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G = C4xC17:C4order 272 = 24·17

Direct product of C4 and C17:C4

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

Aliases: C4xC17:C4, C17:C42, C68:2C4, Dic17:2C4, D34.4C22, C34.3(C2xC4), (C4xD17).6C2, D17.2(C2xC4), C2.2(C2xC17:C4), (C2xC17:C4).3C2, SmallGroup(272,31)

Series: Derived Chief Lower central Upper central

C1C17 — C4xC17:C4
C1C17D17D34C2xC17:C4 — C4xC17:C4
C17 — C4xC17:C4
C1C4

Generators and relations for C4xC17:C4
 G = < a,b,c | a4=b17=c4=1, ab=ba, ac=ca, cbc-1=b4 >

Subgroups: 222 in 30 conjugacy classes, 18 normal (10 characteristic)
Quotients: C1, C2, C4, C22, C2xC4, C42, C17:C4, C2xC17:C4, C4xC17:C4
17C2
17C2
17C4
17C4
17C22
17C4
17C4
17C4
17C2xC4
17C2xC4
17C2xC4
17C42

Smallest permutation representation of C4xC17:C4
On 68 points
Generators in S68
(1 52 18 35)(2 53 19 36)(3 54 20 37)(4 55 21 38)(5 56 22 39)(6 57 23 40)(7 58 24 41)(8 59 25 42)(9 60 26 43)(10 61 27 44)(11 62 28 45)(12 63 29 46)(13 64 30 47)(14 65 31 48)(15 66 32 49)(16 67 33 50)(17 68 34 51)
(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 52 18 35)(2 65 34 39)(3 61 33 43)(4 57 32 47)(5 53 31 51)(6 66 30 38)(7 62 29 42)(8 58 28 46)(9 54 27 50)(10 67 26 37)(11 63 25 41)(12 59 24 45)(13 55 23 49)(14 68 22 36)(15 64 21 40)(16 60 20 44)(17 56 19 48)

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

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

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

32 conjugacy classes

class 1 2A2B2C4A4B4C···4L17A17B17C17D34A34B34C34D68A···68H
order1222444···4171717173434343468···68
size1117171117···17444444444···4

32 irreducible representations

dim111111444
type+++++
imageC1C2C2C4C4C4C17:C4C2xC17:C4C4xC17:C4
kernelC4xC17:C4C4xD17C2xC17:C4Dic17C68C17:C4C4C2C1
# reps112228448

Matrix representation of C4xC17:C4 in GL5(F137)

370000
01000
00100
00010
00001
,
10000
0632763136
01000
00100
00010
,
10000
01000
0136632763
0136174109
0109741136

G:=sub<GL(5,GF(137))| [37,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,63,1,0,0,0,27,0,1,0,0,63,0,0,1,0,136,0,0,0],[1,0,0,0,0,0,1,136,136,109,0,0,63,1,74,0,0,27,74,1,0,0,63,109,136] >;

C4xC17:C4 in GAP, Magma, Sage, TeX

C_4\times C_{17}\rtimes C_4
% in TeX

G:=Group("C4xC17:C4");
// GroupNames label

G:=SmallGroup(272,31);
// by ID

G=gap.SmallGroup(272,31);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,20,46,5204,1614]);
// Polycyclic

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

Export

Subgroup lattice of C4xC17:C4 in TeX

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