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G = C23.669C24  order 128 = 27

386th central stem extension by C23 of C24

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C23.669C24, C22.3352- 1+4, C22.4422+ 1+4, C42⋊8C4.47C2, C42⋊5C4.17C2, (C22×C4).587C23, (C2×C42).701C22, C23.65C23.79C2, C2.C42.373C22, C23.63C23.49C2, C23.81C23.39C2, C23.83C23.36C2, C2.94(C22.33C24), C2.56(C22.35C24), C2.41(C22.57C24), C2.41(C22.49C24), C2.107(C22.46C24), (C2×C4).463(C4○D4), (C2×C4⋊C4).479C22, C22.530(C2×C4○D4), SmallGroup(128,1501)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C23.669C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.65C23 — C23.669C24
Lower central
C1 — C23 — C23.669C24
Upper central
C1 — C23 — C23.669C24
Jennings
C1 — C23 — C23.669C24

Generators and relations for C23.669C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=b, f2=ba=ab, g2=a, ac=ca, ede-1=ad=da, geg-1=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, gdg-1=abd, fg=gf >

Subgroups: 292 in 172 conjugacy classes, 88 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C22×C4, C22×C4, C2.C42, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊8C4, C42⋊5C4, C23.63C23, C23.65C23, C23.81C23, C23.81C23, C23.83C23, C23.83C23, C23.669C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.33C24, C22.35C24, C22.46C24, C22.49C24, C22.57C24, C23.669C24

Smallest permutation representation of C23.669C24
►Regular action on 128 points
Generators in S128
(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)

(1 115 99 85)(2 56 100 26)(3 113 97 87)(4 54 98 28)(5 24 126 50)(6 81 127 111)(7 22 128 52)(8 83 125 109)(9 55 39 25)(10 116 40 86)(11 53 37 27)(12 114 38 88)(13 59 43 29)(14 120 44 90)(15 57 41 31)(16 118 42 92)(17 63 47 33)(18 124 48 94)(19 61 45 35)(20 122 46 96)(21 65 51 71)(23 67 49 69)(30 74 60 104)(32 76 58 102)(34 78 64 108)(36 80 62 106)(66 112 72 82)(68 110 70 84)(73 119 103 89)(75 117 101 91)(77 123 107 93)(79 121 105 95)

(1 47 39 77)(2 18 40 108)(3 45 37 79)(4 20 38 106)(5 120 68 30)(6 91 65 57)(7 118 66 32)(8 89 67 59)(9 107 99 17)(10 78 100 48)(11 105 97 19)(12 80 98 46)(13 111 103 21)(14 82 104 52)(15 109 101 23)(16 84 102 50)(22 44 112 74)(24 42 110 76)(25 121 115 35)(26 96 116 62)(27 123 113 33)(28 94 114 64)(29 125 119 69)(31 127 117 71)(34 54 124 88)(36 56 122 86)(41 83 75 49)(43 81 73 51)(53 93 87 63)(55 95 85 61)(58 128 92 72)(60 126 90 70)

(1 13 9 73)(2 104 10 44)(3 15 11 75)(4 102 12 42)(5 122 70 62)(6 33 71 93)(7 124 72 64)(8 35 69 95)(14 100 74 40)(16 98 76 38)(17 81 77 21)(18 52 78 112)(19 83 79 23)(20 50 80 110)(22 108 82 48)(24 106 84 46)(25 29 85 89)(26 120 86 60)(27 31 87 91)(28 118 88 58)(30 56 90 116)(32 54 92 114)(34 128 94 66)(36 126 96 68)(37 101 97 41)(39 103 99 43)(45 109 105 49)(47 111 107 51)(53 57 113 117)(55 59 115 119)(61 67 121 125)(63 65 123 127)
 
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), (1,115,99,85)(2,56,100,26)(3,113,97,87)(4,54,98,28)(5,24,126,50)(6,81,127,111)(7,22,128,52)(8,83,125,109)(9,55,39,25)(10,116,40,86)(11,53,37,27)(12,114,38,88)(13,59,43,29)(14,120,44,90)(15,57,41,31)(16,118,42,92)(17,63,47,33)(18,124,48,94)(19,61,45,35)(20,122,46,96)(21,65,51,71)(23,67,49,69)(30,74,60,104)(32,76,58,102)(34,78,64,108)(36,80,62,106)(66,112,72,82)(68,110,70,84)(73,119,103,89)(75,117,101,91)(77,123,107,93)(79,121,105,95), (1,47,39,77)(2,18,40,108)(3,45,37,79)(4,20,38,106)(5,120,68,30)(6,91,65,57)(7,118,66,32)(8,89,67,59)(9,107,99,17)(10,78,100,48)(11,105,97,19)(12,80,98,46)(13,111,103,21)(14,82,104,52)(15,109,101,23)(16,84,102,50)(22,44,112,74)(24,42,110,76)(25,121,115,35)(26,96,116,62)(27,123,113,33)(28,94,114,64)(29,125,119,69)(31,127,117,71)(34,54,124,88)(36,56,122,86)(41,83,75,49)(43,81,73,51)(53,93,87,63)(55,95,85,61)(58,128,92,72)(60,126,90,70), (1,13,9,73)(2,104,10,44)(3,15,11,75)(4,102,12,42)(5,122,70,62)(6,33,71,93)(7,124,72,64)(8,35,69,95)(14,100,74,40)(16,98,76,38)(17,81,77,21)(18,52,78,112)(19,83,79,23)(20,50,80,110)(22,108,82,48)(24,106,84,46)(25,29,85,89)(26,120,86,60)(27,31,87,91)(28,118,88,58)(30,56,90,116)(32,54,92,114)(34,128,94,66)(36,126,96,68)(37,101,97,41)(39,103,99,43)(45,109,105,49)(47,111,107,51)(53,57,113,117)(55,59,115,119)(61,67,121,125)(63,65,123,127)>;
 
