p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.251C24, C22.832+ (1+4), C4⋊C4⋊26Q8, C4⋊Q8⋊33C4, C2.2(Q82), C4.32(C4×Q8), C42.193(C2×C4), C2.7(D4⋊3Q8), C22.46(C22×Q8), (C22×C4).770C23, C22.142(C23×C4), (C2×C42).441C22, (C22×Q8).92C22, C2.34(C22.11C24), C2.C42.527C22, C23.67C23.30C2, C23.65C23.36C2, C2.4(C22.49C24), C4⋊C4○7(C4⋊C4), C2.21(C2×C4×Q8), (C4×C4⋊C4).43C2, (C2×C4⋊Q8).27C2, C4⋊C4.158(C2×C4), (C2×C4).305(C2×Q8), (C2×C4).49(C22×C4), (C2×Q8).112(C2×C4), (C2×C4).803(C4○D4), (C2×C4⋊C4).191C22, C22.136(C2×C4○D4), SmallGroup(128,1101)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 380 in 248 conjugacy classes, 164 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×22], C22 [×3], C22 [×4], C2×C4 [×30], C2×C4 [×30], Q8 [×8], C23, C42 [×4], C42 [×12], C4⋊C4 [×24], C4⋊C4 [×8], C22×C4, C22×C4 [×14], C2×Q8 [×8], C2×Q8 [×4], C2.C42 [×12], C2×C42, C2×C42 [×8], C2×C4⋊C4 [×12], C4⋊Q8 [×8], C22×Q8 [×2], C4×C4⋊C4 [×6], C23.65C23 [×4], C23.67C23 [×4], C2×C4⋊Q8, C23.251C24
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×8], C23 [×15], C22×C4 [×14], C2×Q8 [×12], C4○D4 [×4], C24, C4×Q8 [×8], C23×C4, C22×Q8 [×2], C2×C4○D4 [×2], 2+ (1+4) [×2], C2×C4×Q8 [×2], C22.11C24, D4⋊3Q8 [×2], C22.49C24, Q82, C23.251C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=cb=bc, f2=b, g2=a, ab=ba, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
(1 9)(2 10)(3 11)(4 12)(5 70)(6 71)(7 72)(8 69)(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)(41 101)(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)(65 127)(66 128)(67 125)(68 126)
(1 99)(2 100)(3 97)(4 98)(5 126)(6 127)(7 128)(8 125)(9 39)(10 40)(11 37)(12 38)(13 43)(14 44)(15 41)(16 42)(17 47)(18 48)(19 45)(20 46)(21 51)(22 52)(23 49)(24 50)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 63)(34 64)(35 61)(36 62)(65 71)(66 72)(67 69)(68 70)(73 103)(74 104)(75 101)(76 102)(77 107)(78 108)(79 105)(80 106)(81 111)(82 112)(83 109)(84 110)(85 115)(86 116)(87 113)(88 114)(89 119)(90 120)(91 117)(92 118)(93 123)(94 124)(95 121)(96 122)
(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)
(1 113 97 85)(2 54 98 26)(3 115 99 87)(4 56 100 28)(5 24 128 52)(6 81 125 109)(7 22 126 50)(8 83 127 111)(9 53 37 25)(10 114 38 86)(11 55 39 27)(12 116 40 88)(13 57 41 29)(14 118 42 90)(15 59 43 31)(16 120 44 92)(17 61 45 33)(18 122 46 94)(19 63 47 35)(20 124 48 96)(21 67 49 71)(23 65 51 69)(30 74 58 102)(32 76 60 104)(34 78 62 106)(36 80 64 108)(66 112 70 84)(68 110 72 82)(73 117 101 89)(75 119 103 91)(77 121 105 93)(79 123 107 95)
