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G = C2×He34Q8order 432 = 24·33

Direct product of C2 and He34Q8

direct product, non-abelian, supersoluble, monomial

Aliases: C2×He34Q8, C62.44D6, He36(C2×Q8), (C2×He3)⋊4Q8, (C3×C6)⋊4Dic6, (C6×C12).15S3, (C3×C12).54D6, C325(C2×Dic6), (C2×He3).29C23, (C4×He3).39C22, C6.11(C324Q8), He33C4.16C22, (C22×He3).31C22, (C2×C4×He3).9C2, C12.82(C2×C3⋊S3), C6.62(C22×C3⋊S3), (C2×C12).22(C3⋊S3), C3.2(C2×C324Q8), (C2×He33C4).9C2, (C3×C6).39(C22×S3), C4.11(C2×He3⋊C2), (C2×C4).4(He3⋊C2), C22.8(C2×He3⋊C2), C2.3(C22×He3⋊C2), (C2×C6).56(C2×C3⋊S3), SmallGroup(432,384)

Series: Derived Chief Lower central Upper central

C1C3C2×He3 — C2×He34Q8
C1C3C32He3C2×He3He33C4C2×He33C4 — C2×He34Q8
He3C2×He3 — C2×He34Q8
C1C2×C6C2×C12

Generators and relations for C2×He34Q8
 G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, be=eb, fbf-1=b-1, cd=dc, ce=ec, cf=fc, de=ed, fdf-1=d-1, fef-1=e-1 >

Subgroups: 625 in 209 conjugacy classes, 67 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, He3, C3×Dic3, C3×C12, C62, C2×Dic6, C6×Q8, C2×He3, C2×He3, C3×Dic6, C6×Dic3, C6×C12, He33C4, C4×He3, C22×He3, C6×Dic6, He34Q8, C2×He33C4, C2×C4×He3, C2×He34Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C3⋊S3, Dic6, C22×S3, C2×C3⋊S3, C2×Dic6, He3⋊C2, C324Q8, C22×C3⋊S3, C2×He3⋊C2, C2×C324Q8, He34Q8, C22×He3⋊C2, C2×He34Q8

