p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.624C23, (C2×C42)⋊13C4, (C2×C4).38C42, C22.73(C4×D4), C22.19(C4×Q8), (C22×C4).69Q8, C2.C42⋊7C4, (C22×C4).645D4, C23.714(C2×D4), (C23×C4).4C22, (C22×C42).2C2, C23.126(C2×Q8), C2.1(C42⋊5C4), C2.1(C42⋊8C4), C2.3(C42⋊4C4), C22.36(C2×C42), C23.337(C4○D4), C22.93(C4⋊D4), C23.238(C22×C4), C22.57(C22⋊Q8), C2.2(C23.7Q8), C2.2(C23.34D4), C22.43(C4.4D4), C22.16(C42.C2), C22.20(C42⋊2C2), C22.41(C42⋊C2), C2.1(C24.C22), C2.1(C23.63C23), C22.67(C22.D4), C2.4(C4×C4⋊C4), C2.5(C4×C22⋊C4), (C2×C4).73(C4⋊C4), C22.42(C2×C4⋊C4), (C22×C4).94(C2×C4), C22.84(C2×C22⋊C4), (C2×C4).308(C22⋊C4), (C2×C2.C42).2C2, SmallGroup(128,166)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.624C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=c, f2=d, g2=a, ab=ba, ac=ca, ad=da, fef-1=ae=ea, af=fa, ag=ga, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >
Subgroups: 484 in 284 conjugacy classes, 132 normal (24 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C42, C22×C4, C22×C4, C24, C2.C42, C2.C42, C2×C42, C2×C42, C23×C4, C23×C4, C2×C2.C42, C22×C42, C24.624C23
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C42⋊4C4, C4×C22⋊C4, C4×C4⋊C4, C23.7Q8, C23.34D4, C42⋊8C4, C42⋊5C4, C23.63C23, C24.C22, C24.624C23
►Regular action on 128 points
Generators in S128
(1 11)(2 12)(3 9)(4 10)(5 93)(6 94)(7 95)(8 96)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)(29 33)(30 34)(31 35)(32 36)(37 41)(38 42)(39 43)(40 44)(45 49)(46 50)(47 51)(48 52)(53 57)(54 58)(55 59)(56 60)(61 68)(62 65)(63 66)(64 67)(69 73)(70 74)(71 75)(72 76)(77 81)(78 82)(79 83)(80 84)(85 89)(86 90)(87 91)(88 92)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)
(1 71)(2 72)(3 69)(4 70)(5 33)(6 34)(7 35)(8 36)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 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)(65 126)(66 127)(67 128)(68 125)
(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 99)(2 100)(3 97)(4 98)(5 68)(6 65)(7 66)(8 67)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 69)(38 70)(39 71)(40 72)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)
(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)
(1 115 99 23)(2 120 100 28)(3 113 97 21)(4 118 98 26)(5 81 68 49)(6 78 65 46)(7 83 66 51)(8 80 67 48)(9 117 101 25)(10 114 102 22)(11 119 103 27)(12 116 104 24)(13 121 105 29)(14 126 106 34)(15 123 107 31)(16 128 108 36)(17 125 109 33)(18 122 110 30)(19 127 111 35)(20 124 112 32)(37 85 69 53)(38 90 70 58)(39 87 71 55)(40 92 72 60)(41 89 73 57)(42 86 74 54)(43 91 75 59)(44 88 76 56)(45 93 77 61)(47 95 79 63)(50 94 82 62)(52 96 84 64)
(1 19 11 15)(2 84 12 80)(3 17 9 13)(4 82 10 78)(5 89 93 85)(6 26 94 22)(7 91 95 87)(8 28 96 24)(14 70 18 74)(16 72 20 76)(21 33 25 29)(23 35 27 31)(30 86 34 90)(32 88 36 92)(37 49 41 45)(38 110 42 106)(39 51 43 47)(40 112 44 108)(46 98 50 102)(48 100 52 104)(53 68 57 61)(54 126 58 122)(55 66 59 63)(56 128 60 124)(62 114 65 118)(64 116 67 120)(69 81 73 77)(71 83 75 79)(97 109 101 105)(99 111 103 107)(113 125 117 121)(115 127 119 123)
