Copied to
clipboard

## G = C4×D21order 168 = 23·3·7

### Direct product of C4 and D21

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

 Derived series C1 — C21 — C4×D21
 Chief series C1 — C7 — C21 — C42 — D42 — C4×D21
 Lower central C21 — C4×D21
 Upper central C1 — C4

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

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

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

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

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

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 2A 2B 2C 3 4A 4B 4C 4D 6 7A 7B 7C 12A 12B 14A 14B 14C 21A ··· 21F 28A ··· 28F 42A ··· 42F 84A ··· 84L order 1 2 2 2 3 4 4 4 4 6 7 7 7 12 12 14 14 14 21 ··· 21 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 21 21 2 1 1 21 21 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D6 D7 C4×S3 D14 D21 C4×D7 D42 C4×D21 kernel C4×D21 Dic21 C84 D42 D21 C28 C14 C12 C7 C6 C4 C3 C2 C1 # reps 1 1 1 1 4 1 1 3 2 3 6 6 6 12

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

 32 0 0 32
,
 17 8 8 40
,
 1 0 8 40
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

׿
×
𝔽