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G = C23.691C24  order 128 = 27

408th central stem extension by C23 of C24

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

Aliases: C23.691C24, C22.4642+ 1+4, C22.3542- 1+4, C4⋊C4.9Q8, C42⋊9C4.37C2, C2.30(Q8⋊3Q8), C2.64(D4⋊3Q8), (C2×C42).718C22, (C22×C4).217C23, C22.162(C22×Q8), C23.81C23.43C2, C2.C42.395C22, C23.63C23.53C2, C23.65C23.82C2, C23.83C23.40C2, C2.51(C22.57C24), C2.63(C22.34C24), C2.43(C23.41C23), C2.109(C22.33C24), C2.119(C22.46C24), (C2×C4).86(C2×Q8), (C2×C4).233(C4○D4), (C2×C4⋊C4).501C22, C22.552(C2×C4○D4), SmallGroup(128,1523)

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

Derived series
C1 — C23 — C23.691C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.63C23 — C23.691C24
Lower central
C1 — C23 — C23.691C24
Upper central
C1 — C23 — C23.691C24
Jennings
C1 — C23 — C23.691C24

Generators and relations for C23.691C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=cb=bc, e2=a, f2=b, g2=ba=ab, 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, fef-1=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

Subgroups: 308 in 180 conjugacy classes, 96 normal (82 characteristic)
C1, C2, C4, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C42⋊9C4, C23.63C23, C23.65C23, C23.81C23, C23.83C23, C23.691C24
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.33C24, C22.34C24, C23.41C23, C22.46C24, D4⋊3Q8, Q8⋊3Q8, C22.57C24, C23.691C24

Smallest permutation representation of C23.691C24
►Regular action on 128 points
Generators in S128
(1 39)(2 40)(3 37)(4 38)(5 69)(6 70)(7 71)(8 72)(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)(33 97)(34 98)(35 99)(36 100)(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)(65 125)(66 126)(67 127)(68 128)

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

(1 7)(2 8)(3 5)(4 6)(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)(65 97)(66 98)(67 99)(68 100)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111112244
type++++++-+-
imageC1C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC23.691C24C42⋊9C4C23.63C23C23.65C23C23.81C23C23.83C23C4⋊C4C2×C4C22C22
# reps1162514822

Matrix representation of C23.691C24 ►in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
200000
020000
000400
004000
000022
000003
,
400000
010000
003000
000200
000040
000004
,
010000
100000
001000
000100
000030
000042
,
100000
010000
000100
004000
000020
000013

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

C23.691C24 in GAP, Magma, Sage, TeX 

C_2^3._{691}C_2^4
 
% in TeX
 
G:=Group("C2^3.691C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1523);
 
// by ID
 
G=gap.SmallGroup(128,1523);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,120,758,723,184,1571,346,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*b=b*c,e^2=a,f^2=b,g^2=b*a=a*b,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,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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