Smallest permutation representation of C2×He34Q8
On 144 points
Generators in S144
(1 42)(2 43)(3 44)(4 41)(5 31)(6 32)(7 29)(8 30)(9 85)(10 86)(11 87)(12 88)(13 38)(14 39)(15 40)(16 37)(17 47)(18 48)(19 45)(20 46)(21 58)(22 59)(23 60)(24 57)(25 111)(26 112)(27 109)(28 110)(33 71)(34 72)(35 69)(36 70)(49 103)(50 104)(51 101)(52 102)(53 75)(54 76)(55 73)(56 74)(61 132)(62 129)(63 130)(64 131)(65 95)(66 96)(67 93)(68 94)(77 120)(78 117)(79 118)(80 119)(81 124)(82 121)(83 122)(84 123)(89 114)(90 115)(91 116)(92 113)(97 140)(98 137)(99 138)(100 139)(105 134)(106 135)(107 136)(108 133)(125 142)(126 143)(127 144)(128 141)
(5 35 28)(6 36 25)(7 33 26)(8 34 27)(9 141 118)(10 142 119)(11 143 120)(12 144 117)(21 104 95)(22 101 96)(23 102 93)(24 103 94)(29 71 112)(30 72 109)(31 69 110)(32 70 111)(49 68 57)(50 65 58)(51 66 59)(52 67 60)(61 97 108)(62 98 105)(63 99 106)(64 100 107)(77 87 126)(78 88 127)(79 85 128)(80 86 125)(129 137 134)(130 138 135)(131 139 136)(132 140 133)
(1 37 19)(2 38 20)(3 39 17)(4 40 18)(5 35 28)(6 36 25)(7 33 26)(8 34 27)(9 118 141)(10 119 142)(11 120 143)(12 117 144)(13 46 43)(14 47 44)(15 48 41)(16 45 42)(21 95 104)(22 96 101)(23 93 102)(24 94 103)(29 71 112)(30 72 109)(31 69 110)(32 70 111)(49 57 68)(50 58 65)(51 59 66)(52 60 67)(53 114 81)(54 115 82)(55 116 83)(56 113 84)(61 97 108)(62 98 105)(63 99 106)(64 100 107)(73 91 122)(74 92 123)(75 89 124)(76 90 121)(77 126 87)(78 127 88)(79 128 85)(80 125 86)(129 137 134)(130 138 135)(131 139 136)(132 140 133)
(1 96 31)(2 93 32)(3 94 29)(4 95 30)(5 42 66)(6 43 67)(7 44 68)(8 41 65)(9 99 82)(10 100 83)(11 97 84)(12 98 81)(13 52 36)(14 49 33)(15 50 34)(16 51 35)(17 24 112)(18 21 109)(19 22 110)(20 23 111)(25 46 60)(26 47 57)(27 48 58)(28 45 59)(37 101 69)(38 102 70)(39 103 71)(40 104 72)(53 117 105)(54 118 106)(55 119 107)(56 120 108)(61 113 143)(62 114 144)(63 115 141)(64 116 142)(73 80 136)(74 77 133)(75 78 134)(76 79 135)(85 138 121)(86 139 122)(87 140 123)(88 137 124)(89 127 129)(90 128 130)(91 125 131)(92 126 132)
(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)(129 130 131 132)(133 134 135 136)(137 138 139 140)(141 142 143 144)
(1 54 3 56)(2 53 4 55)(5 79 7 77)(6 78 8 80)(9 112 11 110)(10 111 12 109)(13 89 15 91)(14 92 16 90)(17 84 19 82)(18 83 20 81)(21 100 23 98)(22 99 24 97)(25 88 27 86)(26 87 28 85)(29 120 31 118)(30 119 32 117)(33 126 35 128)(34 125 36 127)(37 115 39 113)(38 114 40 116)(41 73 43 75)(42 76 44 74)(45 121 47 123)(46 124 48 122)(49 132 51 130)(50 131 52 129)(57 140 59 138)(58 139 60 137)(61 101 63 103)(62 104 64 102)(65 136 67 134)(66 135 68 133)(69 141 71 143)(70 144 72 142)(93 105 95 107)(94 108 96 106)

