p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.362C24, C22.1242- (1+4), C4⋊C4⋊14Q8, C2.16(D4×Q8), C4⋊C4.330D4, C2.9(Q8⋊3Q8), C2.30(D4⋊6D4), C22.81(C22×Q8), (C22×C4).815C23, (C2×C42).505C22, C22.242(C22×D4), (C22×Q8).429C22, C2.34(C22.19C24), C23.81C23.6C2, C23.78C23.4C2, C23.63C23.16C2, C2.C42.119C22, C23.65C23.42C2, C2.17(C23.37C23), C2.23(C22.46C24), C2.12(C22.35C24), (C4×C4⋊C4).51C2, (C2×C4×Q8).32C2, (C2×C4).34(C2×Q8), (C2×C4).338(C2×D4), (C2×C4).366(C4○D4), (C2×C4⋊C4).243C22, C22.239(C2×C4○D4), (C2×C42.C2).17C2, SmallGroup(128,1194)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 356 in 226 conjugacy classes, 108 normal (82 characteristic)
C1, C2 [×7], C4 [×24], C22 [×7], C2×C4 [×18], C2×C4 [×36], Q8 [×8], C23, C42 [×10], C4⋊C4 [×8], C4⋊C4 [×26], C22×C4 [×15], C2×Q8 [×6], C2.C42 [×14], C2×C42 [×5], C2×C4⋊C4 [×15], C4×Q8 [×4], C42.C2 [×4], C22×Q8, C4×C4⋊C4, C23.63C23 [×6], C23.65C23 [×2], C23.78C23 [×2], C23.81C23 [×2], C2×C4×Q8, C2×C42.C2, C23.362C24
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×6], C24, C22×D4, C22×Q8, C2×C4○D4 [×3], 2- (1+4) [×2], C22.19C24, C23.37C23, C22.35C24, D4⋊6D4, D4×Q8, C22.46C24, Q8⋊3Q8, C23.362C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=f2=a, g2=b, 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 >
(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 73)(2 74)(3 75)(4 76)(5 36)(6 33)(7 34)(8 35)(9 105)(10 106)(11 107)(12 108)(13 41)(14 42)(15 43)(16 44)(17 113)(18 114)(19 115)(20 116)(21 49)(22 50)(23 51)(24 52)(25 121)(26 122)(27 123)(28 124)(29 57)(30 58)(31 59)(32 60)(37 66)(38 67)(39 68)(40 65)(45 77)(46 78)(47 79)(48 80)(53 85)(54 86)(55 87)(56 88)(61 93)(62 94)(63 95)(64 96)(69 104)(70 101)(71 102)(72 103)(81 109)(82 110)(83 111)(84 112)(89 117)(90 118)(91 119)(92 120)(97 125)(98 126)(99 127)(100 128)
(1 11)(2 12)(3 9)(4 10)(5 98)(6 99)(7 100)(8 97)(13 47)(14 48)(15 45)(16 46)(17 111)(18 112)(19 109)(20 110)(21 55)(22 56)(23 53)(24 54)(25 119)(26 120)(27 117)(28 118)(29 63)(30 64)(31 61)(32 62)(33 127)(34 128)(35 125)(36 126)(37 72)(38 69)(39 70)(40 71)(41 79)(42 80)(43 77)(44 78)(49 87)(50 88)(51 85)(52 86)(57 95)(58 96)(59 93)(60 94)(65 102)(66 103)(67 104)(68 101)(73 107)(74 108)(75 105)(76 106)(81 115)(82 116)(83 113)(84 114)(89 123)(90 124)(91 121)(92 122)
(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)
(1 119 3 117)(2 118 4 120)(5 114 7 116)(6 113 8 115)(9 27 11 25)(10 26 12 28)(13 31 15 29)(14 30 16 32)(17 35 19 33)(18 34 20 36)(21 39 23 37)(22 38 24 40)(41 59 43 57)(42 58 44 60)(45 63 47 61)(46 62 48 64)(49 68 51 66)(50 67 52 65)(53 72 55 70)(54 71 56 69)(73 91 75 89)(74 90 76 92)(77 95 79 93)(78 94 80 96)(81 99 83 97)(82 98 84 100)(85 103 87 101)(86 102 88 104)(105 123 107 121)(106 122 108 124)(109 127 111 125)(110 126 112 128)
(1 111 3 109)(2 84 4 82)(5 118 7 120)(6 91 8 89)(9 19 11 17)(10 116 12 114)(13 23 15 21)(14 52 16 50)(18 106 20 108)(22 42 24 44)(25 125 27 127)(26 98 28 100)(29 70 31 72)(30 102 32 104)(33 119 35 117)(34 92 36 90)(37 63 39 61)(38 96 40 94)(41 51 43 49)(45 55 47 53)(46 88 48 86)(54 78 56 80)(57 101 59 103)(58 71 60 69)(62 67 64 65)(66 95 68 93)(73 83 75 81)(74 112 76 110)(77 87 79 85)(97 123 99 121)(105 115 107 113)(122 126 124 128)
