Copied to
clipboard

G = C4×D17order 136 = 23·17

Direct product of C4 and D17

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

Aliases: C4×D17, C682C2, C2.1D34, D34.2C2, Dic172C2, C34.2C22, C172(C2×C4), SmallGroup(136,5)

Series: Derived Chief Lower central Upper central

C1C17 — C4×D17
C1C17C34D34 — C4×D17
C17 — C4×D17
C1C4

Generators and relations for C4×D17
 G = < a,b,c | a4=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >

17C2
17C2
17C22
17C4
17C2×C4

Smallest permutation representation of C4×D17
On 68 points
Generators in S68
(1 60 32 38)(2 61 33 39)(3 62 34 40)(4 63 18 41)(5 64 19 42)(6 65 20 43)(7 66 21 44)(8 67 22 45)(9 68 23 46)(10 52 24 47)(11 53 25 48)(12 54 26 49)(13 55 27 50)(14 56 28 51)(15 57 29 35)(16 58 30 36)(17 59 31 37)
(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)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 28)(19 27)(20 26)(21 25)(22 24)(29 34)(30 33)(31 32)(35 40)(36 39)(37 38)(41 51)(42 50)(43 49)(44 48)(45 47)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 60)

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

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

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

C4×D17 is a maximal subgroup of
C8⋊D17  C68.C4  D34.4C4  C68⋊C4  D685C2  D42D17  D68⋊C2  D512C4
C4×D17 is a maximal quotient of
C8⋊D17  C34.D4  D34⋊C4  D512C4

40 conjugacy classes

class 1 2A2B2C4A4B4C4D17A···17H34A···34H68A···68P
order1222444417···1734···3468···68
size1117171117172···22···22···2

40 irreducible representations

dim11111222
type++++++
imageC1C2C2C2C4D17D34C4×D17
kernelC4×D17Dic17C68D34D17C4C2C1
# reps111148816

Matrix representation of C4×D17 in GL3(𝔽137) generated by

3700
01360
00136
,
100
001
013676
,
13600
001
010
G:=sub<GL(3,GF(137))| [37,0,0,0,136,0,0,0,136],[1,0,0,0,0,136,0,1,76],[136,0,0,0,0,1,0,1,0] >;

C4×D17 in GAP, Magma, Sage, TeX

C_4\times D_{17}
% in TeX

G:=Group("C4xD17");
// GroupNames label

G:=SmallGroup(136,5);
// by ID

G=gap.SmallGroup(136,5);
# by ID

G:=PCGroup([4,-2,-2,-2,-17,21,2051]);
// Polycyclic

G:=Group<a,b,c|a^4=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×D17 in TeX

׿
×
𝔽