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G = C3×D28order 168 = 23·3·7

Direct product of C3 and D28

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

Aliases: C3×D28, C215D4, C285C6, C843C2, C123D7, D144C6, C6.15D14, C42.15C22, C4⋊(C3×D7), C74(C3×D4), (C6×D7)⋊4C2, C2.4(C6×D7), C14.11(C2×C6), SmallGroup(168,26)

Series: Derived Chief Lower central Upper central

C1C14 — C3×D28
C1C7C14C42C6×D7 — C3×D28
C7C14 — C3×D28
C1C6C12

Generators and relations for C3×D28
 G = < a,b,c | a3=b28=c2=1, ab=ba, ac=ca, cbc=b-1 >

14C2
14C2
7C22
7C22
14C6
14C6
2D7
2D7
7D4
7C2×C6
7C2×C6
2C3×D7
2C3×D7
7C3×D4

Smallest permutation representation of C3×D28
On 84 points
Generators in S84
(1 65 39)(2 66 40)(3 67 41)(4 68 42)(5 69 43)(6 70 44)(7 71 45)(8 72 46)(9 73 47)(10 74 48)(11 75 49)(12 76 50)(13 77 51)(14 78 52)(15 79 53)(16 80 54)(17 81 55)(18 82 56)(19 83 29)(20 84 30)(21 57 31)(22 58 32)(23 59 33)(24 60 34)(25 61 35)(26 62 36)(27 63 37)(28 64 38)
(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(49 56)(50 55)(51 54)(52 53)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)

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

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

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

C3×D28 is a maximal subgroup of
C21⋊D8  C3⋊D56  C28.D6  C6.D28  D285S3  D28⋊S3  C28⋊D6  C3×D4×D7

51 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F7A7B7C12A12B14A14B14C21A···21F28A···28F42A···42F84A···84L
order1222334666666777121214141421···2128···2842···4284···84
size1114141121114141414222222222···22···22···22···2

51 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C3C6C6D4D7C3×D4D14C3×D7D28C6×D7C3×D28
kernelC3×D28C84C6×D7D28C28D14C21C12C7C6C4C3C2C1
# reps112224132366612

Matrix representation of C3×D28 in GL2(𝔽337) generated by

1280
0128
,
313275
124222
,
305281
26532
G:=sub<GL(2,GF(337))| [128,0,0,128],[313,124,275,222],[305,265,281,32] >;

C3×D28 in GAP, Magma, Sage, TeX

C_3\times D_{28}
% in TeX

G:=Group("C3xD28");
// GroupNames label

G:=SmallGroup(168,26);
// by ID

G=gap.SmallGroup(168,26);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-7,141,66,3604]);
// Polycyclic

G:=Group<a,b,c|a^3=b^28=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D28 in TeX

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