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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
Jennings
C1 — C27

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

Generators and relations
 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 >

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

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

128 conjugacy classes

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

128 irreducible representations

dim11
type++
imageC1C2
kernelC27C26
# reps1127

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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