(1 13 73 41)(2 16 74 44)(3 15 75 43)(4 14 76 42)(5 102 36 71)(6 101 33 70)(7 104 34 69)(8 103 35 72)(9 45 105 77)(10 48 106 80)(11 47 107 79)(12 46 108 78)(17 53 113 85)(18 56 114 88)(19 55 115 87)(20 54 116 86)(21 81 49 109)(22 84 50 112)(23 83 51 111)(24 82 52 110)(25 61 121 93)(26 64 122 96)(27 63 123 95)(28 62 124 94)(29 89 57 117)(30 92 58 120)(31 91 59 119)(32 90 60 118)(37 97 66 125)(38 100 67 128)(39 99 68 127)(40 98 65 126)
G:=sub<Sym(128)| (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), (1,73)(2,74)(3,75)(4,76)(5,36)(6,33)(7,34)(8,35)(9,105)(10,106)(11,107)(12,108)(13,41)(14,42)(15,43)(16,44)(17,113)(18,114)(19,115)(20,116)(21,49)(22,50)(23,51)(24,52)(25,121)(26,122)(27,123)(28,124)(29,57)(30,58)(31,59)(32,60)(37,66)(38,67)(39,68)(40,65)(45,77)(46,78)(47,79)(48,80)(53,85)(54,86)(55,87)(56,88)(61,93)(62,94)(63,95)(64,96)(69,104)(70,101)(71,102)(72,103)(81,109)(82,110)(83,111)(84,112)(89,117)(90,118)(91,119)(92,120)(97,125)(98,126)(99,127)(100,128), (1,11)(2,12)(3,9)(4,10)(5,98)(6,99)(7,100)(8,97)(13,47)(14,48)(15,45)(16,46)(17,111)(18,112)(19,109)(20,110)(21,55)(22,56)(23,53)(24,54)(25,119)(26,120)(27,117)(28,118)(29,63)(30,64)(31,61)(32,62)(33,127)(34,128)(35,125)(36,126)(37,72)(38,69)(39,70)(40,71)(41,79)(42,80)(43,77)(44,78)(49,87)(50,88)(51,85)(52,86)(57,95)(58,96)(59,93)(60,94)(65,102)(66,103)(67,104)(68,101)(73,107)(74,108)(75,105)(76,106)(81,115)(82,116)(83,113)(84,114)(89,123)(90,124)(91,121)(92,122), (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), (1,119,3,117)(2,118,4,120)(5,114,7,116)(6,113,8,115)(9,27,11,25)(10,26,12,28)(13,31,15,29)(14,30,16,32)(17,35,19,33)(18,34,20,36)(21,39,23,37)(22,38,24,40)(41,59,43,57)(42,58,44,60)(45,63,47,61)(46,62,48,64)(49,68,51,66)(50,67,52,65)(53,72,55,70)(54,71,56,69)(73,91,75,89)(74,90,76,92)(77,95,79,93)(78,94,80,96)(81,99,83,97)(82,98,84,100)(85,103,87,101)(86,102,88,104)(105,123,107,121)(106,122,108,124)(109,127,111,125)(110,126,112,128), (1,111,3,109)(2,84,4,82)(5,118,7,120)(6,91,8,89)(9,19,11,17)(10,116,12,114)(13,23,15,21)(14,52,16,50)(18,106,20,108)(22,42,24,44)(25,125,27,127)(26,98,28,100)(29,70,31,72)(30,102,32,104)(33,119,35,117)(34,92,36,90)(37,63,39,61)(38,96,40,94)(41,51,43,49)(45,55,47,53)(46,88,48,86)(54,78,56,80)(57,101,59,103)(58,71,60,69)(62,67,64,65)(66,95,68,93)(73,83,75,81)(74,112,76,110)(77,87,79,85)(97,123,99,121)(105,115,107,113)(122,126,124,128), (1,13,73,41)(2,16,74,44)(3,15,75,43)(4,14,76,42)(5,102,36,71)(6,101,33,70)(7,104,34,69)(8,103,35,72)(9,45,105,77)(10,48,106,80)(11,47,107,79)(12,46,108,78)(17,53,113,85)(18,56,114,88)(19,55,115,87)(20,54,116,86)(21,81,49,109)(22,84,50,112)(23,83,51,111)(24,82,52,110)(25,61,121,93)(26,64,122,96)(27,63,123,95)(28,62,124,94)(29,89,57,117)(30,92,58,120)(31,91,59,119)(32,90,60,118)(37,97,66,125)(38,100,67,128)(39,99,68,127)(40,98,65,126)>;
