p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.488C24, C22.2702+ (1+4), C22.2002- (1+4), C4⋊C4⋊23Q8, C2.24(Q8⋊3Q8), C2.46(D4⋊3Q8), (C22×C4).114C23, (C2×C42).581C22, C22.123(C22×Q8), (C22×Q8).145C22, C2.61(C22.45C24), C23.63C23.32C2, C23.65C23.63C2, C23.83C23.19C2, C23.78C23.12C2, C23.81C23.22C2, C2.C42.222C22, C23.67C23.46C2, C2.32(C22.33C24), C2.46(C22.50C24), C2.34(C23.37C23), C2.92(C23.36C23), (C4×C4⋊C4).74C2, (C2×C4).64(C2×Q8), (C2×C4).527(C4○D4), (C2×C4⋊C4).879C22, C22.364(C2×C4○D4), SmallGroup(128,1320)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 324 in 194 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C4 [×22], C22 [×7], C2×C4 [×14], C2×C4 [×38], Q8 [×4], C23, C42 [×7], C4⋊C4 [×4], C4⋊C4 [×18], C22×C4 [×15], C2×Q8 [×5], C2.C42 [×18], C2×C42 [×5], C2×C4⋊C4 [×11], C22×Q8, C4×C4⋊C4 [×2], C23.63C23 [×6], C23.65C23, C23.67C23 [×2], C23.78C23, C23.81C23, C23.83C23 [×2], C23.488C24
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×8], C24, C22×Q8, C2×C4○D4 [×4], 2+ (1+4), 2- (1+4), C23.36C23, C23.37C23, C22.33C24, C22.45C24, D4⋊3Q8, C22.50C24, Q8⋊3Q8, C23.488C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=a, e2=abc, f2=c, 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 9)(2 10)(3 11)(4 12)(5 71)(6 72)(7 69)(8 70)(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)(41 103)(42 104)(43 101)(44 102)(45 107)(46 108)(47 105)(48 106)(49 111)(50 112)(51 109)(52 110)(53 115)(54 116)(55 113)(56 114)(57 119)(58 120)(59 117)(60 118)(61 123)(62 124)(63 121)(64 122)(65 126)(66 127)(67 128)(68 125)
(1 101)(2 102)(3 103)(4 104)(5 100)(6 97)(7 98)(8 99)(9 43)(10 44)(11 41)(12 42)(13 47)(14 48)(15 45)(16 46)(17 51)(18 52)(19 49)(20 50)(21 55)(22 56)(23 53)(24 54)(25 59)(26 60)(27 57)(28 58)(29 63)(30 64)(31 61)(32 62)(33 68)(34 65)(35 66)(36 67)(37 72)(38 69)(39 70)(40 71)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 121)(90 122)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)
(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 57 41 85)(2 60 42 88)(3 59 43 87)(4 58 44 86)(5 114 38 24)(6 113 39 23)(7 116 40 22)(8 115 37 21)(9 119 103 25)(10 118 104 28)(11 117 101 27)(12 120 102 26)(13 123 107 29)(14 122 108 32)(15 121 105 31)(16 124 106 30)(17 127 111 33)(18 126 112 36)(19 125 109 35)(20 128 110 34)(45 89 73 61)(46 92 74 64)(47 91 75 63)(48 90 76 62)(49 93 77 66)(50 96 78 65)(51 95 79 68)(52 94 80 67)(53 97 81 70)(54 100 82 69)(55 99 83 72)(56 98 84 71)
(1 109 101 77)(2 52 102 18)(3 111 103 79)(4 50 104 20)(5 32 100 62)(6 89 97 121)(7 30 98 64)(8 91 99 123)(9 51 43 17)(10 110 44 78)(11 49 41 19)(12 112 42 80)(13 55 47 21)(14 114 48 82)(15 53 45 23)(16 116 46 84)(22 74 56 106)(24 76 54 108)(25 33 59 68)(26 94 60 126)(27 35 57 66)(28 96 58 128)(29 37 63 72)(31 39 61 70)(34 118 65 86)(36 120 67 88)(38 122 69 90)(40 124 71 92)(73 113 105 81)(75 115 107 83)(85 93 117 125)(87 95 119 127)
(1 13 9 73)(2 16 10 76)(3 15 11 75)(4 14 12 74)(5 126 71 65)(6 125 72 68)(7 128 69 67)(8 127 70 66)(17 81 77 21)(18 84 78 24)(19 83 79 23)(20 82 80 22)(25 89 85 29)(26 92 86 32)(27 91 87 31)(28 90 88 30)(33 97 93 37)(34 100 94 40)(35 99 95 39)(36 98 96 38)(41 107 103 45)(42 106 104 48)(43 105 101 47)(44 108 102 46)(49 115 111 53)(50 114 112 56)(51 113 109 55)(52 116 110 54)(57 123 119 61)(58 122 120 64)(59 121 117 63)(60 124 118 62)
