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G = C23.711C24  order 128 = 27

428th central stem extension by C23 of C24

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

Aliases: C23.711C24, C22.4842+ (1+4), (C2×Q8)⋊6Q8, C2.12(Q82), (C2×C42).729C22, (C22×C4).893C23, C2.23(C23⋊2Q8), C22.171(C22×Q8), (C22×Q8).230C22, C2.14(C24⋊C22), C23.78C23.29C2, C23.67C23.63C2, C2.C42.415C22, (C2×C4).95(C2×Q8), (C2×C4⋊C4).521C22, SmallGroup(128,1543)

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

Derived series
C1 — C23 — C23.711C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.67C23 — C23.711C24
Lower central
C1 — C23 — C23.711C24
Upper central
C1 — C23 — C23.711C24
Jennings
C1 — C23 — C23.711C24

Subgroups: 420 in 216 conjugacy classes, 112 normal (5 characteristic)
C1, C2, C2 [×6], C4 [×24], C22, C22 [×6], C2×C4 [×18], C2×C4 [×36], Q8 [×20], C23, C42 [×3], C4⋊C4 [×9], C22×C4 [×15], C2×Q8 [×12], C2×Q8 [×12], C2.C42 [×18], C2×C42 [×3], C2×C4⋊C4 [×9], C22×Q8 [×5], C23.67C23 [×9], C23.78C23 [×6], C23.711C24

Quotients:
C1, C2 [×15], C22 [×35], Q8 [×12], C23 [×15], C2×Q8 [×18], C24, C22×Q8 [×3], 2+ (1+4) [×4], C23⋊2Q8 [×3], Q82 [×3], C24⋊C22, C23.711C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=g2=ba=ab, e2=ca=ac, f2=cb=bc, 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 >

Smallest permutation representation
►Regular action on 128 points
Generators in S128
(1 73)(2 74)(3 75)(4 76)(5 126)(6 127)(7 128)(8 125)(9 39)(10 40)(11 37)(12 38)(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 47)(42 48)(43 45)(44 46)(49 55)(50 56)(51 53)(52 54)(57 63)(58 64)(59 61)(60 62)(65 71)(66 72)(67 69)(68 70)(101 107)(102 108)(103 105)(104 106)(109 115)(110 116)(111 113)(112 114)(117 123)(118 124)(119 121)(120 122)

(1 75)(2 76)(3 73)(4 74)(5 128)(6 125)(7 126)(8 127)(9 37)(10 38)(11 39)(12 40)(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)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(101 105)(102 106)(103 107)(104 108)(109 113)(110 114)(111 115)(112 116)(117 121)(118 122)(119 123)(120 124)

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

(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)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
030000
300000
001000
000100
000020
000003
,
200000
030000
002200
000300
000010
000001
,
400000
040000
002000
001300
000001
000040
,
010000
400000
001000
000100
000001
000040

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

32 conjugacy classes

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

32 irreducible representations

dim11124
type+++-+
imageC1C2C2Q82+ (1+4)
kernelC23.711C24C23.67C23C23.78C23C2×Q8C22
# reps196124

In GAP, Magma, Sage, TeX 

C_2^3._{711}C_2^4
 
% in TeX
 
G:=Group("C2^3.711C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1543);
 
// by ID
 
G=gap.SmallGroup(128,1543);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,568,758,723,520,1571,346,192]);
 
// 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=c*a=a*c,f^2=c*b=b*c,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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