,49)(44,50)(65,100)(66,97)(67,98)(68,99)(69,107)(70,108)(71,105)(72,106)(73,79)(74,80)(75,77)(76,78)(81,115)(82,116)(83,113)(84,114)(85,111)(86,112)(87,109)(88,110)(89,123)(90,124)(91,121)(92,122)(93,119)(94,120)(95,117)(96,118), (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), (1,61,33,81)(2,114,34,30)(3,63,35,83)(4,116,36,32)(5,125,105,79)(6,76,106,13)(7,127,107,77)(8,74,108,15)(9,117,97,41)(10,50,98,94)(11,119,99,43)(12,52,100,96)(14,71,73,39)(16,69,75,37)(17,121,109,21)(18,54,110,90)(19,123,111,23)(20,56,112,92)(22,86,122,58)(24,88,124,60)(25,113,101,29)(26,62,102,82)(27,115,103,31)(28,64,104,84)(38,80,70,126)(40,78,72,128)(42,65,118,46)(44,67,120,48)(45,93,68,49)(47,95,66,51)(53,57,89,85)(55,59,91,87), (1,107,103,69)(2,70,104,108)(3,105,101,71)(4,72,102,106)(5,25,39,35)(6,36,40,26)(7,27,37,33)(8,34,38,28)(9,59,47,17)(10,18,48,60)(11,57,45,19)(12,20,46,58)(13,116,128,82)(14,83,125,113)(15,114,126,84)(16,81,127,115)(21,41,55,51)(22,52,56,42)(23,43,53,49)(24,50,54,44)(29,73,63,79)(30,80,64,74)(31,75,61,77)(32,78,62,76)(65,86,100,112)(66,109,97,87)(67,88,98,110)(68,111,99,85)(89,93,123,119)(90,120,124,94)(91,95,121,117)(92,118,122,96)>;
G:=Group( (1,45)(2,46)(3,47)(4,48)(5,109)(6,110)(7,111)(8,112)(9,101)(10,102)(11,103)(12,104)(13,54)(14,55)(15,56)(16,53)(17,105)(18,106)(19,107)(20,108)(21,125)(22,126)(23,127)(24,128)(25,97)(26,98)(27,99)(28,100)(29,117)(30,118)(31,119)(32,120)(33,68)(34,65)(35,66)(36,67)(37,85)(38,86)(39,87)(40,88)(41,113)(42,114)(43,115)(44,116)(49,81)(50,82)(51,83)(52,84)(57,69)(58,70)(59,71)(60,72)(61,93)(62,94)(63,95)(64,96)(73,91)(74,92)(75,89)(76,90)(77,123)(78,124)(79,121)(80,122), (1,25)(2,26)(3,27)(4,28)(5,69)(6,70)(7,71)(8,72)(9,68)(10,65)(11,66)(12,67)(13,80)(14,77)(15,78)(16,79)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(29,81)(30,82)(31,83)(32,84)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,117)(50,118)(51,119)(52,120)(53,121)(54,122)(55,123)(56,124)(57,109)(58,110)(59,111)(60,112)(61,113)(62,114)(63,115)(64,116)(73,127)(74,128)(75,125)(76,126), (1,103)(2,104)(3,101)(4,102)(5,39)(6,40)(7,37)(8,38)(9,47)(10,48)(11,45)(12,46)(13,128)(14,125)(15,126)(16,127)(17,59)(18,60)(19,57)(20,58)(21,55)(22,56)(23,53)(24,54)(25,35)(26,36)(27,33)(28,34)(29,63)(30,64)(31,61)(32,62)(41,51)(42,52)(43,49)(44,50)(65,100)(66,97)(67,98)(68,99)(69,107)(70,108)(71,105)(72,106)(73,79)(74,80)(75,77)(76,78)(81,115)(82,116)(83,113)(84,114)(85,111)(86,112)(87,109)(88,110)(89,123)(90,124)(91,121)(92,122)(93,119)(94,120)(95,117)(96,118), (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), (1,61,33,81)(2,114,34,30)(3,63,35,83)(4,116,36,32)(5,125,105,79)(6,76,106,13)(7,127,107,77)(8,74,108,15)(9,117,97,41)(10,50,98,94)(11,119,99,43)(12,52,100,96)(14,71,73,39)(16,69,75,37)(17,121,109,21)(18,54,110,90)(19,123,111,23)(20,56,112,92)(22,86,122,58)(24,88,124,60)(25,113,101,29)(26,62,102,82)(27,115,103,31)(28,64,104,84)(38,80,70,126)(40,78,72,128)(42,65,118,46)(44,67,120,48)(45,93,68,49)(47,95,66,51)(53,57,89,85)(55,59,91,87), (1,107,103,69)(2,70,104,108)(3,105,101,71)(4,72,102,106)(5,25,39,35)(6,36,40,26)(7,27,37,33)(8,34,38,28)(9,59,47,17)(10,18,48,60)(11,57,45,19)(12,20,46,58)(13,116,128,82)(14,83,125,113)(15,114,126,84)(16,81,127,115)(21,41,55,51)(22,52,56,42)(23,43,53,49)(24,50,54,44)(29,73,63,79)(30,80,64,74)(31,75,61,77)(32,78,62,76)(65,86,100,112)(66,109,97,87)(67,88,98,110)(68,111,99,85)(89,93,123,119)(90,120,124,94)(91,95,121,117)(92,118,122,96) );