(1 107 99 77)(2 78 100 108)(3 105 97 79)(4 80 98 106)(5 118 126 92)(6 89 127 119)(7 120 128 90)(8 91 125 117)(9 47 39 17)(10 18 40 48)(11 45 37 19)(12 20 38 46)(13 51 43 21)(14 22 44 52)(15 49 41 23)(16 24 42 50)(25 63 55 33)(26 34 56 64)(27 61 53 35)(28 36 54 62)(29 65 59 71)(30 72 60 66)(31 67 57 69)(32 70 58 68)(73 111 103 81)(74 82 104 112)(75 109 101 83)(76 84 102 110)(85 123 115 93)(86 94 116 124)(87 121 113 95)(88 96 114 122)
(1 13 9 73)(2 14 10 74)(3 15 11 75)(4 16 12 76)(5 36 70 96)(6 33 71 93)(7 34 72 94)(8 35 69 95)(17 81 77 21)(18 82 78 22)(19 83 79 23)(20 84 80 24)(25 29 85 89)(26 30 86 90)(27 31 87 91)(28 32 88 92)(37 101 97 41)(38 102 98 42)(39 103 99 43)(40 104 100 44)(45 109 105 49)(46 110 106 50)(47 111 107 51)(48 112 108 52)(53 57 113 117)(54 58 114 118)(55 59 115 119)(56 60 116 120)(61 67 121 125)(62 68 122 126)(63 65 123 127)(64 66 124 128)
G:=sub<Sym(128)| (1,9)(2,10)(3,11)(4,12)(5,70)(6,71)(7,72)(8,69)(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)(41,101)(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)(65,127)(66,128)(67,125)(68,126), (1,99)(2,100)(3,97)(4,98)(5,126)(6,127)(7,128)(8,125)(9,39)(10,40)(11,37)(12,38)(13,43)(14,44)(15,41)(16,42)(17,47)(18,48)(19,45)(20,46)(21,51)(22,52)(23,49)(24,50)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,63)(34,64)(35,61)(36,62)(65,71)(66,72)(67,69)(68,70)(73,103)(74,104)(75,101)(76,102)(77,107)(78,108)(79,105)(80,106)(81,111)(82,112)(83,109)(84,110)(85,115)(86,116)(87,113)(88,114)(89,119)(90,120)(91,117)(92,118)(93,123)(94,124)(95,121)(96,122), (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), (1,113,97,85)(2,54,98,26)(3,115,99,87)(4,56,100,28)(5,24,128,52)(6,81,125,109)(7,22,126,50)(8,83,127,111)(9,53,37,25)(10,114,38,86)(11,55,39,27)(12,116,40,88)(13,57,41,29)(14,118,42,90)(15,59,43,31)(16,120,44,92)(17,61,45,33)(18,122,46,94)(19,63,47,35)(20,124,48,96)(21,67,49,71)(23,65,51,69)(30,74,58,102)(32,76,60,104)(34,78,62,106)(36,80,64,108)(66,112,70,84)(68,110,72,82)(73,117,101,89)(75,119,103,91)(77,121,105,93)(79,123,107,95), (1,107,99,77)(2,78,100,108)(3,105,97,79)(4,80,98,106)(5,118,126,92)(6,89,127,119)(7,120,128,90)(8,91,125,117)(9,47,39,17)(10,18,40,48)(11,45,37,19)(12,20,38,46)(13,51,43,21)(14,22,44,52)(15,49,41,23)(16,24,42,50)(25,63,55,33)(26,34,56,64)(27,61,53,35)(28,36,54,62)(29,65,59,71)(30,72,60,66)(31,67,57,69)(32,70,58,68)(73,111,103,81)(74,82,104,112)(75,109,101,83)(76,84,102,110)(85,123,115,93)(86,94,116,124)(87,121,113,95)(88,96,114,122), (1,13,9,73)(2,14,10,74)(3,15,11,75)(4,16,12,76)(5,36,70,96)(6,33,71,93)(7,34,72,94)(8,35,69,95)(17,81,77,21)(18,82,78,22)(19,83,79,23)(20,84,80,24)(25,29,85,89)(26,30,86,90)(27,31,87,91)(28,32,88,92)(37,101,97,41)(38,102,98,42)(39,103,99,43)(40,104,100,44)(45,109,105,49)(46,110,106,50)(47,111,107,51)(48,112,108,52)(53,57,113,117)(54,58,114,118)(55,59,115,119)(56,60,116,120)(61,67,121,125)(62,68,122,126)(63,65,123,127)(64,66,124,128)