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G = C3⋊D21order 126 = 2·32·7

The semidirect product of C3 and D21 acting via D21/C21=C2

metabelian, supersoluble, monomial, A-group

Aliases: C3⋊D21, C211S3, C322D7, C7⋊(C3⋊S3), (C3×C21)⋊1C2, SmallGroup(126,15)

Series: Derived Chief Lower central Upper central

C1C3×C21 — C3⋊D21
C1C7C21C3×C21 — C3⋊D21
C3×C21 — C3⋊D21
C1

Generators and relations for C3⋊D21
 G = < a,b,c | a3=b21=c2=1, ab=ba, cac=a-1, cbc=b-1 >

63C2
21S3
21S3
21S3
21S3
9D7
7C3⋊S3
3D21
3D21
3D21
3D21

Smallest permutation representation of C3⋊D21
On 63 points
Generators in S63
(1 45 24)(2 46 25)(3 47 26)(4 48 27)(5 49 28)(6 50 29)(7 51 30)(8 52 31)(9 53 32)(10 54 33)(11 55 34)(12 56 35)(13 57 36)(14 58 37)(15 59 38)(16 60 39)(17 61 40)(18 62 41)(19 63 42)(20 43 22)(21 44 23)
(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)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 46)(23 45)(24 44)(25 43)(26 63)(27 62)(28 61)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)

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

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

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

C3⋊D21 is a maximal subgroup of   D7×C3⋊S3  S3×D21  C32⋊F7  He3⋊D7  C3⋊D63  C324F7  C33⋊D7
C3⋊D21 is a maximal quotient of   C3⋊Dic21  C3⋊D63  C32⋊D21  C33⋊D7

33 conjugacy classes

class 1  2 3A3B3C3D7A7B7C21A···21X
order12333377721···21
size16322222222···2

33 irreducible representations

dim11222
type+++++
imageC1C2S3D7D21
kernelC3⋊D21C3×C21C21C32C3
# reps114324

Matrix representation of C3⋊D21 in GL4(𝔽43) generated by

4800
353800
00206
002322
,
271800
253900
0010
0001
,
271800
361600
0010
002942
G:=sub<GL(4,GF(43))| [4,35,0,0,8,38,0,0,0,0,20,23,0,0,6,22],[27,25,0,0,18,39,0,0,0,0,1,0,0,0,0,1],[27,36,0,0,18,16,0,0,0,0,1,29,0,0,0,42] >;

C3⋊D21 in GAP, Magma, Sage, TeX

C_3\rtimes D_{21}
% in TeX

G:=Group("C3:D21");
// GroupNames label

G:=SmallGroup(126,15);
// by ID

G=gap.SmallGroup(126,15);
# by ID

G:=PCGroup([4,-2,-3,-3,-7,33,146,1731]);
// Polycyclic

G:=Group<a,b,c|a^3=b^21=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3⋊D21 in TeX

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