G:=sub<Sym(144)| (1,42)(2,43)(3,44)(4,41)(5,31)(6,32)(7,29)(8,30)(9,85)(10,86)(11,87)(12,88)(13,38)(14,39)(15,40)(16,37)(17,47)(18,48)(19,45)(20,46)(21,58)(22,59)(23,60)(24,57)(25,111)(26,112)(27,109)(28,110)(33,71)(34,72)(35,69)(36,70)(49,103)(50,104)(51,101)(52,102)(53,75)(54,76)(55,73)(56,74)(61,132)(62,129)(63,130)(64,131)(65,95)(66,96)(67,93)(68,94)(77,120)(78,117)(79,118)(80,119)(81,124)(82,121)(83,122)(84,123)(89,114)(90,115)(91,116)(92,113)(97,140)(98,137)(99,138)(100,139)(105,134)(106,135)(107,136)(108,133)(125,142)(126,143)(127,144)(128,141), (5,35,28)(6,36,25)(7,33,26)(8,34,27)(9,141,118)(10,142,119)(11,143,120)(12,144,117)(21,104,95)(22,101,96)(23,102,93)(24,103,94)(29,71,112)(30,72,109)(31,69,110)(32,70,111)(49,68,57)(50,65,58)(51,66,59)(52,67,60)(61,97,108)(62,98,105)(63,99,106)(64,100,107)(77,87,126)(78,88,127)(79,85,128)(80,86,125)(129,137,134)(130,138,135)(131,139,136)(132,140,133), (1,37,19)(2,38,20)(3,39,17)(4,40,18)(5,35,28)(6,36,25)(7,33,26)(8,34,27)(9,118,141)(10,119,142)(11,120,143)(12,117,144)(13,46,43)(14,47,44)(15,48,41)(16,45,42)(21,95,104)(22,96,101)(23,93,102)(24,94,103)(29,71,112)(30,72,109)(31,69,110)(32,70,111)(49,57,68)(50,58,65)(51,59,66)(52,60,67)(53,114,81)(54,115,82)(55,116,83)(56,113,84)(61,97,108)(62,98,105)(63,99,106)(64,100,107)(73,91,122)(74,92,123)(75,89,124)(76,90,121)(77,126,87)(78,127,88)(79,128,85)(80,125,86)(129,137,134)(130,138,135)(131,139,136)(132,140,133), (1,96,31)(2,93,32)(3,94,29)(4,95,30)(5,42,66)(6,43,67)(7,44,68)(8,41,65)(9,99,82)(10,100,83)(11,97,84)(12,98,81)(13,52,36)(14,49,33)(15,50,34)(16,51,35)(17,24,112)(18,21,109)(19,22,110)(20,23,111)(25,46,60)(26,47,57)(27,48,58)(28,45,59)(37,101,69)(38,102,70)(39,103,71)(40,104,72)(53,117,105)(54,118,106)(55,119,107)(56,120,108)(61,113,143)(62,114,144)(63,115,141)(64,116,142)(73,80,136)(74,77,133)(75,78,134)(76,79,135)(85,138,121)(86,139,122)(87,140,123)(88,137,124)(89,127,129)(90,128,130)(91,125,131)(92,126,132), (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)(129,130,131,132)(133,134,135,136)(137,138,139,140)(141,142,143,144), (1,54,3,56)(2,53,4,55)(5,79,7,77)(6,78,8,80)(9,112,11,110)(10,111,12,109)(13,89,15,91)(14,92,16,90)(17,84,19,82)(18,83,20,81)(21,100,23,98)(22,99,24,97)(25,88,27,86)(26,87,28,85)(29,120,31,118)(30,119,32,117)(33,126,35,128)(34,125,36,127)(37,115,39,113)(38,114,40,116)(41,73,43,75)(42,76,44,74)(45,121,47,123)(46,124,48,122)(49,132,51,130)(50,131,52,129)(57,140,59,138)(58,139,60,137)(61,101,63,103)(62,104,64,102)(65,136,67,134)(66,135,68,133)(69,141,71,143)(70,144,72,142)(93,105,95,107)(94,108,96,106)>;

