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G = C23.709C24order 128 = 27

426th central stem extension by C23 of C24

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

Aliases: C23.709C24, C22.4822+ 1+4, C22.3692- 1+4, (C2×Q8).14Q8, C2.35(Q83Q8), (C2×C42).728C22, (C22×C4).222C23, C2.22(C232Q8), C22.170(C22×Q8), (C22×Q8).229C22, C23.65C23.87C2, C2.C42.413C22, C23.83C23.45C2, C23.67C23.62C2, C23.81C23.49C2, C23.63C23.60C2, C2.50(C22.49C24), C2.56(C22.57C24), C2.120(C22.33C24), (C2×C4).94(C2×Q8), (C2×C4).250(C4○D4), (C2×C4⋊C4).519C22, C22.570(C2×C4○D4), SmallGroup(128,1541)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.709C24
C1C2C22C23C22×C4C2×C42C23.65C23 — C23.709C24
C1C23 — C23.709C24
C1C23 — C23.709C24
C1C23 — C23.709C24

Generators and relations for C23.709C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=ca=ac, e2=g2=ba=ab, 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 >

Subgroups: 324 in 184 conjugacy classes, 96 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C4 [×20], C22 [×3], C22 [×4], C2×C4 [×10], C2×C4 [×40], Q8 [×4], C23, C42 [×3], C4⋊C4 [×19], C22×C4, C22×C4 [×14], C2×Q8 [×4], C2×Q8 [×2], C2.C42 [×18], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4 [×12], C22×Q8, C23.63C23 [×4], C23.65C23 [×2], C23.67C23, C23.67C23 [×2], C23.81C23 [×4], C23.83C23 [×2], C23.709C24
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C22×Q8, C2×C4○D4 [×2], 2+ 1+4 [×2], 2- 1+4 [×2], C22.33C24 [×2], C232Q8, C22.49C24, Q83Q8 [×2], C22.57C24, C23.709C24

Smallest permutation representation of C23.709C24
Regular action on 128 points
Generators in S128
(1 77)(2 78)(3 79)(4 80)(5 127)(6 128)(7 125)(8 126)(9 39)(10 40)(11 37)(12 38)(13 20)(14 17)(15 18)(16 19)(21 28)(22 25)(23 26)(24 27)(29 36)(30 33)(31 34)(32 35)(41 46)(42 47)(43 48)(44 45)(49 54)(50 55)(51 56)(52 53)(57 62)(58 63)(59 64)(60 61)(65 70)(66 71)(67 72)(68 69)(73 97)(74 98)(75 99)(76 100)(81 86)(82 87)(83 88)(84 85)(89 94)(90 95)(91 96)(92 93)(101 108)(102 105)(103 106)(104 107)(109 116)(110 113)(111 114)(112 115)(117 124)(118 121)(119 122)(120 123)
(1 106)(2 107)(3 108)(4 105)(5 100)(6 97)(7 98)(8 99)(9 66)(10 67)(11 68)(12 65)(13 47)(14 48)(15 45)(16 46)(17 43)(18 44)(19 41)(20 42)(21 55)(22 56)(23 53)(24 54)(25 51)(26 52)(27 49)(28 50)(29 63)(30 64)(31 61)(32 62)(33 59)(34 60)(35 57)(36 58)(37 69)(38 70)(39 71)(40 72)(73 128)(74 125)(75 126)(76 127)(77 103)(78 104)(79 101)(80 102)(81 115)(82 116)(83 113)(84 114)(85 111)(86 112)(87 109)(88 110)(89 123)(90 124)(91 121)(92 122)(93 119)(94 120)(95 117)(96 118)
(1 79)(2 80)(3 77)(4 78)(5 125)(6 126)(7 127)(8 128)(9 37)(10 38)(11 39)(12 40)(13 18)(14 19)(15 20)(16 17)(21 26)(22 27)(23 28)(24 25)(29 34)(30 35)(31 36)(32 33)(41 48)(42 45)(43 46)(44 47)(49 56)(50 53)(51 54)(52 55)(57 64)(58 61)(59 62)(60 63)(65 72)(66 69)(67 70)(68 71)(73 99)(74 100)(75 97)(76 98)(81 88)(82 85)(83 86)(84 87)(89 96)(90 93)(91 94)(92 95)(101 106)(102 107)(103 108)(104 105)(109 114)(110 115)(111 116)(112 113)(117 122)(118 123)(119 124)(120 121)
(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 92)(2 123 104 94)(3 117 101 90)(4 121 102 96)(5 114 76 85)(6 112 73 81)(7 116 74 87)(8 110 75 83)(9 26 71 53)(10 24 72 49)(11 28 69 55)(12 22 70 51)(13 58 42 29)(14 64 43 33)(15 60 44 31)(16 62 41 35)(17 59 48 30)(18 61 45 34)(19 57 46 32)(20 63 47 36)(21 68 50 37)(23 66 52 39)(25 65 56 38)(27 67 54 40)(77 122 106 93)(78 120 107 89)(79 124 108 95)(80 118 105 91)(82 125 109 98)(84 127 111 100)(86 128 115 97)(88 126 113 99)
(1 109 101 84)(2 88 102 115)(3 111 103 82)(4 86 104 113)(5 117 74 92)(6 96 75 123)(7 119 76 90)(8 94 73 121)(9 29 69 60)(10 64 70 35)(11 31 71 58)(12 62 72 33)(13 52 44 21)(14 27 41 56)(15 50 42 23)(16 25 43 54)(17 24 46 51)(18 55 47 26)(19 22 48 49)(20 53 45 28)(30 38 57 67)(32 40 59 65)(34 66 63 37)(36 68 61 39)(77 116 108 85)(78 83 105 112)(79 114 106 87)(80 81 107 110)(89 97 118 126)(91 99 120 128)(93 127 124 98)(95 125 122 100)
(1 42 103 13)(2 14 104 43)(3 44 101 15)(4 16 102 41)(5 37 76 68)(6 65 73 38)(7 39 74 66)(8 67 75 40)(9 98 71 125)(10 126 72 99)(11 100 69 127)(12 128 70 97)(17 107 48 78)(18 79 45 108)(19 105 46 80)(20 77 47 106)(21 84 50 111)(22 112 51 81)(23 82 52 109)(24 110 49 83)(25 115 56 86)(26 87 53 116)(27 113 54 88)(28 85 55 114)(29 93 58 122)(30 123 59 94)(31 95 60 124)(32 121 57 96)(33 120 64 89)(34 90 61 117)(35 118 62 91)(36 92 63 119)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111112244
type++++++-+-
imageC1C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC23.709C24C23.63C23C23.65C23C23.67C23C23.81C23C23.83C23C2×Q8C2×C4C22C22
# reps1423424822

Matrix representation of C23.709C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
030000
300000
002000
000200
000012
000004
,
300000
020000
002300
004300
000030
000003
,
100000
010000
003000
001200
000012
000044
,
010000
400000
004000
000400
000012
000044

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

C23.709C24 in GAP, Magma, Sage, TeX

C_2^3._{709}C_2^4
% in TeX

G:=Group("C2^3.709C2^4");
// GroupNames label

G:=SmallGroup(128,1541);
// by ID

G=gap.SmallGroup(128,1541);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,344,758,723,520,1571,346,192]);
// Polycyclic

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