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G = C23.545C24order 128 = 27

262nd central stem extension by C23 of C24

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

Aliases: C23.545C24, C22.2372- 1+4, C22.3202+ 1+4, C424C4.26C2, C4.10(C422C2), (C2×C42).620C22, (C22×C4).155C23, (C22×Q8).161C22, C23.84C23.3C2, C2.C42.265C22, C23.65C23.67C2, C23.67C23.50C2, C23.83C23.25C2, C2.30(C22.35C24), C2.54(C22.36C24), (C2×C4).666(C4○D4), (C2×C4⋊C4).371C22, C2.24(C2×C422C2), C22.417(C2×C4○D4), SmallGroup(128,1377)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.545C24
C1C2C22C23C22×C4C22×Q8C23.67C23 — C23.545C24
C1C23 — C23.545C24
C1C23 — C23.545C24
C1C23 — C23.545C24

Generators and relations for C23.545C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=cb=bc, e2=b, f2=g2=a, ab=ba, ac=ca, ede-1=ad=da, ae=ea, gfg-1=af=fa, ag=ga, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >

Subgroups: 308 in 176 conjugacy classes, 92 normal (9 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×14], C22, C22 [×6], C2×C4 [×6], C2×C4 [×42], Q8 [×4], C23, C42 [×6], C4⋊C4 [×12], C22×C4, C22×C4 [×14], C2×Q8 [×6], C2.C42, C2.C42 [×21], C2×C42 [×3], C2×C4⋊C4 [×9], C22×Q8, C424C4, C23.65C23 [×3], C23.67C23 [×3], C23.83C23 [×6], C23.84C23 [×2], C23.545C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C422C2 [×4], C2×C4○D4 [×3], 2+ 1+4, 2- 1+4 [×3], C2×C422C2, C22.35C24 [×3], C22.36C24 [×3], C23.545C24

Smallest permutation representation of C23.545C24
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 21 5 115)(2 86 6 56)(3 23 7 113)(4 88 8 54)(9 119 103 25)(10 60 104 90)(11 117 101 27)(12 58 102 92)(13 123 107 29)(14 64 108 94)(15 121 105 31)(16 62 106 96)(17 127 111 33)(18 68 112 98)(19 125 109 35)(20 66 110 100)(22 70 116 40)(24 72 114 38)(26 74 120 44)(28 76 118 42)(30 78 124 48)(32 80 122 46)(34 82 128 52)(36 84 126 50)(37 87 71 53)(39 85 69 55)(41 91 75 57)(43 89 73 59)(45 95 79 61)(47 93 77 63)(49 99 83 65)(51 97 81 67)
(1 47 39 107)(2 78 40 14)(3 45 37 105)(4 80 38 16)(5 77 69 13)(6 48 70 108)(7 79 71 15)(8 46 72 106)(9 81 73 17)(10 52 74 112)(11 83 75 19)(12 50 76 110)(18 104 82 44)(20 102 84 42)(21 61 85 121)(22 96 86 32)(23 63 87 123)(24 94 88 30)(25 65 89 125)(26 100 90 36)(27 67 91 127)(28 98 92 34)(29 113 93 53)(31 115 95 55)(33 117 97 57)(35 119 99 59)(41 109 101 49)(43 111 103 51)(54 124 114 64)(56 122 116 62)(58 128 118 68)(60 126 120 66)
(1 43 39 103)(2 44 40 104)(3 41 37 101)(4 42 38 102)(5 73 69 9)(6 74 70 10)(7 75 71 11)(8 76 72 12)(13 17 77 81)(14 18 78 82)(15 19 79 83)(16 20 80 84)(21 89 85 25)(22 90 86 26)(23 91 87 27)(24 92 88 28)(29 33 93 97)(30 34 94 98)(31 35 95 99)(32 36 96 100)(45 49 105 109)(46 50 106 110)(47 51 107 111)(48 52 108 112)(53 117 113 57)(54 118 114 58)(55 119 115 59)(56 120 116 60)(61 65 121 125)(62 66 122 126)(63 67 123 127)(64 68 124 128)

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

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

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

32 conjugacy classes

class 1 2A···2G4A4B4C4D4E···4P4Q···4X
order12···244444···44···4
size11···122224···48···8

32 irreducible representations

dim111111244
type+++++++-
imageC1C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC23.545C24C424C4C23.65C23C23.67C23C23.83C23C23.84C23C2×C4C22C22
# reps1133621213

Matrix representation of C23.545C24 in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
30000000
03000000
00030000
00200000
00001141
00003441
00001411
00001434
,
10000000
04000000
00010000
00400000
00001024
00000143
00002440
00004304
,
01000000
10000000
00010000
00100000
00000010
00000001
00004000
00000400
,
40000000
04000000
00400000
00040000
00001100
00003400
00000044
00000021

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,3,1,1,0,0,0,0,1,4,4,4,0,0,0,0,4,4,1,3,0,0,0,0,1,1,1,4],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,2,4,0,0,0,0,0,1,4,3,0,0,0,0,2,4,4,0,0,0,0,0,4,3,0,4],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,4,2,0,0,0,0,0,0,4,1] >;

C23.545C24 in GAP, Magma, Sage, TeX

C_2^3._{545}C_2^4
% in TeX

G:=Group("C2^3.545C2^4");
// GroupNames label

G:=SmallGroup(128,1377);
// by ID

G=gap.SmallGroup(128,1377);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,232,758,723,100,185,80]);
// 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=b,f^2=g^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,g*f*g^-1=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,d*g=g*d,e*g=g*e>;
// generators/relations

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