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## G = C27order 128 = 27

### Elementary abelian group of type [2,2,2,2,2,2,2]

Aliases: C27, SmallGroup(128,2328)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C27
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C26 — C27
 Lower central C1 — C27
 Upper central C1 — C27
 Jennings C1 — C27

Generators and relations for C27
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 29212, all normal (2 characteristic)
C1, C2 [×127], C22 [×2667], C23 [×11811], C24 [×11811], C25 [×2667], C26 [×127], C27
Quotients: C1, C2 [×127], C22 [×2667], C23 [×11811], C24 [×11811], C25 [×2667], C26 [×127], C27

Smallest permutation representation of C27
Regular action on 128 points
Generators in S128
```(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)(2 120)(3 112)(4 111)(5 113)(6 114)(7 66)(8 65)(9 108)(10 107)(11 70)(12 69)(13 104)(14 103)(15 30)(16 29)(17 53)(18 54)(19 44)(20 43)(21 27)(22 28)(23 41)(24 42)(25 34)(26 33)(31 59)(32 60)(35 56)(36 55)(37 62)(38 61)(39 92)(40 91)(45 83)(46 84)(47 121)(48 122)(49 58)(50 57)(51 87)(52 88)(63 127)(64 128)(67 85)(68 86)(71 117)(72 118)(73 95)(74 96)(75 81)(76 82)(77 124)(78 123)(79 106)(80 105)(89 93)(90 94)(97 126)(98 125)(99 109)(100 110)(101 116)(102 115)
(1 103)(2 104)(3 96)(4 95)(5 60)(6 59)(7 42)(8 41)(9 78)(10 77)(11 75)(12 76)(13 120)(14 119)(15 19)(16 20)(17 99)(18 100)(21 117)(22 118)(23 65)(24 66)(25 88)(26 87)(27 71)(28 72)(29 43)(30 44)(31 114)(32 113)(33 51)(34 52)(35 48)(36 47)(37 98)(38 97)(39 68)(40 67)(45 63)(46 64)(49 94)(50 93)(53 109)(54 110)(55 121)(56 122)(57 89)(58 90)(61 126)(62 125)(69 82)(70 81)(73 111)(74 112)(79 102)(80 101)(83 127)(84 128)(85 91)(86 92)(105 116)(106 115)(107 124)(108 123)
(1 68)(2 67)(3 8)(4 7)(5 82)(6 81)(9 57)(10 58)(11 31)(12 32)(13 91)(14 92)(15 46)(16 45)(17 118)(18 117)(19 64)(20 63)(21 100)(22 99)(23 74)(24 73)(25 79)(26 80)(27 110)(28 109)(29 83)(30 84)(33 105)(34 106)(35 126)(36 125)(37 121)(38 122)(39 103)(40 104)(41 96)(42 95)(43 127)(44 128)(47 62)(48 61)(49 107)(50 108)(51 116)(52 115)(53 72)(54 71)(55 98)(56 97)(59 70)(60 69)(65 112)(66 111)(75 114)(76 113)(77 90)(78 89)(85 120)(86 119)(87 101)(88 102)(93 123)(94 124)
(1 54)(2 53)(3 77)(4 78)(5 46)(6 45)(7 89)(8 90)(9 95)(10 96)(11 43)(12 44)(13 99)(14 100)(15 82)(16 81)(17 120)(18 119)(19 69)(20 70)(21 92)(22 91)(23 49)(24 50)(25 55)(26 56)(27 39)(28 40)(29 75)(30 76)(31 127)(32 128)(33 35)(34 36)(37 102)(38 101)(41 58)(42 57)(47 52)(48 51)(59 63)(60 64)(61 116)(62 115)(65 94)(66 93)(67 72)(68 71)(73 108)(74 107)(79 98)(80 97)(83 114)(84 113)(85 118)(86 117)(87 122)(88 121)(103 110)(104 109)(105 126)(106 125)(111 123)(112 124)
(1 64)(2 63)(3 38)(4 37)(5 110)(6 109)(7 121)(8 122)(9 79)(10 80)(11 118)(12 117)(13 83)(14 84)(15 39)(16 40)(17 31)(18 32)(19 68)(20 67)(21 76)(22 75)(23 35)(24 36)(25 57)(26 58)(27 82)(28 81)(29 91)(30 92)(33 49)(34 50)(41 56)(42 55)(43 85)(44 86)(45 104)(46 103)(47 66)(48 65)(51 94)(52 93)(53 59)(54 60)(61 112)(62 111)(69 71)(70 72)(73 125)(74 126)(77 101)(78 102)(87 90)(88 89)(95 98)(96 97)(99 114)(100 113)(105 107)(106 108)(115 123)(116 124)(119 128)(120 127)
(1 89)(2 90)(3 72)(4 71)(5 55)(6 56)(7 54)(8 53)(9 39)(10 40)(11 61)(12 62)(13 49)(14 50)(15 79)(16 80)(17 65)(18 66)(19 102)(20 101)(21 73)(22 74)(23 99)(24 100)(25 46)(26 45)(27 95)(28 96)(29 105)(30 106)(31 48)(32 47)(33 83)(34 84)(35 114)(36 113)(37 69)(38 70)(41 109)(42 110)(43 116)(44 115)(51 127)(52 128)(57 103)(58 104)(59 122)(60 121)(63 87)(64 88)(67 77)(68 78)(75 126)(76 125)(81 97)(82 98)(85 124)(86 123)(91 107)(92 108)(93 119)(94 120)(111 117)(112 118)```

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

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

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

128 conjugacy classes

 class 1 2A ··· 2DW order 1 2 ··· 2 size 1 1 ··· 1

128 irreducible representations

 dim 1 1 type + + image C1 C2 kernel C27 C26 # reps 1 127

Matrix representation of C27 in GL7(ℤ)

 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1

`G:=sub<GL(7,Integers())| [-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1] >;`

C27 in GAP, Magma, Sage, TeX

`C_2^7`
`% in TeX`

`G:=Group("C2^7");`
`// GroupNames label`

`G:=SmallGroup(128,2328);`
`// by ID`

`G=gap.SmallGroup(128,2328);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,2,2,2]);`
`// Polycyclic`

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

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