p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.407C24, C22.2022+ (1+4), C2.4(Q82), C4⋊C4.20Q8, C42⋊9C4.26C2, C2.23(D4⋊3Q8), C4.29(C42.C2), C22.90(C22×Q8), (C22×C4).829C23, (C2×C42).527C22, C23.81C23.8C2, C23.65C23.44C2, C2.C42.158C22, C23.63C23.18C2, C2.38(C22.47C24), C2.21(C23.37C23), C2.18(C22.34C24), (C4×C4⋊C4).53C2, (C2×C4).43(C2×Q8), C2.12(C2×C42.C2), (C2×C4).129(C4○D4), (C2×C4⋊C4).273C22, C22.284(C2×C4○D4), SmallGroup(128,1239)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 324 in 198 conjugacy classes, 112 normal (42 characteristic)
C1, C2 [×7], C4 [×4], C4 [×20], C22 [×7], C2×C4 [×18], C2×C4 [×36], C23, C42 [×8], C4⋊C4 [×8], C4⋊C4 [×22], C22×C4 [×7], C22×C4 [×8], C2.C42 [×2], C2.C42 [×10], C2×C42 [×3], C2×C42 [×2], C2×C4⋊C4 [×6], C2×C4⋊C4 [×12], C4×C4⋊C4 [×2], C42⋊9C4, C23.63C23 [×2], C23.65C23 [×4], C23.65C23 [×2], C23.81C23 [×4], C23.407C24
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×8], C23 [×15], C2×Q8 [×12], C4○D4 [×6], C24, C42.C2 [×4], C22×Q8 [×2], C2×C4○D4 [×3], 2+ (1+4) [×2], C2×C42.C2, C23.37C23, C22.34C24, C22.47C24, D4⋊3Q8 [×2], Q82, C23.407C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=ca=ac, e2=f2=b, g2=a, ab=ba, 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 >
(1 10)(2 11)(3 12)(4 9)(5 69)(6 70)(7 71)(8 72)(13 84)(14 81)(15 82)(16 83)(17 86)(18 87)(19 88)(20 85)(21 92)(22 89)(23 90)(24 91)(25 94)(26 95)(27 96)(28 93)(29 100)(30 97)(31 98)(32 99)(33 78)(34 79)(35 80)(36 77)(37 75)(38 76)(39 73)(40 74)(41 102)(42 103)(43 104)(44 101)(45 108)(46 105)(47 106)(48 107)(49 110)(50 111)(51 112)(52 109)(53 116)(54 113)(55 114)(56 115)(57 118)(58 119)(59 120)(60 117)(61 124)(62 121)(63 122)(64 123)(65 128)(66 125)(67 126)(68 127)
(1 42)(2 43)(3 44)(4 41)(5 39)(6 40)(7 37)(8 38)(9 102)(10 103)(11 104)(12 101)(13 108)(14 105)(15 106)(16 107)(17 110)(18 111)(19 112)(20 109)(21 116)(22 113)(23 114)(24 115)(25 118)(26 119)(27 120)(28 117)(29 124)(30 121)(31 122)(32 123)(33 126)(34 127)(35 128)(36 125)(45 84)(46 81)(47 82)(48 83)(49 86)(50 87)(51 88)(52 85)(53 92)(54 89)(55 90)(56 91)(57 94)(58 95)(59 96)(60 93)(61 100)(62 97)(63 98)(64 99)(65 80)(66 77)(67 78)(68 79)(69 73)(70 74)(71 75)(72 76)
(1 12)(2 9)(3 10)(4 11)(5 71)(6 72)(7 69)(8 70)(13 82)(14 83)(15 84)(16 81)(17 88)(18 85)(19 86)(20 87)(21 90)(22 91)(23 92)(24 89)(25 96)(26 93)(27 94)(28 95)(29 98)(30 99)(31 100)(32 97)(33 80)(34 77)(35 78)(36 79)(37 73)(38 74)(39 75)(40 76)(41 104)(42 101)(43 102)(44 103)(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)(65 126)(66 127)(67 128)(68 125)
