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G = C23.676C24order 128 = 27

393rd central stem extension by C23 of C24

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

Aliases: C23.676C24, C22.3422- 1+4, C22.4492+ 1+4, C425C4.18C2, (C22×C4).212C23, (C2×C42).707C22, C23.84C23.7C2, C23.65C23.81C2, C23.83C23.38C2, C23.63C23.51C2, C23.81C23.41C2, C2.C42.380C22, C2.99(C22.33C24), C2.47(C22.57C24), C2.58(C22.35C24), C2.66(C22.50C24), C2.105(C22.47C24), C2.114(C22.46C24), (C2×C4).225(C4○D4), (C2×C4⋊C4).486C22, C22.537(C2×C4○D4), SmallGroup(128,1508)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.676C24
C1C2C22C23C22×C4C2×C42C23.65C23 — C23.676C24
C1C23 — C23.676C24
C1C23 — C23.676C24
C1C23 — C23.676C24

Generators and relations for C23.676C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=abc, 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 (82 characteristic)
C1, C2 [×7], C4 [×18], C22 [×7], C2×C4 [×6], C2×C4 [×42], C23, C42 [×4], C4⋊C4 [×20], C22×C4 [×15], C2.C42 [×20], C2×C42 [×3], C2×C4⋊C4 [×12], C425C4, C23.63C23 [×5], C23.65C23 [×3], C23.81C23 [×2], C23.83C23 [×3], C23.84C23, C23.676C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ 1+4, 2- 1+4 [×3], C22.33C24, C22.35C24 [×2], C22.46C24, C22.47C24, C22.50C24, C22.57C24, C23.676C24

Smallest permutation representation of C23.676C24
Regular action on 128 points
Generators in S128
(1 103)(2 104)(3 101)(4 102)(5 69)(6 70)(7 71)(8 72)(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 42)(38 43)(39 44)(40 41)(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)(65 128)(66 125)(67 126)(68 127)
(1 12)(2 9)(3 10)(4 11)(5 44)(6 41)(7 42)(8 43)(13 47)(14 48)(15 45)(16 46)(17 51)(18 52)(19 49)(20 50)(21 55)(22 56)(23 53)(24 54)(25 59)(26 60)(27 57)(28 58)(29 63)(30 64)(31 61)(32 62)(33 67)(34 68)(35 65)(36 66)(37 71)(38 72)(39 69)(40 70)(73 102)(74 103)(75 104)(76 101)(77 108)(78 105)(79 106)(80 107)(81 110)(82 111)(83 112)(84 109)(85 116)(86 113)(87 114)(88 115)(89 118)(90 119)(91 120)(92 117)(93 124)(94 121)(95 122)(96 123)(97 126)(98 127)(99 128)(100 125)
(1 76)(2 73)(3 74)(4 75)(5 37)(6 38)(7 39)(8 40)(9 102)(10 103)(11 104)(12 101)(13 106)(14 107)(15 108)(16 105)(17 112)(18 109)(19 110)(20 111)(21 114)(22 115)(23 116)(24 113)(25 120)(26 117)(27 118)(28 119)(29 122)(30 123)(31 124)(32 121)(33 128)(34 125)(35 126)(36 127)(41 72)(42 69)(43 70)(44 71)(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 26 12 60)(2 91 9 120)(3 28 10 58)(4 89 11 118)(5 114 44 87)(6 54 41 24)(7 116 42 85)(8 56 43 22)(13 63 47 29)(14 121 48 94)(15 61 45 31)(16 123 46 96)(17 126 51 97)(18 68 52 34)(19 128 49 99)(20 66 50 36)(21 71 55 37)(23 69 53 39)(25 73 59 102)(27 75 57 104)(30 78 64 105)(32 80 62 107)(33 81 67 110)(35 83 65 112)(38 86 72 113)(40 88 70 115)(74 119 103 90)(76 117 101 92)(77 124 108 93)(79 122 106 95)(82 127 111 98)(84 125 109 100)
(1 82 74 52)(2 112 75 19)(3 84 76 50)(4 110 73 17)(5 29 39 124)(6 64 40 94)(7 31 37 122)(8 62 38 96)(9 83 104 49)(10 109 101 20)(11 81 102 51)(12 111 103 18)(13 116 108 21)(14 86 105 56)(15 114 106 23)(16 88 107 54)(22 48 113 78)(24 46 115 80)(25 65 118 99)(26 36 119 125)(27 67 120 97)(28 34 117 127)(30 70 121 41)(32 72 123 43)(33 91 126 57)(35 89 128 59)(42 61 71 95)(44 63 69 93)(45 87 79 53)(47 85 77 55)(58 68 92 98)(60 66 90 100)
(1 108 103 47)(2 14 104 78)(3 106 101 45)(4 16 102 80)(5 66 69 125)(6 97 70 33)(7 68 71 127)(8 99 72 35)(9 48 75 105)(10 79 76 15)(11 46 73 107)(12 77 74 13)(17 54 81 115)(18 85 82 21)(19 56 83 113)(20 87 84 23)(22 112 86 49)(24 110 88 51)(25 123 89 62)(26 29 90 93)(27 121 91 64)(28 31 92 95)(30 57 94 120)(32 59 96 118)(34 37 98 42)(36 39 100 44)(38 65 43 128)(40 67 41 126)(50 114 109 53)(52 116 111 55)(58 61 117 122)(60 63 119 124)

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

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

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

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
kernelC23.676C24C425C4C23.63C23C23.65C23C23.81C23C23.83C23C23.84C23C2×C4C22C22
# reps11532311213

Matrix representation of C23.676C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
010000
400000
003000
000300
000022
000003
,
010000
100000
003300
004200
000030
000003
,
300000
030000
001100
000400
000011
000034
,
200000
030000
004000
000400
000022
000013

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

C23.676C24 in GAP, Magma, Sage, TeX

C_2^3._{676}C_2^4
% in TeX

G:=Group("C2^3.676C2^4");
// GroupNames label

G:=SmallGroup(128,1508);
// by ID

G=gap.SmallGroup(128,1508);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,232,758,723,268,1571,346,80]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=a*b*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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