G:=Group( (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), (1,73)(2,74)(3,75)(4,76)(5,36)(6,33)(7,34)(8,35)(9,105)(10,106)(11,107)(12,108)(13,41)(14,42)(15,43)(16,44)(17,113)(18,114)(19,115)(20,116)(21,49)(22,50)(23,51)(24,52)(25,121)(26,122)(27,123)(28,124)(29,57)(30,58)(31,59)(32,60)(37,66)(38,67)(39,68)(40,65)(45,77)(46,78)(47,79)(48,80)(53,85)(54,86)(55,87)(56,88)(61,93)(62,94)(63,95)(64,96)(69,104)(70,101)(71,102)(72,103)(81,109)(82,110)(83,111)(84,112)(89,117)(90,118)(91,119)(92,120)(97,125)(98,126)(99,127)(100,128), (1,11)(2,12)(3,9)(4,10)(5,98)(6,99)(7,100)(8,97)(13,47)(14,48)(15,45)(16,46)(17,111)(18,112)(19,109)(20,110)(21,55)(22,56)(23,53)(24,54)(25,119)(26,120)(27,117)(28,118)(29,63)(30,64)(31,61)(32,62)(33,127)(34,128)(35,125)(36,126)(37,72)(38,69)(39,70)(40,71)(41,79)(42,80)(43,77)(44,78)(49,87)(50,88)(51,85)(52,86)(57,95)(58,96)(59,93)(60,94)(65,102)(66,103)(67,104)(68,101)(73,107)(74,108)(75,105)(76,106)(81,115)(82,116)(83,113)(84,114)(89,123)(90,124)(91,121)(92,122), (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), (1,119,3,117)(2,118,4,120)(5,114,7,116)(6,113,8,115)(9,27,11,25)(10,26,12,28)(13,31,15,29)(14,30,16,32)(17,35,19,33)(18,34,20,36)(21,39,23,37)(22,38,24,40)(41,59,43,57)(42,58,44,60)(45,63,47,61)(46,62,48,64)(49,68,51,66)(50,67,52,65)(53,72,55,70)(54,71,56,69)(73,91,75,89)(74,90,76,92)(77,95,79,93)(78,94,80,96)(81,99,83,97)(82,98,84,100)(85,103,87,101)(86,102,88,104)(105,123,107,121)(106,122,108,124)(109,127,111,125)(110,126,112,128), (1,111,3,109)(2,84,4,82)(5,118,7,120)(6,91,8,89)(9,19,11,17)(10,116,12,114)(13,23,15,21)(14,52,16,50)(18,106,20,108)(22,42,24,44)(25,125,27,127)(26,98,28,100)(29,70,31,72)(30,102,32,104)(33,119,35,117)(34,92,36,90)(37,63,39,61)(38,96,40,94)(41,51,43,49)(45,55,47,53)(46,88,48,86)(54,78,56,80)(57,101,59,103)(58,71,60,69)(62,67,64,65)(66,95,68,93)(73,83,75,81)(74,112,76,110)(77,87,79,85)(97,123,99,121)(105,115,107,113)(122,126,124,128), (1,13,73,41)(2,16,74,44)(3,15,75,43)(4,14,76,42)(5,102,36,71)(6,101,33,70)(7,104,34,69)(8,103,35,72)(9,45,105,77)(10,48,106,80)(11,47,107,79)(12,46,108,78)(17,53,113,85)(18,56,114,88)(19,55,115,87)(20,54,116,86)(21,81,49,109)(22,84,50,112)(23,83,51,111)(24,82,52,110)(25,61,121,93)(26,64,122,96)(27,63,123,95)(28,62,124,94)(29,89,57,117)(30,92,58,120)(31,91,59,119)(32,90,60,118)(37,97,66,125)(38,100,67,128)(39,99,68,127)(40,98,65,126) );
G=PermutationGroup([(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)], [(1,73),(2,74),(3,75),(4,76),(5,36),(6,33),(7,34),(8,35),(9,105),(10,106),(11,107),(12,108),(13,41),(14,42),(15,43),(16,44),(17,113),(18,114),(19,115),(20,116),(21,49),(22,50),(23,51),(24,52),(25,121),(26,122),(27,123),(28,124),(29,57),(30,58),(31,59),(32,60),(37,66),(38,67),(39,68),(40,65),(45,77),(46,78),(47,79),(48,80),(53,85),(54,86),(55,87),(56,88),(61,93),(62,94),(63,95),(64,96),(69,104),(70,101),(71,102),(72,103),(81,109),(82,110),(83,111),(84,112),(89,117),(90,118),(91,119),(92,120),(97,125),(98,126),(99,127),(100,128)], [(1,11),(2,12),(3,9),(4,10),(5,98),(6,99),(7,100),(8,97),(13,47),(14,48),(15,45),(16,46),(17,111),(18,112),