G:=sub<Sym(128)| (1,11)(2,12)(3,9)(4,10)(5,93)(6,94)(7,95)(8,96)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28)(29,33)(30,34)(31,35)(32,36)(37,41)(38,42)(39,43)(40,44)(45,49)(46,50)(47,51)(48,52)(53,57)(54,58)(55,59)(56,60)(61,68)(62,65)(63,66)(64,67)(69,73)(70,74)(71,75)(72,76)(77,81)(78,82)(79,83)(80,84)(85,89)(86,90)(87,91)(88,92)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128), (1,71)(2,72)(3,69)(4,70)(5,33)(6,34)(7,35)(8,36)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(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)(65,126)(66,127)(67,128)(68,125), (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), (1,99)(2,100)(3,97)(4,98)(5,68)(6,65)(7,66)(8,67)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,69)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96), (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), (1,115,99,23)(2,120,100,28)(3,113,97,21)(4,118,98,26)(5,81,68,49)(6,78,65,46)(7,83,66,51)(8,80,67,48)(9,117,101,25)(10,114,102,22)(11,119,103,27)(12,116,104,24)(13,121,105,29)(14,126,106,34)(15,123,107,31)(16,128,108,36)(17,125,109,33)(18,122,110,30)(19,127,111,35)(20,124,112,32)(37,85,69,53)(38,90,70,58)(39,87,71,55)(40,92,72,60)(41,89,73,57)(42,86,74,54)(43,91,75,59)(44,88,76,56)(45,93,77,61)(47,95,79,63)(50,94,82,62)(52,96,84,64), (1,19,11,15)(2,84,12,80)(3,17,9,13)(4,82,10,78)(5,89,93,85)(6,26,94,22)(7,91,95,87)(8,28,96,24)(14,70,18,74)(16,72,20,76)(21,33,25,29)(23,35,27,31)(30,86,34,90)(32,88,36,92)(37,49,41,45)(38,110,42,106)(39,51,43,47)(40,112,44,108)(46,98,50,102)(48,100,52,104)(53,68,57,61)(54,126,58,122)(55,66,59,63)(56,128,60,124)(62,114,65,118)(64,116,67,120)(69,81,73,77)(71,83,75,79)(97,109,101,105)(99,111,103,107)(113,125,117,121)(115,127,119,123)>;
G:=Group( (1,11)(2,12)(3,9)(4,10)(5,93)(6,94)(7,95)(8,96)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28)(29,33)(30,34)(31,35)(32,36)(37,41)(38,42)(39,43)(40,44)(45,49)(46,50)(47,51)(48,52)(53,57)(54,58)(55,59)(56,60)(61,68)(62,65)(63,66)(64,67)(69,73)(70,74)(71,75)(72,76)(77,81)(78,82)(79,83)(80,84)(85,89)(86,90)(87,91)(88,92)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128), (1,71)(2,72)(3,69)(4,70)(5,33)(6,34)(7,35)(8,36)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(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)(65,126)(66,127)(67,128)(68,125), (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), (1,99)(2,100)(3,97)(4,98)(5,68)(6,65)(7,66)(8,67)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,69)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96), (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), (1,115,99,23)(2,120,100,28)(3,113,97,21)(4,118,98,26)(5,81,68,49)(6,78,65,46)(7,83,66,51)(8,80,67,48)(9,117,101,25)(10,114,102,22)(11,119,103,27)(12,116,104,24)(13,121,105,29)(14,126,106,34)(15,123,107,31)(16,128,108,36)(17,125,109,33)(18,122,110,30)(19,127,111,35)(20,124,112,32)(37,85,69,53)(38,90,70,58)(39,87,71,55)(40,92,72,60)(41,89,73,57)(42,86,74,54)(43,91,75,59)(44,88,76,56)(45,93,77,61)(47,95,79,63)(50,94,82,62)(52,96,84,64), (1,19,11,15)(2,84,12,80)(3,17,9,13)(4,82,10,78)(5,89,93,85)(6,26,94,22)(7,91,95,87)(8,28,96,24)(14,70,18,74)(16,72,20,76)(21,33,25,29)(23,35,27,31)(30,86,34,90)(32,88,36,92)(37,49,41,45)(38,110,42,106)(39,51,43,47)(40,112,44,108)(46,98,50,102)(48,100,52,104)(53,68,57,61)(54,126,58,122)(55,66,59,63)(56,128,60,124)(62,114,65,118)(64,116,67,120)(69,81,73,77)(71,83,75,79)(97,109,101,105)(99,111,103,107)(113,125,117,121)(115,127,119,123) );
