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G = C217D4order 168 = 23·3·7

1st semidirect product of C21 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C217D4, D422C2, C2.5D42, C14.12D6, C6.12D14, C222D21, Dic211C2, C42.12C22, (C2×C6)⋊2D7, (C2×C14)⋊4S3, (C2×C42)⋊2C2, C33(C7⋊D4), C73(C3⋊D4), SmallGroup(168,38)

Series: Derived Chief Lower central Upper central

C1C42 — C217D4
C1C7C21C42D42 — C217D4
C21C42 — C217D4
C1C2C22

Generators and relations for C217D4
 G = < a,b,c | a21=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
42C2
21C22
21C4
2C6
14S3
2C14
6D7
21D4
7D6
7Dic3
3D14
3Dic7
2C42
2D21
7C3⋊D4
3C7⋊D4

Smallest permutation representation of C217D4
On 84 points
Generators in S84
(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 63 28 80)(2 62 29 79)(3 61 30 78)(4 60 31 77)(5 59 32 76)(6 58 33 75)(7 57 34 74)(8 56 35 73)(9 55 36 72)(10 54 37 71)(11 53 38 70)(12 52 39 69)(13 51 40 68)(14 50 41 67)(15 49 42 66)(16 48 22 65)(17 47 23 64)(18 46 24 84)(19 45 25 83)(20 44 26 82)(21 43 27 81)
(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(11 12)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(35 42)(36 41)(37 40)(38 39)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 84)(60 83)(61 82)(62 81)(63 80)

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

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

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

C217D4 is a maximal subgroup of   Dic7.D6  Dic3.D14  D7×C3⋊D4  S3×C7⋊D4  D8411C2  D4×D21  D42D21
C217D4 is a maximal quotient of   C42.4Q8  C2.D84  D4⋊D21  D4.D21  Q82D21  C217Q16  C42.38D4

45 conjugacy classes

class 1 2A2B2C 3  4 6A6B6C7A7B7C14A···14I21A···21F42A···42R
order12223466677714···1421···2142···42
size112422422222222···22···22···2

45 irreducible representations

dim11112222222222
type+++++++++++
imageC1C2C2C2S3D4D6D7C3⋊D4D14D21C7⋊D4D42C217D4
kernelC217D4Dic21D42C2×C42C2×C14C21C14C2×C6C7C6C22C3C2C1
# reps111111132366612

Matrix representation of C217D4 in GL2(𝔽43) generated by

3531
3141
,
01
420
,
3634
347
G:=sub<GL(2,GF(43))| [35,31,31,41],[0,42,1,0],[36,34,34,7] >;

C217D4 in GAP, Magma, Sage, TeX

C_{21}\rtimes_7D_4
% in TeX

G:=Group("C21:7D4");
// GroupNames label

G:=SmallGroup(168,38);
// by ID

G=gap.SmallGroup(168,38);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,61,323,3604]);
// Polycyclic

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

Export

Subgroup lattice of C217D4 in TeX

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