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G = C2×C37⋊C3order 222 = 2·3·37

Direct product of C2 and C37⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C2×C37⋊C3, C74⋊C3, C372C6, SmallGroup(222,2)

Series: Derived Chief Lower central Upper central

C1C37 — C2×C37⋊C3
C1C37C37⋊C3 — C2×C37⋊C3
C37 — C2×C37⋊C3
C1C2

Generators and relations for C2×C37⋊C3
 G = < a,b,c | a2=b37=c3=1, ab=ba, ac=ca, cbc-1=b10 >

37C3
37C6

Character table of C2×C37⋊C3

 class 123A3B6A6B37A37B37C37D37E37F37G37H37I37J37K37L74A74B74C74D74E74F74G74H74I74J74K74L
 size 1137373737333333333333333333333333
ρ1111111111111111111111111111111    trivial
ρ21-111-1-1111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ31-1ζ32ζ3ζ65ζ6111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 6
ρ411ζ32ζ3ζ3ζ32111111111111111111111111    linear of order 3
ρ51-1ζ3ζ32ζ6ζ65111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 6
ρ611ζ3ζ32ζ32ζ3111111111111111111111111    linear of order 3
ρ7330000ζ3726371037ζ3723378376ζ373237243718ζ37343733377ζ373637273711ζ37163712379ζ373137293714ζ37203715372ζ372837253721ζ37193713375ζ3730374373ζ373537223717ζ37203715372ζ372837253721ζ3730374373ζ373537223717ζ3723378376ζ373237243718ζ37343733377ζ373637273711ζ37163712379ζ373137293714ζ37193713375ζ3726371037    complex lifted from C37⋊C3
ρ8330000ζ37203715372ζ37163712379ζ373637273711ζ373137293714ζ373537223717ζ373237243718ζ372837253721ζ3730374373ζ37193713375ζ3726371037ζ3723378376ζ37343733377ζ3730374373ζ37193713375ζ3723378376ζ37343733377ζ37163712379ζ373637273711ζ373137293714ζ373537223717ζ373237243718ζ372837253721ζ3726371037ζ37203715372    complex lifted from C37⋊C3
ρ93-30000ζ37203715372ζ37163712379ζ373637273711ζ373137293714ζ373537223717ζ373237243718ζ372837253721ζ3730374373ζ37193713375ζ3726371037ζ3723378376ζ3734373337737303743733719371337537233783763734373337737163712379373637273711373137293714373537223717373237243718372837253721372637103737203715372    complex faithful
ρ103-30000ζ3730374373ζ373237243718ζ373537223717ζ372837253721ζ37343733377ζ373637273711ζ37193713375ζ3723378376ζ3726371037ζ37203715372ζ37163712379ζ37313729371437233783763726371037371637123793731372937143732372437183735372237173728372537213734373337737363727371137193713375372037153723730374373    complex faithful
ρ11330000ζ3723378376ζ373637273711ζ37343733377ζ37193713375ζ373137293714ζ373537223717ζ3726371037ζ37163712379ζ37203715372ζ3730374373ζ373237243718ζ372837253721ζ37163712379ζ37203715372ζ373237243718ζ372837253721ζ373637273711ζ37343733377ζ37193713375ζ373137293714ζ373537223717ζ3726371037ζ3730374373ζ3723378376    complex lifted from C37⋊C3
ρ123-30000ζ373237243718ζ37343733377ζ372837253721ζ37203715372ζ37193713375ζ373137293714ζ3730374373ζ373637273711ζ3723378376ζ37163712379ζ373537223717ζ372637103737363727371137233783763735372237173726371037373437333773728372537213720371537237193713375373137293714373037437337163712379373237243718    complex faithful
ρ133-30000ζ3723378376ζ373637273711ζ37343733377ζ37193713375ζ373137293714ζ373537223717ζ3726371037ζ37163712379ζ37203715372ζ3730374373ζ373237243718ζ37283725372137163712379372037153723732372437183728372537213736372737113734373337737193713375373137293714373537223717372637103737303743733723378376    complex faithful
ρ14330000ζ3730374373ζ373237243718ζ373537223717ζ372837253721ζ37343733377ζ373637273711ζ37193713375ζ3723378376ζ3726371037ζ37203715372ζ37163712379ζ373137293714ζ3723378376ζ3726371037ζ37163712379ζ373137293714ζ373237243718ζ373537223717ζ372837253721ζ37343733377ζ373637273711ζ37193713375ζ37203715372ζ3730374373    complex lifted from C37⋊C3
ρ15330000ζ37343733377ζ37193713375ζ37203715372ζ37163712379ζ3730374373ζ3726371037ζ373237243718ζ373137293714ζ373637273711ζ373537223717ζ372837253721ζ3723378376ζ373137293714ζ373637273711ζ372837253721ζ3723378376ζ37193713375ζ37203715372ζ37163712379ζ3730374373ζ3726371037ζ373237243718ζ373537223717ζ37343733377    complex lifted from C37⋊C3
ρ16330000ζ37193713375ζ3730374373ζ37163712379ζ373537223717ζ373237243718ζ3723378376ζ37343733377ζ3726371037ζ373137293714ζ372837253721ζ37203715372ζ373637273711ζ3726371037ζ373137293714ζ37203715372ζ373637273711ζ3730374373ζ37163712379ζ373537223717ζ373237243718ζ3723378376ζ37343733377ζ372837253721ζ37193713375    complex lifted from C37⋊C3