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), (1,9)(2,10)(3,11)(4,12)(5,71)(6,72)(7,69)(8,70)(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)(41,103)(42,104)(43,101)(44,102)(45,107)(46,108)(47,105)(48,106)(49,111)(50,112)(51,109)(52,110)(53,115)(54,116)(55,113)(56,114)(57,119)(58,120)(59,117)(60,118)(61,123)(62,124)(63,121)(64,122)(65,126)(66,127)(67,128)(68,125), (1,101)(2,102)(3,103)(4,104)(5,100)(6,97)(7,98)(8,99)(9,43)(10,44)(11,41)(12,42)(13,47)(14,48)(15,45)(16,46)(17,51)(18,52)(19,49)(20,50)(21,55)(22,56)(23,53)(24,54)(25,59)(26,60)(27,57)(28,58)(29,63)(30,64)(31,61)(32,62)(33,68)(34,65)(35,66)(36,67)(37,72)(38,69)(39,70)(40,71)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128), (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), (1,57,41,85)(2,60,42,88)(3,59,43,87)(4,58,44,86)(5,114,38,24)(6,113,39,23)(7,116,40,22)(8,115,37,21)(9,119,103,25)(10,118,104,28)(11,117,101,27)(12,120,102,26)(13,123,107,29)(14,122,108,32)(15,121,105,31)(16,124,106,30)(17,127,111,33)(18,126,112,36)(19,125,109,35)(20,128,110,34)(45,89,73,61)(46,92,74,64)(47,91,75,63)(48,90,76,62)(49,93,77,66)(50,96,78,65)(51,95,79,68)(52,94,80,67)(53,97,81,70)(54,100,82,69)(55,99,83,72)(56,98,84,71), (1,109,101,77)(2,52,102,18)(3,111,103,79)(4,50,104,20)(5,32,100,62)(6,89,97,121)(7,30,98,64)(8,91,99,123)(9,51,43,17)(10,110,44,78)(11,49,41,19)(12,112,42,80)(13,55,47,21)(14,114,48,82)(15,53,45,23)(16,116,46,84)(22,74,56,106)(24,76,54,108)(25,33,59,68)(26,94,60,126)(27,35,57,66)(28,96,58,128)(29,37,63,72)(31,39,61,70)(34,118,65,86)(36,120,67,88)(38,122,69,90)(40,124,71,92)(73,113,105,81)(75,115,107,83)(85,93,117,125)(87,95,119,127), (1,13,9,73)(2,16,10,76)(3,15,11,75)(4,14,12,74)(5,126,71,65)(6,125,72,68)(7,128,69,67)(8,127,70,66)(17,81,77,21)(18,84,78,24)(19,83,79,23)(20,82,80,22)(25,89,85,29)(26,92,86,32)(27,91,87,31)(28,90,88,30)(33,97,93,37)(34,100,94,40)(35,99,95,39)(36,98,96,38)(41,107,103,45)(42,106,104,48)(43,105,101,47)(44,108,102,46)(49,115,111,53)(50,114,112,56)(51,113,109,55)(52,116,110,54)(57,123,119,61)(58,122,120,64)(59,121,117,63)(60,124,118,62)>;
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), (1,9)(2,10)(3,11)(4,12)(5,71)(6,72)(7,69)(8,70)(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)(41,103)(42,104)(43,101)(44,102)(45,107)(46,108)(47,105)(48,106)(49,111)(50,112)(51,109)(52,110)(53,115)(54,116)(55,113)(56,114)(57,119)(58,120)(59,117)(60,118)(61,123)(62,124)(63,121)(64,122)(65,126)(66,127)(67,128)(68,125), (1,101)(2,102)(3,103)(4,104)(5,100)(6,97)(7,98)(8,99)(9,43)(10,44)(11,41)(12,42)(13,47)(14,48)(15,45)(16,46)(17,51)(18,52)(19,49)(20,50)(21,55)(22,56)(23,53)(24,54)(25,59)(26,60)(27,57)(28,58)(29,63)(30,64)(31,61)(32,62)(33,68)(34,65)(35,66)(36,67)(37,72)(38,69)(39,70)(40,71)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128), (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), (1,57,41,85)(2,60,42,88)(3,59,43,87)(4,58,44,86)(5,114,38,24)(6,113,39,23)(7,116,40,22)(8,115,37,21)(9,119,103,25)(10,118,104,28)(11,117,101,27)(12,120,102,26)(13,123,107,29)(14,122,108,32)(15,121,105,31)(16,124,106,30)(17,127,111,33)(18,126,112,36)(19,125,109,35)(20,128,110,34)(45,89,73,61)(46,92,74,64)(47,91,75,63)(48,90,76,62)(49,93,77,66)(50,96,78,65)(51,95,79,68)(52,94,80,67)(53,97,81,70)(54,100,82,69)(55,99,83,72)(56,98,84,71), (1,109,101,77)(2,52,102,18)(3,111,103,79)(4,50,104,20)(5,32,100,62)(6,89,97,121)(7,30,98,64)(8,91,99,123)(9,51,43,17)(10,110,44,78)(11,49,41,19)(12,112,42,80)(13,55,47,21)(14,114,48,82)(15,53,45,23)(16,116,46,84)(22,74,56,106)(24,76,54,108)(25,33,59,68)(26,94,60,126)(27,35,57,66)(28,96,58,128)(29,37,63,72)(31,39,61,70)(34,118,65,86)(36,120,67,88)(38,122,69,90)(40,124,71,92)(73,113,105,81)(75,115,107,83)(85,93,117,125)(87,95,119,127), (1,13,9,73)(2,16,10,76)(3,15,11,75)(4,14,12,74)(5,126,71,65)(6,125,72,68)(7,128,69,67)(8,127,70,66)(17,81,77,21)(18,84,78,24)(19,83,79,23)(20,82,80,22)(25,89,85,29)(26,92,86,32)(27,91,87,31)(28,90,88,30)(33,97,93,37)(34,100,94,40)(35,99,95,39)(36,98,96,38)(41,107,103,45)(42,106,104,48)(43,105,101,47)(44,108,102,46)(49,115,111,53)(50,114,112,56)(51,113,109,55)(52,116,110,54)(57,123,119,61)(58,122,120,64)(59,121,117,63)(60,124,118,62) );
