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G = A4xD17order 408 = 23·3·17

Direct product of A4 and D17

direct product, metabelian, soluble, monomial, A-group

Aliases: A4xD17, C17:(C2xA4), (C2xC34):C6, (C22xD17):C3, C22:(C3xD17), (A4xC17):2C2, SmallGroup(408,38)

Series: Derived Chief Lower central Upper central

C1C2xC34 — A4xD17
C1C17C2xC34A4xC17 — A4xD17
C2xC34 — A4xD17
C1

Generators and relations for A4xD17
 G = < a,b,c,d,e | a2=b2=c3=d17=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 308 in 24 conjugacy classes, 9 normal (all characteristic)
Quotients: C1, C2, C3, C6, A4, C2xA4, D17, C3xD17, A4xD17
3C2
17C2
51C2
4C3
51C22
51C22
68C6
3C34
3D17
4C51
17C23
3D34
3D34
4C3xD17
17C2xA4

Smallest permutation representation of A4xD17
On 68 points
Generators in S68
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(35 66)(36 67)(37 68)(38 52)(39 53)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 52)(32 53)(33 54)(34 55)
(18 42 56)(19 43 57)(20 44 58)(21 45 59)(22 46 60)(23 47 61)(24 48 62)(25 49 63)(26 50 64)(27 51 65)(28 35 66)(29 36 67)(30 37 68)(31 38 52)(32 39 53)(33 40 54)(34 41 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)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(25 27)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(49 51)(52 59)(53 58)(54 57)(55 56)(60 68)(61 67)(62 66)(63 65)

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

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

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

40 conjugacy classes

class 1 2A2B2C3A3B6A6B17A···17H34A···34H51A···51P
order1222336617···1734···3451···51
size1317514468682···26···68···8

40 irreducible representations

dim111122336
type++++++
imageC1C2C3C6D17C3xD17A4C2xA4A4xD17
kernelA4xD17A4xC17C22xD17C2xC34A4C22D17C17C1
# reps1122816118

Matrix representation of A4xD17 in GL5(F103)

10000
01000
00010
00100
00102102102
,
10000
01000
00001
00102102102
00100
,
460000
046000
00100
00102102102
00010
,
341000
1020000
00100
00010
00001
,
01000
10000
00100
00010
00001

G:=sub<GL(5,GF(103))| [1,0,0,0,0,0,1,0,0,0,0,0,0,1,102,0,0,1,0,102,0,0,0,0,102],[1,0,0,0,0,0,1,0,0,0,0,0,0,102,1,0,0,0,102,0,0,0,1,102,0],[46,0,0,0,0,0,46,0,0,0,0,0,1,102,0,0,0,0,102,1,0,0,0,102,0],[34,102,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4xD17 in GAP, Magma, Sage, TeX

A_4\times D_{17}
% in TeX

G:=Group("A4xD17");
// GroupNames label

G:=SmallGroup(408,38);
// by ID

G=gap.SmallGroup(408,38);
# by ID

G:=PCGroup([5,-2,-3,-2,2,-17,142,68,9604]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^17=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

Export

Subgroup lattice of A4xD17 in TeX

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