ρ17330000ζ372837253721ζ37203715372ζ3723378376ζ373637273711ζ37163712379ζ3730374373ζ373537223717ζ37193713375ζ37343733377ζ373137293714ζ3726371037ζ373237243718ζ37193713375ζ37343733377ζ3726371037ζ373237243718ζ37203715372ζ3723378376ζ373637273711ζ37163712379ζ3730374373ζ373537223717ζ373137293714ζ372837253721    complex lifted from C37⋊C3
ρ18330000ζ37163712379ζ373537223717ζ373137293714ζ3726371037ζ372837253721ζ37343733377ζ37203715372ζ373237243718ζ3730374373ζ3723378376ζ373637273711ζ37193713375ζ373237243718ζ3730374373ζ373637273711ζ37193713375ζ373537223717ζ373137293714ζ3726371037ζ372837253721ζ37343733377ζ37203715372ζ3723378376ζ37163712379    complex lifted from C37⋊C3
ρ193-30000ζ373637273711ζ373137293714ζ37193713375ζ3730374373ζ3726371037ζ372837253721ζ3723378376ζ373537223717ζ37163712379ζ373237243718ζ37343733377ζ3720371537237353722371737163712379373437333773720371537237313729371437193713375373037437337263710373728372537213723378376373237243718373637273711    complex faithful
ρ203-30000ζ372837253721ζ37203715372ζ3723378376ζ373637273711ζ37163712379ζ3730374373ζ373537223717ζ37193713375ζ37343733377ζ373137293714ζ3726371037ζ37323724371837193713375373437333773726371037373237243718372037153723723378376373637273711371637123793730374373373537223717373137293714372837253721    complex faithful
ρ213-30000ζ373537223717ζ372837253721ζ3726371037ζ3723378376ζ37203715372ζ37193713375ζ37163712379ζ37343733377ζ373237243718ζ373637273711ζ373137293714ζ373037437337343733377373237243718373137293714373037437337283725372137263710373723378376372037153723719371337537163712379373637273711373537223717    complex faithful
ρ22330000ζ373137293714ζ3726371037ζ3730374373ζ373237243718ζ3723378376ζ37203715372ζ373637273711ζ372837253721ζ373537223717ζ37343733377ζ37193713375ζ37163712379ζ372837253721ζ373537223717ζ37193713375ζ37163712379ζ3726371037ζ3730374373ζ373237243718ζ3723378376ζ37203715372ζ373637273711ζ37343733377ζ373137293714    complex lifted from C37⋊C3
ρ233-30000ζ373137293714ζ3726371037ζ3730374373ζ373237243718ζ3723378376ζ37203715372ζ373637273711ζ372837253721ζ373537223717ζ37343733377ζ37193713375ζ3716371237937283725372137353722371737193713375371637123793726371037373037437337323724371837233783763720371537237363727371137343733377373137293714    complex faithful
ρ24330000ζ373537223717ζ372837253721ζ3726371037ζ3723378376ζ37203715372ζ37193713375ζ37163712379ζ37343733377ζ373237243718ζ373637273711ζ373137293714ζ3730374373ζ37343733377ζ373237243718ζ373137293714ζ3730374373ζ372837253721ζ3726371037ζ3723378376ζ37203715372ζ37193713375ζ37163712379ζ373637273711ζ373537223717    complex lifted from C37⋊C3
ρ253-30000ζ3726371037ζ3723378376ζ373237243718ζ37343733377ζ373637273711ζ37163712379ζ373137293714ζ37203715372ζ372837253721ζ37193713375ζ3730374373ζ37353722371737203715372372837253721373037437337353722371737233783763732372437183734373337737363727371137163712379373137293714371937133753726371037    complex faithful
ρ263-30000ζ37343733377ζ37193713375ζ37203715372ζ37163712379ζ3730374373ζ3726371037ζ373237243718ζ373137293714ζ373637273711ζ373537223717ζ372837253721ζ372337837637313729371437363727371137283725372137233783763719371337537203715372371637123793730374373372637103737323724371837353722371737343733377    complex faithful
ρ273-30000ζ37163712379ζ373537223717ζ373137293714ζ3726371037ζ372837253721ζ37343733377ζ37203715372ζ373237243718ζ3730374373ζ3723378376ζ373637273711ζ3719371337537323724371837303743733736372737113719371337537353722371737313729371437263710373728372537213734373337737203715372372337837637163712379    complex faithful
ρ283-30000ζ37193713375ζ3730374373ζ37163712379ζ373537223717ζ373237243718ζ3723378376ζ37343733377ζ3726371037ζ373137293714ζ372837253721ζ37203715372ζ37363727371137263710373731372937143720371537237363727371137303743733716371237937353722371737323724371837233783763734373337737283725372137193713375    complex faithful
ρ29330000ζ373637273711ζ373137293714ζ37193713375ζ3730374373ζ3726371037ζ372837253721ζ3723378376ζ373537223717ζ37163712379ζ373237243718ζ37343733377ζ37203715372ζ373537223717ζ37163712379ζ37343733377ζ37203715372ζ373137293714ζ37193713375ζ3730374373ζ3726371037ζ372837253721ζ3723378376ζ373237243718ζ373637273711    complex lifted from C37⋊C3
ρ30330000ζ373237243718ζ37343733377ζ372837253721ζ37203715372ζ37193713375ζ373137293714ζ3730374373ζ373637273711ζ3723378376ζ37163712379ζ373537223717ζ3726371037ζ373637273711ζ3723378376ζ373537223717ζ3726371037ζ37343733377ζ372837253721ζ37203715372ζ37193713375ζ373137293714ζ3730374373ζ37163712379ζ373237243718    complex lifted from C37⋊C3

