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G = C4xD17order 136 = 23·17

Direct product of C4 and D17

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

Aliases: C4xD17, C68:2C2, C2.1D34, D34.2C2, Dic17:2C2, C34.2C22, C17:2(C2xC4), SmallGroup(136,5)

Series: Derived Chief Lower central Upper central

C1C17 — C4xD17
C1C17C34D34 — C4xD17
C17 — C4xD17
C1C4

Generators and relations for C4xD17
 G = < a,b,c | a4=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 96 in 16 conjugacy classes, 11 normal (9 characteristic)
Quotients: C1, C2, C4, C22, C2xC4, D17, D34, C4xD17
17C2
17C2
17C22
17C4
17C2xC4

Smallest permutation representation of C4xD17
On 68 points
Generators in S68
(1 63 20 38)(2 64 21 39)(3 65 22 40)(4 66 23 41)(5 67 24 42)(6 68 25 43)(7 52 26 44)(8 53 27 45)(9 54 28 46)(10 55 29 47)(11 56 30 48)(12 57 31 49)(13 58 32 50)(14 59 33 51)(15 60 34 35)(16 61 18 36)(17 62 19 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 21)(19 20)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(35 40)(36 39)(37 38)(41 51)(42 50)(43 49)(44 48)(45 47)(52 56)(53 55)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)

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

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

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

C4xD17 is a maximal subgroup of
C8:D17  C68.C4  D34.4C4  C68:C4  D68:5C2  D4:2D17  D68:C2  D51:2C4
C4xD17 is a maximal quotient of
C8:D17  C34.D4  D34:C4  D51:2C4

40 conjugacy classes

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

40 irreducible representations

dim11111222
type++++++
imageC1C2C2C2C4D17D34C4xD17
kernelC4xD17Dic17C68D34D17C4C2C1
# reps111148816

Matrix representation of C4xD17 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] >;

C4xD17 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 C4xD17 in TeX

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