(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 58 42 95)(2 120 43 27)(3 60 44 93)(4 118 41 25)(5 114 39 23)(6 56 40 91)(7 116 37 21)(8 54 38 89)(9 57 102 94)(10 119 103 26)(11 59 104 96)(12 117 101 28)(13 124 108 29)(14 62 105 97)(15 122 106 31)(16 64 107 99)(17 67 110 78)(18 127 111 34)(19 65 112 80)(20 125 109 36)(22 72 113 76)(24 70 115 74)(30 81 121 46)(32 83 123 48)(33 86 126 49)(35 88 128 51)(45 100 84 61)(47 98 82 63)(50 79 87 68)(52 77 85 66)(53 75 92 71)(55 73 90 69)
(1 50 42 87)(2 88 43 51)(3 52 44 85)(4 86 41 49)(5 61 39 100)(6 97 40 62)(7 63 37 98)(8 99 38 64)(9 17 102 110)(10 111 103 18)(11 19 104 112)(12 109 101 20)(13 116 108 21)(14 22 105 113)(15 114 106 23)(16 24 107 115)(25 80 118 65)(26 66 119 77)(27 78 120 67)(28 68 117 79)(29 69 124 73)(30 74 121 70)(31 71 122 75)(32 76 123 72)(33 59 126 96)(34 93 127 60)(35 57 128 94)(36 95 125 58)(45 92 84 53)(46 54 81 89)(47 90 82 55)(48 56 83 91)
(1 15 10 82)(2 83 11 16)(3 13 12 84)(4 81 9 14)(5 127 69 68)(6 65 70 128)(7 125 71 66)(8 67 72 126)(17 22 86 89)(18 90 87 23)(19 24 88 91)(20 92 85 21)(25 30 94 97)(26 98 95 31)(27 32 96 99)(28 100 93 29)(33 38 78 76)(34 73 79 39)(35 40 80 74)(36 75 77 37)(41 46 102 105)(42 106 103 47)(43 48 104 107)(44 108 101 45)(49 54 110 113)(50 114 111 55)(51 56 112 115)(52 116 109 53)(57 62 118 121)(58 122 119 63)(59 64 120 123)(60 124 117 61)
G:=sub<Sym(128)| (1,10)(2,11)(3,12)(4,9)(5,69)(6,70)(7,71)(8,72)(13,84)(14,81)(15,82)(16,83)(17,86)(18,87)(19,88)(20,85)(21,92)(22,89)(23,90)(24,91)(25,94)(26,95)(27,96)(28,93)(29,100)(30,97)(31,98)(32,99)(33,78)(34,79)(35,80)(36,77)(37,75)(38,76)(39,73)(40,74)(41,102)(42,103)(43,104)(44,101)(45,108)(46,105)(47,106)(48,107)(49,110)(50,111)(51,112)(52,109)(53,116)(54,113)(55,114)(56,115)(57,118)(58,119)(59,120)(60,117)(61,124)(62,121)(63,122)(64,123)(65,128)(66,125)(67,126)(68,127), (1,42)(2,43)(3,44)(4,41)(5,39)(6,40)(7,37)(8,38)(9,102)(10,103)(11,104)(12,101)(13,108)(14,105)(15,106)(16,107)(17,110)(18,111)(19,112)(20,109)(21,116)(22,113)(23,114)(24,115)(25,118)(26,119)(27,120)(28,117)(29,124)(30,121)(31,122)(32,123)(33,126)(34,127)(35,128)(36,125)(45,84)(46,81)(47,82)(48,83)(49,86)(50,87)(51,88)(52,85)(53,92)(54,89)(55,90)(56,91)(57,94)(58,95)(59,96)(60,93)(61,100)(62,97)(63,98)(64,99)(65,80)(66,77)(67,78)(68,79)(69,73)(70,74)(71,75)(72,76), (1,12)(2,9)(3,10)(4,11)(5,71)(6,72)(7,69)(8,70)(13,82)(14,83)(15,84)(16,81)(17,88)(18,85)(19,86)(20,87)(21,90)(22,91)(23,92)(24,89)(25,96)(26,93)(27,94)(28,95)(29,98)(30,99)(31,100)(32,97)(33,80)(34,77)(35,78)(36,79)(37,73)(38,74)(39,75)(40,76)(41,104)(42,101)(43,102)(44,103)(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)(65,126)(66,127)(67,128)(68,125), (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), (1,58,42,95)(2,120,43,27)(3,60,44,93)(4,118,41,25)(5,114,39,23)(6,56,40,91)(7,116,37,21)(8,54,38,89)(9,57,102,94)(10,119,103,26)(11,59,104,96)(12,117,101,