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)], [(1,9),(2,10),(3,11),(4,12),(5,71),(6,72),(7,69),(8,70),(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),(41,103),(42,104),(43,101),(44,102),(45,107),(46,108),(47,105),(48,106),(49,111),(50,112),(51,109),(52,110),(53,115),(54,116),(55,113),(56,114),(57,119),(58,120),(59,117),(60,118),(61,123),(62,124),(63,121),(64,122),(65,126),(66,127),(67,128),(68,125)], [(1,101),(2,102),(3,103),(4,104),(5,100),(6,97),(7,98),(8,99),(9,43),(10,44),(11,41),(12,42),(13,47),(14,48),(15,45),(16,46),(17,51),(18,52),(19,49),(20,50),(21,55),(22,56),(23,53),(24,54),(25,59),(26,60),(27,57),(28,58),(29,63),(30,64),(31,61),(32,62),(33,68),(34,65),(35,66),(36,67),(37,72),(38,69),(39,70),(40,71),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112),(81,113),(82,114),(83,115),(84,116),(85,117),(86,118),(87,119),(88,120),(89,121),(90,122),(91,123),(92,124),(93,125),(94,126),(95,127),(96,128)], [(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)], [(1,57,41,85),(2,60,42,88),(3,59,43,87),(4,58,44,86),(5,114,38,24),(6,113,39,23),(7,116,40,22),(8,115,37,21),(9,119,103,25),(10,118,104,28),(11,117,101,27),(12,120,102,26),(13,123,107,29),(14,122,108,32),(15,121,105,31),(16,124,106,30),(17,127,111,33),(18,126,112,36),(19,125,109,35),(20,128,110,34),(45,89,73,61),(46,92,74,64),(47,91,75,63),(48,90,76,62),(49,93,77,66),(50,96,78,65),(51,95,79,68),(52,94,80,67),(53,97,81,70),(54,100,82,69),(55,99,83,72),(56,98,84,71)], [(1,109,101,77),(2,52,102,18),(3,111,103,79),(4,50,104,20),(5,32,100,62),(6,89,97,121),(7,30,98,64),(8,91,99,123),(9,51,43,17),(10,110,44,78),(11,49,41,19),(12,112,42,80),(13,55,47,21),(14,114,48,82),(15,53,45,23),(16,116,46,84),(22,74,56,106),(24,76,54,108),(25,33,59,68),(26,94,60,126),(27,35,57,66),(28,96,58,128),(29,37,63,72),(31,39,61,70),(34,118,65,86),(36,120,67,88),(38,122,69,90),(40,124,71,92),(73,113,105,81),(75,115,107,83),(85,93,117,125),(87,95,119,127)], [(1,13,9,73),(2,16,10,76),(3,15,11,75),(4,14,12,74),(5,126,71,65),(6,125,72,68),(7,128,69,67),(8,127,70,66),(17,81,77,21),(18,84,78,24),(19,83,79,23),(20,82,80,22),(25,89,85,29),(26,92,86,32),(27,91,87,31),(28,90,88,30),(33,97,93,37),(34,100,94,40),(35,99,95,39),(36,98,96,38),(41,107,103,45),(42,106,104,48),(43,105,101,47),(44,108,102,46),(49,115,111,53),(50,114,112,56),(51,113,109,55),(52,116,110,54),(57,123,119,61),(58,122,120,64),(59,121,117,63),(60,124,118,62)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 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 | 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 |
| 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 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 3 | 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 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[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],[4,1,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,4,0,0,0,0,0,4,0,0,0,0,0,0,3,2,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[4,1,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,4,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,3] >;
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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C23.488C24 | C4×C4⋊C4 | C23.63C23 | C23.65C23 | C23.67C23 | C23.78C23 | C23.81C23 | C23.83C23 | C4⋊C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 6 | 1 | 2 | 1 | 1 | 2 | 4 | 16 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_2^3._{488}C_2^4 % in TeX
G:=Group("C2^3.488C2^4"); // GroupNames label
G:=SmallGroup(128,1320);
// by ID
G=gap.SmallGroup(128,1320);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,792,758,723,352,675,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=a,e^2=a*b*c,f^2=c,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