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G = C23.253C24order 128 = 27

106th central extension by C23 of C24

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

Aliases: C23.253C24, C22.622- 1+4, C42.C216C4, C42.195(C2×C4), C424C4.17C2, (C2×C42).443C22, (C22×C4).772C23, C22.144(C23×C4), C2.C42.480C22, C23.63C23.10C2, C2.10(C22.46C24), C2.19(C23.32C23), (C4×C4⋊C4).45C2, C2.40(C4×C4○D4), C4⋊C4.110(C2×C4), (C2×C4).51(C22×C4), (C2×C4).725(C4○D4), (C2×C4⋊C4).830C22, C22.138(C2×C4○D4), (C2×C42.C2).14C2, SmallGroup(128,1103)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.253C24
C1C2C22C23C22×C4C2×C42C424C4 — C23.253C24
C1C22 — C23.253C24
C1C23 — C23.253C24
C1C23 — C23.253C24

Generators and relations for C23.253C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=c, f2=a, g2=b, ab=ba, 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, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 316 in 224 conjugacy classes, 140 normal (8 characteristic)
C1, C2, C2 [×6], C4 [×26], C22, C22 [×6], C2×C4 [×22], C2×C4 [×34], C23, C42 [×4], C42 [×16], C4⋊C4 [×24], C4⋊C4 [×8], C22×C4, C22×C4 [×14], C2.C42 [×16], C2×C42, C2×C42 [×8], C2×C4⋊C4 [×10], C42.C2 [×8], C424C4 [×2], C4×C4⋊C4 [×4], C23.63C23 [×8], C2×C42.C2, C23.253C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×8], C24, C23×C4, C2×C4○D4 [×4], 2- 1+4 [×2], C4×C4○D4 [×2], C23.32C23, C22.46C24 [×4], C23.253C24

Smallest permutation representation of C23.253C24
Regular action on 128 points
Generators in S128
(1 11)(2 12)(3 9)(4 10)(5 70)(6 71)(7 72)(8 69)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(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)(65 127)(66 128)(67 125)(68 126)
(1 99)(2 100)(3 97)(4 98)(5 68)(6 65)(7 66)(8 67)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)(69 125)(70 126)(71 127)(72 128)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 119)(92 120)(93 121)(94 122)(95 123)(96 124)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)(65 67)(66 68)(69 71)(70 72)(73 75)(74 76)(77 79)(78 80)(81 83)(82 84)(85 87)(86 88)(89 91)(90 92)(93 95)(94 96)(97 99)(98 100)(101 103)(102 104)(105 107)(106 108)(109 111)(110 112)(113 115)(114 116)(117 119)(118 120)(121 123)(122 124)(125 127)(126 128)
(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 85 3 87)(2 26 4 28)(5 82 7 84)(6 23 8 21)(9 27 11 25)(10 88 12 86)(13 31 15 29)(14 92 16 90)(17 35 19 33)(18 96 20 94)(22 72 24 70)(30 74 32 76)(34 78 36 80)(37 55 39 53)(38 116 40 114)(41 59 43 57)(42 120 44 118)(45 63 47 61)(46 124 48 122)(49 65 51 67)(50 128 52 126)(54 98 56 100)(58 102 60 104)(62 106 64 108)(66 112 68 110)(69 81 71 83)(73 91 75 89)(77 95 79 93)(97 115 99 113)(101 119 103 117)(105 123 107 121)(109 127 111 125)
(1 19 11 79)(2 48 12 108)(3 17 9 77)(4 46 10 106)(5 58 70 118)(6 31 71 91)(7 60 72 120)(8 29 69 89)(13 81 73 21)(14 110 74 50)(15 83 75 23)(16 112 76 52)(18 38 78 98)(20 40 80 100)(22 42 82 102)(24 44 84 104)(25 93 85 33)(26 122 86 62)(27 95 87 35)(28 124 88 64)(30 126 90 68)(32 128 92 66)(34 54 94 114)(36 56 96 116)(37 105 97 45)(39 107 99 47)(41 109 101 49)(43 111 103 51)(53 121 113 61)(55 123 115 63)(57 125 117 67)(59 127 119 65)
(1 103 99 75)(2 104 100 76)(3 101 97 73)(4 102 98 74)(5 94 68 122)(6 95 65 123)(7 96 66 124)(8 93 67 121)(9 41 37 13)(10 42 38 14)(11 43 39 15)(12 44 40 16)(17 49 45 21)(18 50 46 22)(19 51 47 23)(20 52 48 24)(25 117 53 89)(26 118 54 90)(27 119 55 91)(28 120 56 92)(29 85 57 113)(30 86 58 114)(31 87 59 115)(32 88 60 116)(33 125 61 69)(34 126 62 70)(35 127 63 71)(36 128 64 72)(77 109 105 81)(78 110 106 82)(79 111 107 83)(80 112 108 84)

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

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

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

50 conjugacy classes

class 1 2A···2G4A···4X4Y···4AP
order12···24···44···4
size11···12···24···4

50 irreducible representations

dim11111124
type+++++-
imageC1C2C2C2C2C4C4○D42- 1+4
kernelC23.253C24C424C4C4×C4⋊C4C23.63C23C2×C42.C2C42.C2C2×C4C22
# reps1248116162

Matrix representation of C23.253C24 in GL5(𝔽5)

10000
01000
00100
00040
00004
,
10000
04000
00400
00010
00001
,
40000
01000
00100
00040
00004
,
30000
00100
01000
00003
00030
,
20000
04000
00400
00004
00010
,
10000
04000
00100
00030
00003
,
40000
02000
00200
00001
00010

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

C23.253C24 in GAP, Magma, Sage, TeX

C_2^3._{253}C_2^4
% in TeX

G:=Group("C2^3.253C2^4");
// GroupNames label

G:=SmallGroup(128,1103);
// by ID

G=gap.SmallGroup(128,1103);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,100,346,136]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=c,f^2=a,g^2=b,a*b=b*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,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
// generators/relations

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