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G = C4×D21order 168 = 23·3·7

Direct product of C4 and D21

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×D21, C842C2, C282S3, C122D7, C6.9D14, C2.1D42, C14.9D6, D42.2C2, Dic215C2, C42.9C22, C72(C4×S3), C32(C4×D7), C214(C2×C4), SmallGroup(168,35)

Series: Derived Chief Lower central Upper central

C1C21 — C4×D21
C1C7C21C42D42 — C4×D21
C21 — C4×D21
C1C4

Generators and relations for C4×D21
 G = < a,b,c | a4=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >

21C2
21C2
21C4
21C22
7S3
7S3
3D7
3D7
21C2×C4
7Dic3
7D6
3D14
3Dic7
7C4×S3
3C4×D7

Smallest permutation representation of C4×D21
On 84 points
Generators in S84
(1 70 32 63)(2 71 33 43)(3 72 34 44)(4 73 35 45)(5 74 36 46)(6 75 37 47)(7 76 38 48)(8 77 39 49)(9 78 40 50)(10 79 41 51)(11 80 42 52)(12 81 22 53)(13 82 23 54)(14 83 24 55)(15 84 25 56)(16 64 26 57)(17 65 27 58)(18 66 28 59)(19 67 29 60)(20 68 30 61)(21 69 31 62)
(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)
(1 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(43 68)(44 67)(45 66)(46 65)(47 64)(48 84)(49 83)(50 82)(51 81)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(61 71)(62 70)(63 69)

G:=sub<Sym(84)| (1,70,32,63)(2,71,33,43)(3,72,34,44)(4,73,35,45)(5,74,36,46)(6,75,37,47)(7,76,38,48)(8,77,39,49)(9,78,40,50)(10,79,41,51)(11,80,42,52)(12,81,22,53)(13,82,23,54)(14,83,24,55)(15,84,25,56)(16,64,26,57)(17,65,27,58)(18,66,28,59)(19,67,29,60)(20,68,30,61)(21,69,31,62), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(43,68)(44,67)(45,66)(46,65)(47,64)(48,84)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)>;

G:=Group( (1,70,32,63)(2,71,33,43)(3,72,34,44)(4,73,35,45)(5,74,36,46)(6,75,37,47)(7,76,38,48)(8,77,39,49)(9,78,40,50)(10,79,41,51)(11,80,42,52)(12,81,22,53)(13,82,23,54)(14,83,24,55)(15,84,25,56)(16,64,26,57)(17,65,27,58)(18,66,28,59)(19,67,29,60)(20,68,30,61)(21,69,31,62), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(43,68)(44,67)(45,66)(46,65)(47,64)(48,84)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69) );

G=PermutationGroup([(1,70,32,63),(2,71,33,43),(3,72,34,44),(4,73,35,45),(5,74,36,46),(6,75,37,47),(7,76,38,48),(8,77,39,49),(9,78,40,50),(10,79,41,51),(11,80,42,52),(12,81,22,53),(13,82,23,54),(14,83,24,55),(15,84,25,56),(16,64,26,57),(17,65,27,58),(18,66,28,59),(19,67,29,60),(20,68,30,61),(21,69,31,62)], [(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)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,36),(18,35),(19,34),(20,33),(21,32),(43,68),(44,67),(45,66),(46,65),(47,64),(48,84),(49,83),(50,82),(51,81),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,72),(61,71),(62,70),(63,69)])

C4×D21 is a maximal subgroup of
D21⋊C8  D42.C4  C56⋊S3  D28⋊S3  D12⋊D7  D21⋊Q8  D6.D14  C4×S3×D7  C28⋊D6  D8411C2  D42D21  Q83D21
C4×D21 is a maximal quotient of
C56⋊S3  C42.4Q8  C2.D84

48 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 7A7B7C12A12B14A14B14C21A···21F28A···28F42A···42F84A···84L
order1222344446777121214141421···2128···2842···4284···84
size11212121121212222222222···22···22···22···2

48 irreducible representations

dim11111222222222
type++++++++++
imageC1C2C2C2C4S3D6D7C4×S3D14D21C4×D7D42C4×D21
kernelC4×D21Dic21C84D42D21C28C14C12C7C6C4C3C2C1
# reps111141132366612

Matrix representation of C4×D21 in GL2(𝔽41) generated by

320
032
,
178
840
,
10
840
G:=sub<GL(2,GF(41))| [32,0,0,32],[17,8,8,40],[1,8,0,40] >;

C4×D21 in GAP, Magma, Sage, TeX

C_4\times D_{21}
% in TeX

G:=Group("C4xD21");
// GroupNames label

G:=SmallGroup(168,35);
// by ID

G=gap.SmallGroup(168,35);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,26,323,3604]);
// Polycyclic

G:=Group<a,b,c|a^4=b^21=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×D21 in TeX

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