G:=Group( (1,42)(2,43)(3,44)(4,41)(5,31)(6,32)(7,29)(8,30)(9,85)(10,86)(11,87)(12,88)(13,38)(14,39)(15,40)(16,37)(17,47)(18,48)(19,45)(20,46)(21,58)(22,59)(23,60)(24,57)(25,111)(26,112)(27,109)(28,110)(33,71)(34,72)(35,69)(36,70)(49,103)(50,104)(51,101)(52,102)(53,75)(54,76)(55,73)(56,74)(61,132)(62,129)(63,130)(64,131)(65,95)(66,96)(67,93)(68,94)(77,120)(78,117)(79,118)(80,119)(81,124)(82,121)(83,122)(84,123)(89,114)(90,115)(91,116)(92,113)(97,140)(98,137)(99,138)(100,139)(105,134)(106,135)(107,136)(108,133)(125,142)(126,143)(127,144)(128,141), (5,35,28)(6,36,25)(7,33,26)(8,34,27)(9,141,118)(10,142,119)(11,143,120)(12,144,117)(21,104,95)(22,101,96)(23,102,93)(24,103,94)(29,71,112)(30,72,109)(31,69,110)(32,70,111)(49,68,57)(50,65,58)(51,66,59)(52,67,60)(61,97,108)(62,98,105)(63,99,106)(64,100,107)(77,87,126)(78,88,127)(79,85,128)(80,86,125)(129,137,134)(130,138,135)(131,139,136)(132,140,133), (1,37,19)(2,38,20)(3,39,17)(4,40,18)(5,35,28)(6,36,25)(7,33,26)(8,34,27)(9,118,141)(10,119,142)(11,120,143)(12,117,144)(13,46,43)(14,47,44)(15,48,41)(16,45,42)(21,95,104)(22,96,101)(23,93,102)(24,94,103)(29,71,112)(30,72,109)(31,69,110)(32,70,111)(49,57,68)(50,58,65)(51,59,66)(52,60,67)(53,114,81)(54,115,82)(55,116,83)(56,113,84)(61,97,108)(62,98,105)(63,99,106)(64,100,107)(73,91,122)(74,92,123)(75,89,124)(76,90,121)(77,126,87)(78,127,88)(79,128,85)(80,125,86)(129,137,134)(130,138,135)(131,139,136)(132,140,133), (1,96,31)(2,93,32)(3,94,29)(4,95,30)(5,42,66)(6,43,67)(7,44,68)(8,41,65)(9,99,82)(10,100,83)(11,97,84)(12,98,81)(13,52,36)(14,49,33)(15,50,34)(16,51,35)(17,24,112)(18,21,109)(19,22,110)(20,23,111)(25,46,60)(26,47,57)(27,48,58)(28,45,59)(37,101,69)(38,102,70)(39,103,71)(40,104,72)(53,117,105)(54,118,106)(55,119,107)(56,120,108)(61,113,143)(62,114,144)(63,115,141)(64,116,142)(73,80,136)(74,77,133)(75,78,134)(76,79,135)(85,138,121)(86,139,122)(87,140,123)(88,137,124)(89,127,129)(90,128,130)(91,125,131)(92,126,132), (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)(129,130,131,132)(133,134,135,136)(137,138,139,140)(141,142,143,144), (1,54,3,56)(2,53,4,55)(5,79,7,77)(6,78,8,80)(9,112,11,110)(10,111,12,109)(13,89,15,91)(14,92,16,90)(17,84,19,82)(18,83,20,81)(21,100,23,98)(22,99,24,97)(25,88,27,86)(26,87,28,85)(29,120,31,118)(30,119,32,117)(33,126,35,128)(34,125,36,127)(37,115,39,113)(38,114,40,116)(41,73,43,75)(42,76,44,74)(45,121,47,123)(46,124,48,122)(49,132,51,130)(50,131,52,129)(57,140,59,138)(58,139,60,137)(61,101,63,103)(62,104,64,102)(65,136,67,134)(66,135,68,133)(69,141,71,143)(70,144,72,142)(93,105,95,107)(94,108,96,106) );