28)(13,124,108,29)(14,62,105,97)(15,122,106,31)(16,64,107,99)(17,67,110,78)(18,127,111,34)(19,65,112,80)(20,125,109,36)(22,72,113,76)(24,70,115,74)(30,81,121,46)(32,83,123,48)(33,86,126,49)(35,88,128,51)(45,100,84,61)(47,98,82,63)(50,79,87,68)(52,77,85,66)(53,75,92,71)(55,73,90,69), (1,50,42,87)(2,88,43,51)(3,52,44,85)(4,86,41,49)(5,61,39,100)(6,97,40,62)(7,63,37,98)(8,99,38,64)(9,17,102,110)(10,111,103,18)(11,19,104,112)(12,109,101,20)(13,116,108,21)(14,22,105,113)(15,114,106,23)(16,24,107,115)(25,80,118,65)(26,66,119,77)(27,78,120,67)(28,68,117,79)(29,69,124,73)(30,74,121,70)(31,71,122,75)(32,76,123,72)(33,59,126,96)(34,93,127,60)(35,57,128,94)(36,95,125,58)(45,92,84,53)(46,54,81,89)(47,90,82,55)(48,56,83,91), (1,15,10,82)(2,83,11,16)(3,13,12,84)(4,81,9,14)(5,127,69,68)(6,65,70,128)(7,125,71,66)(8,67,72,126)(17,22,86,89)(18,90,87,23)(19,24,88,91)(20,92,85,21)(25,30,94,97)(26,98,95,31)(27,32,96,99)(28,100,93,29)(33,38,78,76)(34,73,79,39)(35,40,80,74)(36,75,77,37)(41,46,102,105)(42,106,103,47)(43,48,104,107)(44,108,101,45)(49,54,110,113)(50,114,111,55)(51,56,112,115)(52,116,109,53)(57,62,118,121)(58,122,119,63)(59,64,120,123)(60,124,117,61)>;
G:=Group( (1,10)(2,11)(3,12)(4,9)(5,69)(6,70)(7,71)(8,72)(13,84)(14,81)(15,82)(16,83)(17,86)(18,87)(19,88)(20,85)(21,92)(22,89)(23,90)(24,91)(25,94)(26,95)(27,96)(28,93)(29,100)(30,97)(31,98)(32,99)(33,78)(34,79)(35,80)(36,77)(37,75)(38,76)(39,73)(40,74)(41,102)(42,103)(43,104)(44,101)(45,108)(46,105)(47,106)(48,107)(49,110)(50,111)(51,112)(52,109)(53,116)(54,113)(55,114)(56,115)(57,118)(58,119)(59,120)(60,117)(61,124)(62,121)(63,122)(64,123)(65,128)(66,125)(67,126)(68,127), (1,42)(2,43)(3,44)(4,41)(5,39)(6,40)(7,37)(8,38)(9,102)(10,103)(11,104)(12,101)(13,108)(14,105)(15,106)(16,107)(17,110)(18,111)(19,112)(20,109)(21,116)(22,113)(23,114)(24,115)(25,118)(26,119)(27,120)(28,117)(29,124)(30,121)(31,122)(32,123)(33,126)(34,127)(35,128)(36,125)(45,84)(46,81)(47,82)(48,83)(49,86)(50,87)(51,88)(52,85)(53,92)(54,89)(55,90)(56,91)(57,94)(58,95)(59,96)(60,93)(61,100)(62,97)(63,98)(64,99)(65,80)(66,77)(67,78)(68,79)(69,73)(70,74)(71,75)(72,76), (1,12)(2,9)(3,10)(4,11)(5,71)(6,72)(7,69)(8,70)(13,82)(14,83)(15,84)(16,81)(17,88)(18,85)(19,86)(20,87)(21,90)(22,91)(23,92)(24,89)(25,96)(26,93)(27,94)(28,95)(29,98)(30,99)(31,100)(32,97)(33,80)(34,77)(35,78)(36,79)(37,73)(38,74)(39,75)(40,76)(41,104)(42,101)(43,102)(44,103)(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)(65,126)(66,127)(67,128)(68,125), (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), (1,58,42,95)(2,120,43,27)(3,60,44,93)(4,118,41,25)(5,114,39,23)(6,56,40,91)(7,116,37,21)(8,54,38,89)(9,57,102,94)(10,119,103,26)(11,59,104,96)(12,117,101,28)(13,124,108,29)(14,62,105,97)