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G = C27  order 128 = 27

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

direct product, p-group, elementary abelian, monomial, rational

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, C22, C23, C24, C25, C26, C27
Quotients: C1, C2, C22, C23, C24, C25, C26, 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 74)(2 73)(3 106)(4 105)(5 54)(6 53)(7 104)(8 103)(9 55)(10 56)(11 16)(12 15)(13 98)(14 97)(17 92)(18 91)(19 38)(20 37)(21 108)(22 107)(23 34)(24 33)(25 121)(26 122)(27 124)(28 123)(29 120)(30 119)(31 62)(32 61)(35 85)(36 86)(39 100)(40 99)(41 117)(42 118)(43 101)(44 102)(45 83)(46 84)(47 114)(48 113)(49 125)(50 126)(51 93)(52 94)(57 77)(58 78)(59 76)(60 75)(63 95)(64 96)(65 109)(66 110)(67 69)(68 70)(71 115)(72 116)(79 128)(80 127)(81 89)(82 90)(87 111)(88 112)

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

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

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

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

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

128 conjugacy classes

class 1 2A···2DW
order12···2
size11···1

128 irreducible representations

dim11
type++
imageC1C2
kernelC27C26
# reps1127

Matrix representation of C27 ►in GL7(ℤ)

-1000000
0100000
00-10000
0001000
0000100
00000-10
0000001
,
-1000000
0100000
00-10000
000-1000
0000100
00000-10
0000001
,
1000000
0-100000
0010000
0001000
0000-100
00000-10
000000-1
,
1000000
0100000
00-10000
000-1000
0000100
00000-10
0000001
,
1000000
0100000
0010000
0001000
0000-100
0000010
0000001
,
-1000000
0100000
00-10000
000-1000
0000100
00000-10
000000-1
,
1000000
0100000
0010000
0001000
0000-100
00000-10
0000001

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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