Copied to
clipboard

G = C23.634C24  order 128 = 27

351st central stem extension by C23 of C24

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

Aliases: C23.634C24, C22.4072+ 1+4, C22.3082- 1+4, C2.9Q82, C4⋊C4⋊4Q8, C2.35(D4×Q8), (C2×Q8).122D4, C2.58(C23⋊3D4), (C22×C4).892C23, (C2×C42).685C22, C22.443(C22×D4), C22.151(C22×Q8), (C22×Q8).203C22, C2.C42.340C22, C23.65C23.74C2, C23.81C23.34C2, C23.67C23.53C2, C23.63C23.43C2, C23.78C23.19C2, C2.35(C23.41C23), C2.32(C22.57C24), (C2×C4⋊Q8).39C2, (C2×C4).75(C2×Q8), (C2×C4).128(C2×D4), (C2×C4⋊C4).447C22, SmallGroup(128,1466)

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

Derived series
C1 — C23 — C23.634C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.63C23 — C23.634C24
Lower central
C1 — C23 — C23.634C24
Upper central
C1 — C23 — C23.634C24
Jennings
C1 — C23 — C23.634C24

Generators and relations for C23.634C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=g2=ba=ab, e2=a, f2=b, 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: 404 in 228 conjugacy classes, 112 normal (14 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C22×Q8, C22×Q8, C23.63C23, C23.65C23, C23.67C23, C23.78C23, C23.81C23, C2×C4⋊Q8, C23.634C24
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C22×D4, C22×Q8, 2+ 1+4, 2- 1+4, C23⋊3D4, C23.41C23, D4×Q8, Q82, C22.57C24, C23.634C24

Smallest permutation representation of C23.634C24
►Regular action on 128 points
Generators in S128
(1 9)(2 10)(3 11)(4 12)(5 69)(6 70)(7 71)(8 72)(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 75)(38 76)(39 73)(40 74)(41 103)(42 104)(43 101)(44 102)(45 107)(46 108)(47 105)(48 106)(49 111)(50 112)(51 109)(52 110)(53 115)(54 116)(55 113)(56 114)(57 119)(58 120)(59 117)(60 118)(61 123)(62 124)(63 121)(64 122)(65 128)(66 125)(67 126)(68 127)

(1 11)(2 12)(3 9)(4 10)(5 71)(6 72)(7 69)(8 70)(13 79)(14 80)(15 77)(16 78)(17 83)(18 84)(19 81)(20 82)(21 87)(22 88)(23 85)(24 86)(25 91)(26 92)(27 89)(28 90)(29 95)(30 96)(31 93)(32 94)(33 99)(34 100)(35 97)(36 98)(37 73)(38 74)(39 75)(40 76)(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)(65 126)(66 127)(67 128)(68 125)

(1 41)(2 42)(3 43)(4 44)(5 38)(6 39)(7 40)(8 37)(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)(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 98)(66 99)(67 100)(68 97)(69 76)(70 73)(71 74)(72 75)

(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 25 9 89)(2 90 10 26)(3 27 11 91)(4 92 12 28)(5 54 69 116)(6 113 70 55)(7 56 71 114)(8 115 72 53)(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)(21 75 85 37)(22 38 86 76)(23 73 87 39)(24 40 88 74)(41 119 103 57)(42 58 104 120)(43 117 101 59)(44 60 102 118)(45 123 107 61)(46 62 108 124)(47 121 105 63)(48 64 106 122)(49 127 111 68)(50 65 112 128)(51 125 109 66)(52 67 110 126)

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111112244
type+++++++-++-
imageC1C2C2C2C2C2C2Q8D42+ 1+42- 1+4
kernelC23.634C24C23.63C23C23.65C23C23.67C23C23.78C23C23.81C23C2×C4⋊Q8C4⋊C4C2×Q8C22C22
# reps14214228422

Matrix representation of C23.634C24 ►in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
400000
040000
002000
000300
000020
000013
,
010000
100000
001000
000100
000022
000003
,
400000
010000
000400
001000
000040
000004
,
100000
010000
000100
004000
000044
000021

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

C23.634C24 in GAP, Magma, Sage, TeX 

C_2^3._{634}C_2^4
 
% in TeX
 
G:=Group("C2^3.634C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1466);
 
// by ID
 
G=gap.SmallGroup(128,1466);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723,184,1571,346,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=g^2=b*a=a*b,e^2=a,f^2=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,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
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