(15,122,106,31)(16,64,107,99)(17,67,110,78)(18,127,111,34)(19,65,112,80)(20,125,109,36)(22,72,113,76)(24,70,115,74)(30,81,121,46)(32,83,123,48)(33,86,126,49)(35,88,128,51)(45,100,84,61)(47,98,82,63)(50,79,87,68)(52,77,85,66)(53,75,92,71)(55,73,90,69), (1,50,42,87)(2,88,43,51)(3,52,44,85)(4,86,41,49)(5,61,39,100)(6,97,40,62)(7,63,37,98)(8,99,38,64)(9,17,102,110)(10,111,103,18)(11,19,104,112)(12,109,101,20)(13,116,108,21)(14,22,105,113)(15,114,106,23)(16,24,107,115)(25,80,118,65)(26,66,119,77)(27,78,120,67)(28,68,117,79)(29,69,124,73)(30,74,121,70)(31,71,122,75)(32,76,123,72)(33,59,126,96)(34,93,127,60)(35,57,128,94)(36,95,125,58)(45,92,84,53)(46,54,81,89)(47,90,82,55)(48,56,83,91), (1,15,10,82)(2,83,11,16)(3,13,12,84)(4,81,9,14)(5,127,69,68)(6,65,70,128)(7,125,71,66)(8,67,72,126)(17,22,86,89)(18,90,87,23)(19,24,88,91)(20,92,85,21)(25,30,94,97)(26,98,95,31)(27,32,96,99)(28,100,93,29)(33,38,78,76)(34,73,79,39)(35,40,80,74)(36,75,77,37)(41,46,102,105)(42,106,103,47)(43,48,104,107)(44,108,101,45)(49,54,110,113)(50,114,111,55)(51,56,112,115)(52,116,109,53)(57,62,118,121)(58,122,119,63)(59,64,120,123)(60,124,117,61) );
G=PermutationGroup([(1,10),(2,11),(3,12),(4,9),(5,69),(6,70),(7,71),(8,72),(13,84),(14,81),(15,82),(16,83),(17,86),(18,87),(19,88),(20,85),(21,92),(22,89),(23,90),(24,91),(25,94),(26,95),(27,96),(28,93),(29,100),(30,97),(31,98),(32,99),(33,78),(34,79),(35,80),(36,77),(37,75),(38,76),(39,73),(40,74),(41,102),(42,103),(43,104),(44,101),(45,108),(46,105),(47,106),(48,107),(49,110),(50,111),(51,112),(52,109),(53,116),(54,113),(55,114),(56,115),(57,118),(58,119),(59,120),(60,117),(61,124),(62,121),(63,122),(64,123),(65,128),(66,125),(67,126),(68,127)], [(1,42),(2,43),(3,44),(4,41),(5,39),(6,40),(7,37),(8,38),(9,102),(10,103),(11,104),(12,101),(13,108),(14,105),(15,106),(16,107),(17,110),(18,111),(19,112),(20,109),(21,116),(22,113),(23,114),(24,115),(25,118),(26,119),(27,120),(28,117),(29,124),(30,121),(31,122),(32,123),(33,126),(34,127),(35,128),(36,125),(45,84),(46,81),(47,82),(48,83),(49,86),(50,87),(51,88),(52,85),(53,92),(54,89),(55,90),(56,91),(57,94),(58,95),(59,96),(60,93),(61,100),(62,97),(63,98),(64,99),(65,80),(66,77),(67,78),(68,79),(69,73),(70,74),(71,75),(72,76)], [(1,12),(2,9),(3,10),(4,11),(5,71),(6,72),(7,69),(8,70),(13,82),(14,83),(15,84),(16,81),(17,88),(18,85),(19,86),(20,87),(21,90),(22,91),(23,92),(24,89),(25,96),(26,93),(27,94),(28,95),(29,98),(30,99),(31,100),(32,97),(33,80),(34,77),(35,78),(36,79),(37,73),(38,74),(39,75),(40,76),(41,104),(42,101),(43,102),(44,103),(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),(65,126),(66,127),(67,128),(68,125)], [(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)], [(1,58,42,95),(2,120,43,27),(3,