G=PermutationGroup([(1,45),(2,46),(3,47),(4,48),(5,109),(6,110),(7,111),(8,112),(9,101),(10,102),(11,103),(12,104),(13,54),(14,55),(15,56),(16,53),(17,105),(18,106),(19,107),(20,108),(21,125),(22,126),(23,127),(24,128),(25,97),(26,98),(27,99),(28,100),(29,117),(30,118),(31,119),(32,120),(33,68),(34,65),(35,66),(36,67),(37,85),(38,86),(39,87),(40,88),(41,113),(42,114),(43,115),(44,116),(49,81),(50,82),(51,83),(52,84),(57,69),(58,70),(59,71),(60,72),(61,93),(62,94),(63,95),(64,96),(73,91),(74,92),(75,89),(76,90),(77,123),(78,124),(79,121),(80,122)], [(1,25),(2,26),(3,27),(4,28),(5,69),(6,70),(7,71),(8,72),(9,68),(10,65),(11,66),(12,67),(13,80),(14,77),(15,78),(16,79),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(29,81),(30,82),(31,83),(32,84),(33,101),(34,102),(35,103),(36,104),(37,105),(38,106),(39,107),(40,108),(41,93),(42,94),(43,95),(44,96),(45,97),(46,98),(47,99),(48,100),(49,117),(50,118),(51,119),(52,120),(53,121),(54,122),(55,123),(56,124),(57,109),(58,110),(59,111),(60,112),(61,113),(62,114),(63,115),(64,116),(73,127),(74,128),(75,125),(76,126)], [(1,103),(2,104),(3,101),(4,102),(5,39),(6,40),(7,37),(8,38),(9,47),(10,48),(11,45),(12,46),(13,128),(14,125),(15,126),(16,127),(17,59),(18,60),(19,57),(20,58),(21,55),(22,56),(23,53),(24,54),(25,35),(26,36),(27,33),(28,34),(29,63),(30,64),(31,61),(32,62),(41,51),(42,52),(43,49),(44,50),(65,100),(66,97),(67,98),(68,99),(69,107),(70,108),(71,105),(72,106),(73,79),(74,80),(75,77),(76,78),(81,115),(82,116),(83,113),(84,114),(85,111),(86,112),(87,109),(88,110),(89,123),(90,124),(91,121),(92,122),(93,119),(94,120),(95,117),(96,118)], [(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)], [(1,61,33,81),(2,114,34,30),(3,63,35,83),(4,116,36,32),(5,125,105,79),(6,76,106,13),(7,127,107,77),(8,74,108,15),(9,117,97,41),(10,50,98,94),(11,119,99,43),(12,52,100,96),(14,71,73,39),(16,69,75,37),(17,121,109,21),(18,54,110,90),(19,123,111,23),(20,56,112,92),(22,86,122,58),(24,88,124,60),(25,113,101,29),(26,62,102,82),(27,115,103,31),(28,64,104,84),(38,80,70,126),(40,78,72,128),(42,65,118,46),(44,67,120,48),(45,93,68,49),(47,95,66,51),(53,57,89,85),(55,59,91,87)], [(1,107,103,69),(2,70,104,108),(3,105,101,71),(4,72,102,106),(5,25,39,35),(6,36,40,26),(7,27,37,33),(8,34,38,28),(9,59,47,17),(10,18,48,60),(11,57,45,19),(12,20,46,58),(13,116,128,82),(14,83,125,113),(15,114,126,84),(16,81,127,115),(21,41,55,51),(22,52,56,42),(23,43,53,49),(24,50,54,44),(29,73,63,79),(30,80,64,74),(31,75,61,77),(32,78,62,76),(65,86,100,112),(66,109,97,87),(67,88,98,110),(68,111,99,85),(89,93,123,119),(90,120,124,94),(91,95,121,117),(92,118,122,96)])
Matrix representation ►G ⊆ 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 | 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 | 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 |
| 3 | 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 | 2 | 3 |
| 0 | 0 | 0 | 0 | 4 | 3 |
| 2 | 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 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 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 | 1 | 2 |
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,4,0,0,0,0,3,3],[2,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,2] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4X | 4Y | ··· | 4AN |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | C4○D4 |
| kernel | C2×C23.67C23 | C2×C2.C42 | C23.67C23 | C22×C42 | C22×C4⋊C4 | Q8×C23 | C22×Q8 | C22×C4 | C22×C4 | C23 |
| # reps | 1 | 4 | 8 | 1 | 1 | 1 | 16 | 8 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3._{67}C_2^3 % in TeX
G:=Group("C2xC2^3.67C2^3"); // GroupNames label
G:=SmallGroup(128,1026);
// by ID
G=gap.SmallGroup(128,1026);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,792,758,184]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=b*c*d,g^2=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*c=c*b,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,d*f=f*d,d*g=g*d,f*g=g*f>;
// generators/relations