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G = C4×C19⋊C3order 228 = 22·3·19

Direct product of C4 and C19⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C4×C19⋊C3, C76⋊C3, C192C12, C38.2C6, C2.(C2×C19⋊C3), (C2×C19⋊C3).2C2, SmallGroup(228,2)

Series: Derived Chief Lower central Upper central

C1C19 — C4×C19⋊C3
C1C19C38C2×C19⋊C3 — C4×C19⋊C3
C19 — C4×C19⋊C3
C1C4

Generators and relations for C4×C19⋊C3
 G = < a,b,c | a4=b19=c3=1, ab=ba, ac=ca, cbc-1=b11 >

19C3
19C6
19C12

Smallest permutation representation of C4×C19⋊C3
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 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)(21 27 31)(22 34 23)(24 29 26)(25 36 37)(28 38 32)(30 33 35)(40 46 50)(41 53 42)(43 48 45)(44 55 56)(47 57 51)(49 52 54)(59 65 69)(60 72 61)(62 67 64)(63 74 75)(66 76 70)(68 71 73)

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

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

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

C4×C19⋊C3 is a maximal subgroup of   C19⋊C24  Dic38⋊C3  D76⋊C3

36 conjugacy classes

class 1  2 3A3B4A4B6A6B12A12B12C12D19A···19F38A···38F76A···76L
order123344661212121219···1938···3876···76
size111919111919191919193···33···33···3

36 irreducible representations

dim111111333
type++
imageC1C2C3C4C6C12C19⋊C3C2×C19⋊C3C4×C19⋊C3
kernelC4×C19⋊C3C2×C19⋊C3C76C19⋊C3C38C19C4C2C1
# reps1122246612

Matrix representation of C4×C19⋊C3 in GL3(𝔽229) generated by

12200
01220
00122
,
962241
100
010
,
100
102223184
11335
G:=sub<GL(3,GF(229))| [122,0,0,0,122,0,0,0,122],[96,1,0,224,0,1,1,0,0],[1,102,1,0,223,133,0,184,5] >;

C4×C19⋊C3 in GAP, Magma, Sage, TeX

C_4\times C_{19}\rtimes C_3
% in TeX

G:=Group("C4xC19:C3");
// GroupNames label

G:=SmallGroup(228,2);
// by ID

G=gap.SmallGroup(228,2);
# by ID

G:=PCGroup([4,-2,-3,-2,-19,24,679]);
// Polycyclic

G:=Group<a,b,c|a^4=b^19=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^11>;
// generators/relations

Export

Subgroup lattice of C4×C19⋊C3 in TeX

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