Smallest permutation representation of C2×C37⋊C3
On 74 points
Generators in S74
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)
(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)
(2 27 11)(3 16 21)(4 5 31)(6 20 14)(7 9 24)(8 35 34)(10 13 17)(12 28 37)(15 32 30)(18 36 23)(19 25 33)(22 29 26)(39 64 48)(40 53 58)(41 42 68)(43 57 51)(44 46 61)(45 72 71)(47 50 54)(49 65 74)(52 69 67)(55 73 60)(56 62 70)(59 66 63)

G:=sub<Sym(74)| (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74), (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), (2,27,11)(3,16,21)(4,5,31)(6,20,14)(7,9,24)(8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)(39,64,48)(40,53,58)(41,42,68)(43,57,51)(44,46,61)(45,72,71)(47,50,54)(49,65,74)(52,69,67)(55,73,60)(56,62,70)(59,66,63)>;

G:=Group( (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74), (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), (2,27,11)(3,16,21)(4,5,31)(6,20,14)(7,9,24)(8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)(39,64,48)(40,53,58)(41,42,68)(43,57,51)(44,46,61)(45,72,71)(47,50,54)(49,65,74)(52,69,67)(55,73,60)(56,62,70)(59,66,63) );

G=PermutationGroup([[(1,38),(2,39),(3,40),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,73),(37,74)], [(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)], [(2,27,11),(3,16,21),(4,5,31),(6,20,14),(7,9,24),(8,35,34),(10,13,17),(12,28,37),(15,32,30),(18,36,23),(19,25,33),(22,29,26),(39,64,48),(40,53,58),(41,42,68),(43,57,51),(44,46,61),(45,72,71),(47,50,54),(49,65,74),(52,69,67),(55,73,60),(56,62,70),(59,66,63)]])

C2×C37⋊C3 is a maximal subgroup of   C74.C6

Matrix representation of C2×C37⋊C3 in GL4(𝔽223) generated by

222000
0100
0010
0001
,
1000
01242021
0100
0010
,
39000
0100
06817338
01514749
G:=sub<GL(4,GF(223))| [222,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,124,1,0,0,202,0,1,0,1,0,0],[39,0,0,0,0,1,68,151,0,0,173,47,0,0,38,49] >;

C2×C37⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_{37}\rtimes C_3
% in TeX

G:=Group("C2xC37:C3");
// GroupNames label

G:=SmallGroup(222,2);
// by ID

G=gap.SmallGroup(222,2);
# by ID

G:=PCGroup([3,-2,-3,-37,707]);
// Polycyclic

G:=Group<a,b,c|a^2=b^37=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations

Export

Subgroup lattice of C2×C37⋊C3 in TeX
Character table of C2×C37⋊C3 in TeX

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