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G = C23.699C24  order 128 = 27

416th central stem extension by C23 of C24

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

Aliases: C23.699C24, C22.3612- 1+4, C22.4722+ 1+4, (C2×Q8)⋊4Q8, C2.31(Q8⋊3Q8), (C2×C42).721C22, (C22×C4).218C23, C2.19(C23⋊2Q8), C22.164(C22×Q8), (C22×Q8).224C22, C23.78C23.26C2, C2.C42.403C22, C23.67C23.59C2, C23.83C23.42C2, C23.81C23.46C2, C23.63C23.56C2, C2.124(C22.45C24), C2.54(C22.57C24), C2.121(C22.36C24), (C2×C4).88(C2×Q8), (C2×C4).240(C4○D4), (C2×C4⋊C4).509C22, C22.560(C2×C4○D4), SmallGroup(128,1531)

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

Derived series
C1 — C23 — C23.699C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.67C23 — C23.699C24
Lower central
C1 — C23 — C23.699C24
Upper central
C1 — C23 — C23.699C24
Jennings
C1 — C23 — C23.699C24

Generators and relations for C23.699C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=ca=ac, e2=g2=a, f2=cb=bc, ab=ba, 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: 340 in 188 conjugacy classes, 96 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×Q8, C23.63C23, C23.67C23, C23.67C23, C23.78C23, C23.81C23, C23.83C23, C23.83C23, C23.699C24
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.36C24, C23⋊2Q8, C22.45C24, Q8⋊3Q8, C22.57C24, C23.699C24

Smallest permutation representation of C23.699C24
►Regular action on 128 points
Generators in S128
(1 10)(2 11)(3 12)(4 9)(5 69)(6 70)(7 71)(8 72)(13 84)(14 81)(15 82)(16 83)(17 86)(18 87)(19 88)(20 85)(21 92)(22 89)(23 90)(24 91)(25 94)(26 95)(27 96)(28 93)(29 100)(30 97)(31 98)(32 99)(33 78)(34 79)(35 80)(36 77)(37 75)(38 76)(39 73)(40 74)(41 102)(42 103)(43 104)(44 101)(45 108)(46 105)(47 106)(48 107)(49 110)(50 111)(51 112)(52 109)(53 116)(54 113)(55 114)(56 115)(57 118)(58 119)(59 120)(60 117)(61 124)(62 121)(63 122)(64 123)(65 128)(66 125)(67 126)(68 127)

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

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

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

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

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

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
kernelC23.699C24C23.63C23C23.67C23C23.78C23C23.81C23C23.83C23C2×Q8C2×C4C22C22
# reps1451234822

Matrix representation of C23.699C24 ►in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
200000
020000
001400
002400
000003
000020
,
430000
010000
003200
000200
000040
000004
,
300000
220000
004000
000400
000002
000020
,
400000
040000
002000
004300
000001
000010

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

C23.699C24 in GAP, Magma, Sage, TeX 

C_2^3._{699}C_2^4
 
% in TeX
 
G:=Group("C2^3.699C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1531);
 
// by ID
 
G=gap.SmallGroup(128,1531);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,520,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*a=a*c,e^2=g^2=a,f^2=c*b=b*c,a*b=b*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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