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G = C4×C19⋊C6order 456 = 23·3·19

Direct product of C4 and C19⋊C6

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

Aliases: C4×C19⋊C6, C762C6, D19⋊C12, D38.C6, Dic192C6, (C4×D19)⋊C3, C19⋊C122C2, C191(C2×C12), C38.2(C2×C6), C19⋊C31(C2×C4), (C2×C19⋊C6).C2, (C4×C19⋊C3)⋊2C2, C2.1(C2×C19⋊C6), (C2×C19⋊C3).2C22, SmallGroup(456,8)

Series: Derived Chief Lower central Upper central

C1C19 — C4×C19⋊C6
C1C19C38C2×C19⋊C3C2×C19⋊C6 — C4×C19⋊C6
C19 — C4×C19⋊C6
C1C4

Generators and relations for C4×C19⋊C6
 G = < a,b,c | a4=b19=c6=1, ab=ba, ac=ca, cbc-1=b12 >

19C2
19C2
19C3
19C4
19C22
19C6
19C6
19C6
19C2×C4
19C12
19C2×C6
19C12
19C2×C12

Smallest permutation representation of C4×C19⋊C6
On 76 points
Generators in S76
(1 58 20 39)(2 59 21 40)(3 60 22 41)(4 61 23 42)(5 62 24 43)(6 63 25 44)(7 64 26 45)(8 65 27 46)(9 66 28 47)(10 67 29 48)(11 68 30 49)(12 69 31 50)(13 70 32 51)(14 71 33 52)(15 72 34 53)(16 73 35 54)(17 74 36 55)(18 75 37 56)(19 76 38 57)
(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 69 70 71 72 73 74 75 76)
(2 9 8 19 12 13)(3 17 15 18 4 6)(5 14 10 16 7 11)(21 28 27 38 31 32)(22 36 34 37 23 25)(24 33 29 35 26 30)(40 47 46 57 50 51)(41 55 53 56 42 44)(43 52 48 54 45 49)(59 66 65 76 69 70)(60 74 72 75 61 63)(62 71 67 73 64 68)

G:=sub<Sym(76)| (1,58,20,39)(2,59,21,40)(3,60,22,41)(4,61,23,42)(5,62,24,43)(6,63,25,44)(7,64,26,45)(8,65,27,46)(9,66,28,47)(10,67,29,48)(11,68,30,49)(12,69,31,50)(13,70,32,51)(14,71,33,52)(15,72,34,53)(16,73,35,54)(17,74,36,55)(18,75,37,56)(19,76,38,57), (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)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76), (2,9,8,19,12,13)(3,17,15,18,4,6)(5,14,10,16,7,11)(21,28,27,38,31,32)(22,36,34,37,23,25)(24,33,29,35,26,30)(40,47,46,57,50,51)(41,55,53,56,42,44)(43,52,48,54,45,49)(59,66,65,76,69,70)(60,74,72,75,61,63)(62,71,67,73,64,68)>;

G:=Group( (1,58,20,39)(2,59,21,40)(3,60,22,41)(4,61,23,42)(5,62,24,43)(6,63,25,44)(7,64,26,45)(8,65,27,46)(9,66,28,47)(10,67,29,48)(11,68,30,49)(12,69,31,50)(13,70,32,51)(14,71,33,52)(15,72,34,53)(16,73,35,54)(17,74,36,55)(18,75,37,56)(19,76,38,57), (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)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76), (2,9,8,19,12,13)(3,17,15,18,4,6)(5,14,10,16,7,11)(21,28,27,38,31,32)(22,36,34,37,23,25)(24,33,29,35,26,30)(40,47,46,57,50,51)(41,55,53,56,42,44)(43,52,48,54,45,49)(59,66,65,76,69,70)(60,74,72,75,61,63)(62,71,67,73,64,68) );

G=PermutationGroup([(1,58,20,39),(2,59,21,40),(3,60,22,41),(4,61,23,42),(5,62,24,43),(6,63,25,44),(7,64,26,45),(8,65,27,46),(9,66,28,47),(10,67,29,48),(11,68,30,49),(12,69,31,50),(13,70,32,51),(14,71,33,52),(15,72,34,53),(16,73,35,54),(17,74,36,55),(18,75,37,56),(19,76,38,57)], [(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),(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76)], [(2,9,8,19,12,13),(3,17,15,18,4,6),(5,14,10,16,7,11),(21,28,27,38,31,32),(22,36,34,37,23,25),(24,33,29,35,26,30),(40,47,46,57,50,51),(41,55,53,56,42,44),(43,52,48,54,45,49),(59,66,65,76,69,70),(60,74,72,75,61,63),(62,71,67,73,64,68)])

36 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A···6F12A···12H19A19B19C38A38B38C76A···76F
order12223344446···612···1219191938383876···76
size111919191911191919···1919···196666666···6

36 irreducible representations

dim1111111111666
type++++++
imageC1C2C2C2C3C4C6C6C6C12C19⋊C6C2×C19⋊C6C4×C19⋊C6
kernelC4×C19⋊C6C19⋊C12C4×C19⋊C3C2×C19⋊C6C4×D19C19⋊C6Dic19C76D38D19C4C2C1
# reps1111242228336

Matrix representation of C4×C19⋊C6 in GL6(𝔽229)

12200000
01220000
00122000
00012200
00001220
00000122
,
919220138109228
929220138109228
9110220138109228
919221138109228
919220139109228
919220138110228
,
22012762148221120
000010
14817315619228109
29811272011391
001000
1392011278129110

G:=sub<GL(6,GF(229))| [122,0,0,0,0,0,0,122,0,0,0,0,0,0,122,0,0,0,0,0,0,122,0,0,0,0,0,0,122,0,0,0,0,0,0,122],[91,92,91,91,91,91,9,9,10,9,9,9,220,220,220,221,220,220,138,138,138,138,139,138,109,109,109,109,109,110,228,228,228,228,228,228],[220,0,148,29,0,139,127,0,173,81,0,201,62,0,156,127,1,127,148,0,192,201,0,81,221,1,28,139,0,29,120,0,109,1,0,110] >;

C4×C19⋊C6 in GAP, Magma, Sage, TeX

C_4\times C_{19}\rtimes C_6
% in TeX

G:=Group("C4xC19:C6");
// GroupNames label

G:=SmallGroup(456,8);
// by ID

G=gap.SmallGroup(456,8);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-19,66,10804,1064]);
// Polycyclic

G:=Group<a,b,c|a^4=b^19=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^12>;
// generators/relations

Export

Subgroup lattice of C4×C19⋊C6 in TeX

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