(19,109),(20,110),(21,55),(22,56),(23,53),(24,54),(25,119),(26,120),(27,117),(28,118),(29,63),(30,64),(31,61),(32,62),(33,127),(34,128),(35,125),(36,126),(37,72),(38,69),(39,70),(40,71),(41,79),(42,80),(43,77),(44,78),(49,87),(50,88),(51,85),(52,86),(57,95),(58,96),(59,93),(60,94),(65,102),(66,103),(67,104),(68,101),(73,107),(74,108),(75,105),(76,106),(81,115),(82,116),(83,113),(84,114),(89,123),(90,124),(91,121),(92,122)], [(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)], [(1,119,3,117),(2,118,4,120),(5,114,7,116),(6,113,8,115),(9,27,11,25),(10,26,12,28),(13,31,15,29),(14,30,16,32),(17,35,19,33),(18,34,20,36),(21,39,23,37),(22,38,24,40),(41,59,43,57),(42,58,44,60),(45,63,47,61),(46,62,48,64),(49,68,51,66),(50,67,52,65),(53,72,55,70),(54,71,56,69),(73,91,75,89),(74,90,76,92),(77,95,79,93),(78,94,80,96),(81,99,83,97),(82,98,84,100),(85,103,87,101),(86,102,88,104),(105,123,107,121),(106,122,108,124),(109,127,111,125),(110,126,112,128)], [(1,111,3,109),(2,84,4,82),(5,118,7,120),(6,91,8,89),(9,19,11,17),(10,116,12,114),(13,23,15,21),(14,52,16,50),(18,106,20,108),(22,42,24,44),(25,125,27,127),(26,98,28,100),(29,70,31,72),(30,102,32,104),(33,119,35,117),(34,92,36,90),(37,63,39,61),(38,96,40,94),(41,51,43,49),(45,55,47,53),(46,88,48,86),(54,78,56,80),(57,101,59,103),(58,71,60,69),(62,67,64,65),(66,95,68,93),(73,83,75,81),(74,112,76,110),(77,87,79,85),(97,123,99,121),(105,115,107,113),(122,126,124,128)], [(1,13,73,41),(2,16,74,44),(3,15,75,43),(4,14,76,42),(5,102,36,71),(6,101,33,70),(7,104,34,69),(8,103,35,72),(9,45,105,77),(10,48,106,80),(11,47,107,79),(12,46,108,78),(17,53,113,85),(18,56,114,88),(19,55,115,87),(20,54,116,86),(21,81,49,109),(22,84,50,112),(23,83,51,111),(24,82,52,110),(25,61,121,93),(26,64,122,96),(27,63,123,95),(28,62,124,94),(29,89,57,117),(30,92,58,120),(31,91,59,119),(32,90,60,118),(37,97,66,125),(38,100,67,128),(39,99,68,127),(40,98,65,126)])
Matrix representation ►G ⊆ 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 |
| 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 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 4 | 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 | 0 | 2 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,2,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[4,2,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
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 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | 2- (1+4) |
| kernel | C23.362C24 | C4×C4⋊C4 | C23.63C23 | C23.65C23 | C23.78C23 | C23.81C23 | C2×C4×Q8 | C2×C42.C2 | C4⋊C4 | C4⋊C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 6 | 2 | 2 | 2 | 1 | 1 | 4 | 4 | 12 | 2 |
In GAP, Magma, Sage, TeX
C_2^3._{362}C_2^4 % in TeX
G:=Group("C2^3.362C2^4"); // GroupNames label
G:=SmallGroup(128,1194);
// by ID
G=gap.SmallGroup(128,1194);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,100,675,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=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,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