G=PermutationGroup([[(1,42),(2,43),(3,44),(4,41),(5,31),(6,32),(7,29),(8,30),(9,85),(10,86),(11,87),(12,88),(13,38),(14,39),(15,40),(16,37),(17,47),(18,48),(19,45),(20,46),(21,58),(22,59),(23,60),(24,57),(25,111),(26,112),(27,109),(28,110),(33,71),(34,72),(35,69),(36,70),(49,103),(50,104),(51,101),(52,102),(53,75),(54,76),(55,73),(56,74),(61,132),(62,129),(63,130),(64,131),(65,95),(66,96),(67,93),(68,94),(77,120),(78,117),(79,118),(80,119),(81,124),(82,121),(83,122),(84,123),(89,114),(90,115),(91,116),(92,113),(97,140),(98,137),(99,138),(100,139),(105,134),(106,135),(107,136),(108,133),(125,142),(126,143),(127,144),(128,141)], [(5,35,28),(6,36,25),(7,33,26),(8,34,27),(9,141,118),(10,142,119),(11,143,120),(12,144,117),(21,104,95),(22,101,96),(23,102,93),(24,103,94),(29,71,112),(30,72,109),(31,69,110),(32,70,111),(49,68,57),(50,65,58),(51,66,59),(52,67,60),(61,97,108),(62,98,105),(63,99,106),(64,100,107),(77,87,126),(78,88,127),(79,85,128),(80,86,125),(129,137,134),(130,138,135),(131,139,136),(132,140,133)], [(1,37,19),(2,38,20),(3,39,17),(4,40,18),(5,35,28),(6,36,25),(7,33,26),(8,34,27),(9,118,141),(10,119,142),(11,120,143),(12,117,144),(13,46,43),(14,47,44),(15,48,41),(16,45,42),(21,95,104),(22,96,101),(23,93,102),(24,94,103),(29,71,112),(30,72,109),(31,69,110),(32,70,111),(49,57,68),(50,58,65),(51,59,66),(52,60,67),(53,114,81),(54,115,82),(55,116,83),(56,113,84),(61,97,108),(62,98,105),(63,99,106),(64,100,107),(73,91,122),(74,92,123),(75,89,124),(76,90,121),(77,126,87),(78,127,88),(79,128,85),(80,125,86),(129,137,134),(130,138,135),(131,139,136),(132,140,133)], [(1,96,31),(2,93,32),(3,94,29),(4,95,30),(5,42,66),(6,43,67),(7,44,68),(8,41,65),(9,99,82),(10,100,83),(11,97,84),(12,98,81),(13,52,36),(14,49,33),(15,50,34),(16,51,35),(17,24,112),(18,21,109),(19,22,110),(20,23,111),(25,46,60),(26,47,57),(27,48,58),(28,45,59),(37,101,69),(38,102,70),(39,103,71),(40,104,72),(53,117,105),(54,118,106),(55,119,107),(56,120,108),(61,113,143),(62,114,144),(63,115,141),(64,116,142),(73,80,136),(74,77,133),(75,78,134),(76,79,135),(85,138,121),(86,139,122),(87,140,123),(88,137,124),(89,127,129),(90,128,130),(91,125,131),(92,126,132)], [(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),(129,130,131,132),(133,134,135,136),(137,138,139,140),(141,142,143,144)], [(1,54,3,56),(2,53,4,55),(5,79,7,77),(6,78,8,80),(9,112,11,110),(10,111,12,109),(13,89,15,91),(14,92,16,90),(17,84,19,82),(18,83,20,81),(21,100,23,98),(22,99,24,97),(25,88,27,86),(26,87,28,85),(29,120,31,118),(30,119,32,117),(33,126,35,128),(34,125,36,127),(37,115,39,113),(38,114,40,116),(41,73,43,75),(42,76,44,74),(45,121,47,123),(46,124,48,122),(49,132,51,130),(50,131,52,129),(57,140,59,138),(58,139,60,137),(61,101,63,103),(62,104,64,102),(65,136,67,134),(66,135,68,133),(69,141,71,143),(70,144,72,142),(93,105,95,107),(94,108,96,106)]])

62 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F4A4B4C4D4E4F6A···6F6G···6R12A12B12C12D12E···12T12U···12AB
order12223333334444446···66···61212121212···1212···12
size111111666622181818181···16···622226···618···18

62 irreducible representations

dim1111222223336
type+++++-++-
imageC1C2C2C2S3Q8D6D6Dic6He3⋊C2C2×He3⋊C2C2×He3⋊C2He34Q8
kernelC2×He34Q8He34Q8C2×He33C4C2×C4×He3C6×C12C2×He3C3×C12C62C3×C6C2×C4C4C22C2
# reps14214284164844

Matrix representation of C2×He34Q8 in GL7(𝔽13)

1000000
0100000
00120000
00012000
0000100
0000010
0000001
,
01200000
11200000
00012000
00112000
0000100
0000030
0000009
,
1000000
0100000
0010000
0001000
0000300
0000030
0000003
,
01200000
11200000
00121000
00120000
0000010
0000001
0000100
,
10600000
7300000
00106000
0073000
00001200
00000120
00000012
,
5800000
0800000
0092000
00114000
00001200
00000012
00000120

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3],[0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[10,7,0,0,0,0,0,6,3,0,0,0,0,0,0,0,10,7,0,0,0,0,0,6,3,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[5,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,9,11,0,0,0,0,0,2,4,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0] >;

C2×He34Q8 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_4Q_8
% in TeX

G:=Group("C2xHe3:4Q8");
// GroupNames label

G:=SmallGroup(432,384);
// by ID

G=gap.SmallGroup(432,384);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,254,58,1124,4037,537]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,d*b*d^-1=b*c^-1,b*e=e*b,f*b*f^-1=b^-1,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f^-1=d^-1,f*e*f^-1=e^-1>;
// generators/relations

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