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,70)(6,71)(7,72)(8,69)(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)(41,101)(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)(65,127)(66,128)(67,125)(68,126), (1,99)(2,100)(3,97)(4,98)(5,126)(6,127)(7,128)(8,125)(9,39)(10,40)(11,37)(12,38)(13,43)(14,44)(15,41)(16,42)(17,47)(18,48)(19,45)(20,46)(21,51)(22,52)(23,49)(24,50)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,63)(34,64)(35,61)(36,62)(65,71)(66,72)(67,69)(68,70)(73,103)(74,104)(75,101)(76,102)(77,107)(78,108)(79,105)(80,106)(81,111)(82,112)(83,109)(84,110)(85,115)(86,116)(87,113)(88,114)(89,119)(90,120)(91,117)(92,118)(93,123)(94,124)(95,121)(96,122), (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), (1,113,97,85)(2,54,98,26)(3,115,99,87)(4,56,100,28)(5,24,128,52)(6,81,125,109)(7,22,126,50)(8,83,127,111)(9,53,37,25)(10,114,38,86)(11,55,39,27)(12,116,40,88)(13,57,41,29)(14,118,42,90)(15,59,43,31)(16,120,44,92)(17,61,45,33)(18,122,46,94)(19,63,47,35)(20,124,48,96)(21,67,49,71)(23,65,51,69)(30,74,58,102)(32,76,60,104)(34,78,62,106)(36,80,64,108)(66,112,70,84)(68,110,72,82)(73,117,101,89)(75,119,103,91)(77,121,105,93)(79,123,107,95), (1,107,99,77)(2,78,100,108)(3,105,97,79)(4,80,98,106)(5,118,126,92)(6,89,127,119)(7,120,128,90)(8,91,125,117)(9,47,39,17)(10,18,40,48)(11,45,37,19)(12,20,38,46)(13,51,43,21)(14,22,44,52)(15,49,41,23)(16,24,42,50)(25,63,55,33)(26,34,56,64)(27,61,53,35)(28,36,54,62)(29,65,59,71)(30,72,60,66)(31,67,57,69)(32,70,58,68)(73,111,103,81)(74,82,104,112)(75,109,101,83)(76,84,102,110)(85,123,115,93)(86,94,116,124)(87,121,113,95)(88,96,114,122), (1,13,9,73)(2,14,10,74)(3,15,11,75)(4,16,12,76)(5,36,70,96)(6,33,71,93)(7,34,72,94)(8,35,69,95)(17,81,77,21)(18,82,78,22)(19,83,79,23)(20,84,80,24)(25,29,85,89)(26,30,86,90)(27,31,87,91)(28,32,88,92)(37,101,97,41)(38,102,98,42)(39,103,99,43)(40,104,100,44)(45,109,105,49)(46,110,106,50)(47,111,107,51)(48,112,108,52)(53,57,113,117)(54,58,114,118)(55,59,115,119)(56,60,116,120)(61,67,121,125)(62,68,122,126)(63,65,123,127)(64,66,124,128) );
G=PermutationGroup([(1,9),(2,10),(3,11),(4,12),(5,70),(6,71),(7,72),(8,69),(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),(41,101),(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),(65,127),(66,128),(67,125),(68,126)], [(1,99),(2,100),(3,97),(4,98),(5,126),(6,127),(7,128),(8,125),(9,39),(10,40),(11,37),(12,38),(13,43),(14,44),(15,41),(16,42),(17,47),(18,48),(19,45),(20,46),(21,51),(22,52),(23,49),(24,50),(25,55),(26,56),(27,53),(28,54),(29,59),(30,60),(31,57),(32,58),(33,63),(34,64),(35,61),(36,62),(65,71),(66,72),(67,69),(68,70),(73,103),(74,104),(75,101),(76,102),(77,107),(78,108),(79,105),(80,106),(81,111),(82,112),(83,109),(84,110),(85,115),