G=PermutationGroup([[(1,11),(2,12),(3,9),(4,10),(5,93),(6,94),(7,95),(8,96),(13,17),(14,18),(15,19),(16,20),(21,25),(22,26),(23,27),(24,28),(29,33),(30,34),(31,35),(32,36),(37,41),(38,42),(39,43),(40,44),(45,49),(46,50),(47,51),(48,52),(53,57),(54,58),(55,59),(56,60),(61,68),(62,65),(63,66),(64,67),(69,73),(70,74),(71,75),(72,76),(77,81),(78,82),(79,83),(80,84),(85,89),(86,90),(87,91),(88,92),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128)], [(1,71),(2,72),(3,69),(4,70),(5,33),(6,34),(7,35),(8,36),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,81),(18,82),(19,83),(20,84),(21,85),(22,86),(23,87),(24,88),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(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),(65,126),(66,127),(67,128),(68,125)], [(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)], [(1,99),(2,100),(3,97),(4,98),(5,68),(6,65),(7,66),(8,67),(9,101),(10,102),(11,103),(12,104),(13,105),(14,106),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,113),(22,114),(23,115),(24,116),(25,117),(26,118),(27,119),(28,120),(29,121),(30,122),(31,123),(32,124),(33,125),(34,126),(35,127),(36,128),(37,69),(38,70),(39,71),(40,72),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,85),(54,86),(55,87),(56,88),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96)], [(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)], [(1,115,99,23),(2,120,100,28),(3,113,97,21),(4,118,98,26),(5,81,68,49),(6,78,65,46),(7,83,66,51),(8,80,67,48),(9,117,101,25),(10,114,102,22),(11,119,103,27),(12,116,104,24),(13,121,105,29),(14,126,106,34),(15,123,107,31),(16,128,108,36),(17,125,109,33),(18,122,110,30),(19,127,111,35),(20,124,112,32),(37,85,69,53),(38,90,70,58),(39,87,71,55),(40,92,72,60),(41,89,73,57),(42,86,74,54),(43,91,75,59),(44,88,76,56),(45,93,77,61),(47,95,79,63),(50,94,82,62),(52,96,84,64)], [(1,19,11,15),(2,84,12,80),(3,17,9,13),(4,82,10,78),(5,89,93,85),(6,26,94,22),(7,91,95,87),(8,28,96,24),(14,70,18,74),(16,72,20,76),(21,33,25,29),(23,35,27,31),(30,86,34,90),(32,88,36,92),(37,49,41,45),(38,110,42,106),(39,51,43,47),(40,112,44,108),(46,98,50,102),(48,100,52,104),(53,68,57,61),(54,126,58,122),(55,66,59,63),(56,128,60,124),(62,114,65,118),(64,116,67,120),(69,81,73,77),(71,83,75,79),(97,109,101,105),(99,111,103,107),(113,125,117,121),(115,127,119,123)]])
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 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | C4○D4 |
| kernel | C24.624C23 | C2×C2.C42 | C22×C42 | C2.C42 | C2×C42 | C22×C4 | C22×C4 | C23 |
| # reps | 1 | 6 | 1 | 16 | 8 | 6 | 2 | 16 |
Matrix representation of C24.624C23 ►in 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 |
| 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 | 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 |
| 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 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 2 | 1 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 4 | 0 | 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 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
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],[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],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,2,0,0,0,0,4,1],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,3,2],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;
C24.624C23 in GAP, Magma, Sage, TeX
C_2^4._{624}C_2^3 % in TeX
G:=Group("C2^4.624C2^3"); // GroupNames label
G:=SmallGroup(128,166);
// by ID
G=gap.SmallGroup(128,166);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,2,224,141,456,422,58]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=c,f^2=d,g^2=a,a*b=b*a,a*c=c*a,a*d=d*a,f*e*f^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,g*e*g^-1=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,d*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
// generators/relations