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), (1,115,99,85)(2,56,100,26)(3,113,97,87)(4,54,98,28)(5,24,126,50)(6,81,127,111)(7,22,128,52)(8,83,125,109)(9,55,39,25)(10,116,40,86)(11,53,37,27)(12,114,38,88)(13,59,43,29)(14,120,44,90)(15,57,41,31)(16,118,42,92)(17,63,47,33)(18,124,48,94)(19,61,45,35)(20,122,46,96)(21,65,51,71)(23,67,49,69)(30,74,60,104)(32,76,58,102)(34,78,64,108)(36,80,62,106)(66,112,72,82)(68,110,70,84)(73,119,103,89)(75,117,101,91)(77,123,107,93)(79,121,105,95), (1,47,39,77)(2,18,40,108)(3,45,37,79)(4,20,38,106)(5,120,68,30)(6,91,65,57)(7,118,66,32)(8,89,67,59)(9,107,99,17)(10,78,100,48)(11,105,97,19)(12,80,98,46)(13,111,103,21)(14,82,104,52)(15,109,101,23)(16,84,102,50)(22,44,112,74)(24,42,110,76)(25,121,115,35)(26,96,116,62)(27,123,113,33)(28,94,114,64)(29,125,119,69)(31,127,117,71)(34,54,124,88)(36,56,122,86)(41,83,75,49)(43,81,73,51)(53,93,87,63)(55,95,85,61)(58,128,92,72)(60,126,90,70), (1,13,9,73)(2,104,10,44)(3,15,11,75)(4,102,12,42)(5,122,70,62)(6,33,71,93)(7,124,72,64)(8,35,69,95)(14,100,74,40)(16,98,76,38)(17,81,77,21)(18,52,78,112)(19,83,79,23)(20,50,80,110)(22,108,82,48)(24,106,84,46)(25,29,85,89)(26,120,86,60)(27,31,87,91)(28,118,88,58)(30,56,90,116)(32,54,92,114)(34,128,94,66)(36,126,96,68)(37,101,97,41)(39,103,99,43)(45,109,105,49)(47,111,107,51)(53,57,113,117)(55,59,115,119)(61,67,121,125)(63,65,123,127) );
 
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)], [(1,115,99,85),(2,56,100,26),(3,113,97,87),(4,54,98,28),(5,24,126,50),(6,81,127,111),(7,22,128,52),(8,83,125,109),(9,55,39,25),(10,116,40,86),(11,53,37,27),(12,114,38,88),(13,59,43,29),(14,120,44,90),(15,57,41,31),(16,118,42,92),(17,63,47,33),(18,124,48,94),(19,61,45,35),(20,122,46,96),(21,65,51,71),(23,67,49,69),(30,74,60,104),(32,76,58,102),(34,78,64,108),(36,80,62,106),(66,112,72,82),(68,110,70,84),(73,119,103,89),(75,117,101,91),(77,123,107,93),(79,121,105,95)], [(1,47,39,77),(2,18,40,108),(3,45,37,79),(4,20,38,106),(5,120,68,30),(6,91,65,57),(7,118,66,32),(8,89,67,59),(9,107,99,17),(10,78,100,48),(11,105,97,19),(12,80,98,46),(13,111,103,21),(14,82,104,52),(15,109,101,23),(16,84,102,50),(22,44,112,74),(24,42,110,76),(25,121,115,35),(26,96,116,62),(27,123,113,33),(28,94,114,64),(29,125,119,69),(31,127,117,71),(34,54,124,88),(36,56,122,86),(41,83,75,49),(43,81,73,51),(53,93,87,63),(55,95,85,61),(58,128,92,72),(60,126,90,70)], [(1,13,9,73),(2,104,10,44),(3,15,11,75),(4,102,12,42),(5,122,70,62),(6,33,71,93),(7,124,72,64),(8,35,69,95),(14,100,74,40),(16,98,76,38),(17,81,77,21),(18,52,78,112),(19,83,79,23),(20,50,80,110),(22,108,82,48),(24,106,84,46),(25,29,85,89),(26,120,86,60),(27,31,87,91),(28,118,88,58),(30,56,90,116),(32,54,92,114),(34,128,94,66),(36,126,96,68),(37,101,97,41),(39,103,99,43),(45,109,105,49),(47,111,107,51),(53,57,113,117),(55,59,115,119),(61,67,121,125),(63,65,123,127)]])
 

32 conjugacy classes

class 1 2A···2G4A···4R4S···4X
order12···24···44···4
size11···14···48···8

32 irreducible representations

dim1111111244
type++++++++-
imageC1C2C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC23.669C24C42⋊8C4C42⋊5C4C23.63C23C23.65C23C23.81C23C23.83C23C2×C4C22C22
# reps11242331213

Matrix representation of C23.669C24 ►in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
010000
100000
003000
000300
000004
000040
,
200000
020000
002100
002300
000010
000004
,
300000
020000
004000
004100
000030
000003
,
100000
040000
004000
000400
000001
000040

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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,2,0,0,0,0,1,3,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,4,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
 

C23.669C24 in GAP, Magma, Sage, TeX 

C_2^3._{669}C_2^4
 
% in TeX
 
G:=Group("C2^3.669C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1501);
 
// by ID
 
G=gap.SmallGroup(128,1501);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,268,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=b,f^2=b*a=a*b,g^2=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,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,g*d*g^-1=a*b*d,f*g=g*f>;
 
// generators/relations
 

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