(86,116),(87,113),(88,114),(89,119),(90,120),(91,117),(92,118),(93,123),(94,124),(95,121),(96,122)], [(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)], [(1,113,97,85),(2,54,98,26),(3,115,99,87),(4,56,100,28),(5,24,128,52),(6,81,125,109),(7,22,126,50),(8,83,127,111),(9,53,37,25),(10,114,38,86),(11,55,39,27),(12,116,40,88),(13,57,41,29),(14,118,42,90),(15,59,43,31),(16,120,44,92),(17,61,45,33),(18,122,46,94),(19,63,47,35),(20,124,48,96),(21,67,49,71),(23,65,51,69),(30,74,58,102),(32,76,60,104),(34,78,62,106),(36,80,64,108),(66,112,70,84),(68,110,72,82),(73,117,101,89),(75,119,103,91),(77,121,105,93),(79,123,107,95)], [(1,107,99,77),(2,78,100,108),(3,105,97,79),(4,80,98,106),(5,118,126,92),(6,89,127,119),(7,120,128,90),(8,91,125,117),(9,47,39,17),(10,18,40,48),(11,45,37,19),(12,20,38,46),(13,51,43,21),(14,22,44,52),(15,49,41,23),(16,24,42,50),(25,63,55,33),(26,34,56,64),(27,61,53,35),(28,36,54,62),(29,65,59,71),(30,72,60,66),(31,67,57,69),(32,70,58,68),(73,111,103,81),(74,82,104,112),(75,109,101,83),(76,84,102,110),(85,123,115,93),(86,94,116,124),(87,121,113,95),(88,96,114,122)], [(1,13,9,73),(2,14,10,74),(3,15,11,75),(4,16,12,76),(5,36,70,96),(6,33,71,93),(7,34,72,94),(8,35,69,95),(17,81,77,21),(18,82,78,22),(19,83,79,23),(20,84,80,24),(25,29,85,89),(26,30,86,90),(27,31,87,91),(28,32,88,92),(37,101,97,41),(38,102,98,42),(39,103,99,43),(40,104,100,44),(45,109,105,49),(46,110,106,50),(47,111,107,51),(48,112,108,52),(53,57,113,117),(54,58,114,118),(55,59,115,119),(56,60,116,120),(61,67,121,125),(62,68,122,126),(63,65,123,127),(64,66,124,128)])
Matrix representation ►G ⊆ GL5(𝔽5)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 4 | 0 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[3,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,1,0],[3,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,3],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4] >;
50 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4X | 4Y | ··· | 4AP |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | Q8 | C4○D4 | 2+ (1+4) |
| kernel | C23.251C24 | C4×C4⋊C4 | C23.65C23 | C23.67C23 | C2×C4⋊Q8 | C4⋊Q8 | C4⋊C4 | C2×C4 | C22 |
| # reps | 1 | 6 | 4 | 4 | 1 | 16 | 8 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_2^3._{251}C_2^4 % in TeX
G:=Group("C2^3.251C2^4"); // GroupNames label
G:=SmallGroup(128,1101);
// by ID
G=gap.SmallGroup(128,1101);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,680,758,184,346,80]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=c*b=b*c,f^2=b,g^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g^-1=a*e=e*a,a*f=f*a,a*g=g*a,f*d*f^-1=b*d=d*b,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*g=g*d,e*f=f*e,f*g=g*f>;
// generators/relations