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G = C23.674C24  order 128 = 27

391st central stem extension by C23 of C24

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

Aliases: C23.674C24, C22.4472+ 1+4, C22.3402- 1+4, C4⋊C4⋊6Q8, C42⋊8C4.48C2, C2.59(D4⋊3Q8), C2.29(Q8⋊3Q8), (C22×C4).211C23, (C2×C42).705C22, C22.157(C22×Q8), (C22×Q8).216C22, C2.96(C22.32C24), C23.63C23.50C2, C2.C42.378C22, C23.67C23.57C2, C23.65C23.80C2, C23.81C23.40C2, C23.78C23.23C2, C23.83C23.37C2, C2.40(C23.41C23), C2.45(C22.57C24), C2.112(C22.46C24), C2.112(C22.36C24), (C2×C4).81(C2×Q8), (C2×C4).465(C4○D4), (C2×C4⋊C4).484C22, C22.535(C2×C4○D4), SmallGroup(128,1506)

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

Derived series
C1 — C23 — C23.674C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.65C23 — C23.674C24
Lower central
C1 — C23 — C23.674C24
Upper central
C1 — C23 — C23.674C24
Jennings
C1 — C23 — C23.674C24

Generators and relations for C23.674C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=abc, e2=ba=ab, f2=b, 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: 324 in 184 conjugacy classes, 96 normal (82 characteristic)
C1, C2, C4, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, C42⋊8C4, C23.63C23, C23.65C23, C23.67C23, C23.78C23, C23.81C23, C23.83C23, C23.674C24
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.32C24, C22.36C24, C23.41C23, C22.46C24, D4⋊3Q8, Q8⋊3Q8, C22.57C24, C23.674C24

Smallest permutation representation of C23.674C24
►Regular action on 128 points
Generators in S128
(1 103)(2 104)(3 101)(4 102)(5 125)(6 126)(7 127)(8 128)(9 75)(10 76)(11 73)(12 74)(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)(33 97)(34 98)(35 99)(36 100)(37 106)(38 107)(39 108)(40 105)(41 48)(42 45)(43 46)(44 47)(49 56)(50 53)(51 54)(52 55)(57 64)(58 61)(59 62)(60 63)(65 72)(66 69)(67 70)(68 71)(109 114)(110 115)(111 116)(112 113)(117 122)(118 123)(119 124)(120 121)

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

(1 12)(2 9)(3 10)(4 11)(5 34)(6 35)(7 36)(8 33)(13 42)(14 43)(15 44)(16 41)(17 113)(18 114)(19 115)(20 116)(21 50)(22 51)(23 52)(24 49)(25 121)(26 122)(27 123)(28 124)(29 58)(30 59)(31 60)(32 57)(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)(69 106)(70 107)(71 108)(72 105)(73 102)(74 103)(75 104)(76 101)(81 112)(82 109)(83 110)(84 111)(89 120)(90 117)(91 118)(92 119)(97 128)(98 125)(99 126)(100 127)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111112244
type++++++++-+-
imageC1C2C2C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC23.674C24C42⋊8C4C23.63C23C23.65C23C23.67C23C23.78C23C23.81C23C23.83C23C4⋊C4C2×C4C22C22
# reps114312134822

Matrix representation of C23.674C24 ►in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
030000
300000
002000
000200
000001
000040
,
300000
020000
003100
002200
000020
000002
,
100000
010000
001200
000400
000002
000020
,
010000
400000
001000
000100
000001
000010

G:=sub<GL(6,GF(5))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,2,0,0,0,0,1,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,4,0,0,0,0,0,0,0,2,0,0,0,0,2,0],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C23.674C24 in GAP, Magma, Sage, TeX 

C_2^3._{674}C_2^4
 
% in TeX
 
G:=Group("C2^3.674C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1506);
 
// by ID
 
G=gap.SmallGroup(128,1506);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,344,758,723,184,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=a*b*c,e^2=b*a=a*b,f^2=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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