60,44,93),(4,118,41,25),(5,114,39,23),(6,56,40,91),(7,116,37,21),(8,54,38,89),(9,57,102,94),(10,119,103,26),(11,59,104,96),(12,117,101,28),(13,124,108,29),(14,62,105,97),(15,122,106,31),(16,64,107,99),(17,67,110,78),(18,127,111,34),(19,65,112,80),(20,125,109,36),(22,72,113,76),(24,70,115,74),(30,81,121,46),(32,83,123,48),(33,86,126,49),(35,88,128,51),(45,100,84,61),(47,98,82,63),(50,79,87,68),(52,77,85,66),(53,75,92,71),(55,73,90,69)], [(1,50,42,87),(2,88,43,51),(3,52,44,85),(4,86,41,49),(5,61,39,100),(6,97,40,62),(7,63,37,98),(8,99,38,64),(9,17,102,110),(10,111,103,18),(11,19,104,112),(12,109,101,20),(13,116,108,21),(14,22,105,113),(15,114,106,23),(16,24,107,115),(25,80,118,65),(26,66,119,77),(27,78,120,67),(28,68,117,79),(29,69,124,73),(30,74,121,70),(31,71,122,75),(32,76,123,72),(33,59,126,96),(34,93,127,60),(35,57,128,94),(36,95,125,58),(45,92,84,53),(46,54,81,89),(47,90,82,55),(48,56,83,91)], [(1,15,10,82),(2,83,11,16),(3,13,12,84),(4,81,9,14),(5,127,69,68),(6,65,70,128),(7,125,71,66),(8,67,72,126),(17,22,86,89),(18,90,87,23),(19,24,88,91),(20,92,85,21),(25,30,94,97),(26,98,95,31),(27,32,96,99),(28,100,93,29),(33,38,78,76),(34,73,79,39),(35,40,80,74),(36,75,77,37),(41,46,102,105),(42,106,103,47),(43,48,104,107),(44,108,101,45),(49,54,110,113),(50,114,111,55),(51,56,112,115),(52,116,109,53),(57,62,118,121),(58,122,119,63),(59,64,120,123),(60,124,117,61)])
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 |
| 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 |
| 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 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 4 | 2 | 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 |
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],[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],[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],[3,0,0,0,0,0,3,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[2,1,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[2,1,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,4,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
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 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2+ (1+4) |
| kernel | C23.407C24 | C4×C4⋊C4 | C42⋊9C4 | C23.63C23 | C23.65C23 | C23.81C23 | C4⋊C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 2 | 6 | 4 | 8 | 12 | 2 |
In GAP, Magma, Sage, TeX
C_2^3._{407}C_2^4 % in TeX
G:=Group("C2^3.407C2^4"); // GroupNames label
G:=SmallGroup(128,1239);
// by ID
G=gap.SmallGroup(128,1239);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,184,675,80]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*a=a*c,e^2=f^2=b,g^2=a,a*b=b*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