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## G = D4×C19⋊C3order 456 = 23·3·19

### Direct product of D4 and C19⋊C3

Aliases: D4×C19⋊C3, C763C6, (D4×C19)⋊C3, C193(C3×D4), (C2×C38)⋊5C6, C38.7(C2×C6), C4⋊(C2×C19⋊C3), (C4×C19⋊C3)⋊3C2, C222(C2×C19⋊C3), (C22×C19⋊C3)⋊3C2, C2.2(C22×C19⋊C3), (C2×C19⋊C3).7C22, SmallGroup(456,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C38 — D4×C19⋊C3
 Chief series C1 — C19 — C38 — C2×C19⋊C3 — C22×C19⋊C3 — D4×C19⋊C3
 Lower central C19 — C38 — D4×C19⋊C3
 Upper central C1 — C2 — D4

Generators and relations for D4×C19⋊C3
G = < a,b,c,d | a4=b2=c19=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c11 >

Smallest permutation representation of D4×C19⋊C3
On 76 points
Generators in S76
(1 39 20 58)(2 40 21 59)(3 41 22 60)(4 42 23 61)(5 43 24 62)(6 44 25 63)(7 45 26 64)(8 46 27 65)(9 47 28 66)(10 48 29 67)(11 49 30 68)(12 50 31 69)(13 51 32 70)(14 52 33 71)(15 53 34 72)(16 54 35 73)(17 55 36 74)(18 56 37 75)(19 57 38 76)
(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)
(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)| (1,39,20,58)(2,40,21,59)(3,41,22,60)(4,42,23,61)(5,43,24,62)(6,44,25,63)(7,45,26,64)(8,46,27,65)(9,47,28,66)(10,48,29,67)(11,49,30,68)(12,50,31,69)(13,51,32,70)(14,52,33,71)(15,53,34,72)(16,54,35,73)(17,55,36,74)(18,56,37,75)(19,57,38,76), (39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76), (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( (1,39,20,58)(2,40,21,59)(3,41,22,60)(4,42,23,61)(5,43,24,62)(6,44,25,63)(7,45,26,64)(8,46,27,65)(9,47,28,66)(10,48,29,67)(11,49,30,68)(12,50,31,69)(13,51,32,70)(14,52,33,71)(15,53,34,72)(16,54,35,73)(17,55,36,74)(18,56,37,75)(19,57,38,76), (39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76), (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([[(1,39,20,58),(2,40,21,59),(3,41,22,60),(4,42,23,61),(5,43,24,62),(6,44,25,63),(7,45,26,64),(8,46,27,65),(9,47,28,66),(10,48,29,67),(11,49,30,68),(12,50,31,69),(13,51,32,70),(14,52,33,71),(15,53,34,72),(16,54,35,73),(17,55,36,74),(18,56,37,75),(19,57,38,76)], [(39,58),(40,59),(41,60),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,68),(50,69),(51,70),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76)], [(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)]])

45 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 12A 12B 19A ··· 19F 38A ··· 38F 38G ··· 38R 76A ··· 76F order 1 2 2 2 3 3 4 6 6 6 6 6 6 12 12 19 ··· 19 38 ··· 38 38 ··· 38 76 ··· 76 size 1 1 2 2 19 19 2 19 19 38 38 38 38 38 38 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 3 3 3 6 type + + + + image C1 C2 C2 C3 C6 C6 D4 C3×D4 C19⋊C3 C2×C19⋊C3 C2×C19⋊C3 D4×C19⋊C3 kernel D4×C19⋊C3 C4×C19⋊C3 C22×C19⋊C3 D4×C19 C76 C2×C38 C19⋊C3 C19 D4 C4 C22 C1 # reps 1 1 2 2 2 4 1 2 6 6 12 6

Matrix representation of D4×C19⋊C3 in GL5(𝔽229)

 228 16 0 0 0 143 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 86 228 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 51 25 1 0 0 13 181 108 0 0 102 51 2
,
 94 0 0 0 0 0 94 0 0 0 0 0 73 138 157 0 0 91 39 90 0 0 56 197 117

G:=sub<GL(5,GF(229))| [228,143,0,0,0,16,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,86,0,0,0,0,228,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,51,13,102,0,0,25,181,51,0,0,1,108,2],[94,0,0,0,0,0,94,0,0,0,0,0,73,91,56,0,0,138,39,197,0,0,157,90,117] >;

D4×C19⋊C3 in GAP, Magma, Sage, TeX

D_4\times C_{19}\rtimes C_3
% in TeX

G:=Group("D4xC19:C3");
// GroupNames label

G:=SmallGroup(456,20);
// by ID

G=gap.SmallGroup(456,20);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-19,141,1064]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^19=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^11>;
// generators/relations

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