Copied to
clipboard

## G = D4×D17order 272 = 24·17

### Direct product of D4 and D17

Aliases: D4×D17, C41D34, C68⋊C22, D683C2, C221D34, D342C22, C34.5C23, Dic171C22, C172(C2×D4), (C2×C34)⋊C22, (C4×D17)⋊1C2, (D4×C17)⋊2C2, C17⋊D41C2, (C22×D17)⋊2C2, C2.6(C22×D17), SmallGroup(272,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C34 — D4×D17
 Chief series C1 — C17 — C34 — D34 — C22×D17 — D4×D17
 Lower central C17 — C34 — D4×D17
 Upper central C1 — C2 — D4

Generators and relations for D4×D17
G = < a,b,c,d | a4=b2=c17=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 470 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, D4, D4, C23, C2×D4, C17, D17, D17, C34, C34, Dic17, C68, D34, D34, D34, C2×C34, C4×D17, D68, C17⋊D4, D4×C17, C22×D17, D4×D17
Quotients: C1, C2, C22, D4, C23, C2×D4, D17, D34, C22×D17, D4×D17

Smallest permutation representation of D4×D17
On 68 points
Generators in S68
(1 37 25 52)(2 38 26 53)(3 39 27 54)(4 40 28 55)(5 41 29 56)(6 42 30 57)(7 43 31 58)(8 44 32 59)(9 45 33 60)(10 46 34 61)(11 47 18 62)(12 48 19 63)(13 49 20 64)(14 50 21 65)(15 51 22 66)(16 35 23 67)(17 36 24 68)
(35 67)(36 68)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(35 53)(36 52)(37 68)(38 67)(39 66)(40 65)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 17A ··· 17H 34A ··· 34H 34I ··· 34X 68A ··· 68H order 1 2 2 2 2 2 2 2 4 4 17 ··· 17 34 ··· 34 34 ··· 34 68 ··· 68 size 1 1 2 2 17 17 34 34 2 34 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D17 D34 D34 D4×D17 kernel D4×D17 C4×D17 D68 C17⋊D4 D4×C17 C22×D17 D17 D4 C4 C22 C1 # reps 1 1 1 2 1 2 2 8 8 16 8

Matrix representation of D4×D17 in GL4(𝔽137) generated by

 1 0 0 0 0 1 0 0 0 0 0 136 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 136
,
 7 1 0 0 23 23 0 0 0 0 1 0 0 0 0 1
,
 88 40 0 0 77 49 0 0 0 0 136 0 0 0 0 136
G:=sub<GL(4,GF(137))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,136,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,136],[7,23,0,0,1,23,0,0,0,0,1,0,0,0,0,1],[88,77,0,0,40,49,0,0,0,0,136,0,0,0,0,136] >;

D4×D17 in GAP, Magma, Sage, TeX

D_4\times D_{17}
% in TeX

G:=Group("D4xD17");
// GroupNames label

G:=SmallGroup(272,40);
// by ID

G=gap.SmallGroup(272,40);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,97,6404]);
// Polycyclic

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

׿
×
𝔽