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## G = D4×C13⋊C3order 312 = 23·3·13

### Direct product of D4 and C13⋊C3

Aliases: D4×C13⋊C3, C523C6, (D4×C13)⋊C3, C133(C3×D4), (C2×C26)⋊5C6, C26.7(C2×C6), C4⋊(C2×C13⋊C3), (C4×C13⋊C3)⋊3C2, C222(C2×C13⋊C3), (C22×C13⋊C3)⋊3C2, C2.2(C22×C13⋊C3), (C2×C13⋊C3).7C22, SmallGroup(312,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D4×C13⋊C3
 Chief series C1 — C13 — C26 — C2×C13⋊C3 — C22×C13⋊C3 — D4×C13⋊C3
 Lower central C13 — C26 — D4×C13⋊C3
 Upper central C1 — C2 — D4

Generators and relations for D4×C13⋊C3
G = < a,b,c,d | a4=b2=c13=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >

Smallest permutation representation of D4×C13⋊C3
On 52 points
Generators in S52
(1 40 14 27)(2 41 15 28)(3 42 16 29)(4 43 17 30)(5 44 18 31)(6 45 19 32)(7 46 20 33)(8 47 21 34)(9 48 22 35)(10 49 23 36)(11 50 24 37)(12 51 25 38)(13 52 26 39)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)
(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)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)(41 43 49)(42 46 45)(44 52 50)(47 48 51)

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

G:=Group( (1,40,14,27)(2,41,15,28)(3,42,16,29)(4,43,17,30)(5,44,18,31)(6,45,19,32)(7,46,20,33)(8,47,21,34)(9,48,22,35)(10,49,23,36)(11,50,24,37)(12,51,25,38)(13,52,26,39), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52), (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), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51) );

G=PermutationGroup([[(1,40,14,27),(2,41,15,28),(3,42,16,29),(4,43,17,30),(5,44,18,31),(6,45,19,32),(7,46,20,33),(8,47,21,34),(9,48,22,35),(10,49,23,36),(11,50,24,37),(12,51,25,38),(13,52,26,39)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52)], [(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)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12),(15,17,23),(16,20,19),(18,26,24),(21,22,25),(28,30,36),(29,33,32),(31,39,37),(34,35,38),(41,43,49),(42,46,45),(44,52,50),(47,48,51)]])

35 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 12A 12B 13A 13B 13C 13D 26A 26B 26C 26D 26E ··· 26L 52A 52B 52C 52D order 1 2 2 2 3 3 4 6 6 6 6 6 6 12 12 13 13 13 13 26 26 26 26 26 ··· 26 52 52 52 52 size 1 1 2 2 13 13 2 13 13 26 26 26 26 26 26 3 3 3 3 3 3 3 3 6 ··· 6 6 6 6 6

35 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 C13⋊C3 C2×C13⋊C3 C2×C13⋊C3 D4×C13⋊C3 kernel D4×C13⋊C3 C4×C13⋊C3 C22×C13⋊C3 D4×C13 C52 C2×C26 C13⋊C3 C13 D4 C4 C22 C1 # reps 1 1 2 2 2 4 1 2 4 4 8 4

Matrix representation of D4×C13⋊C3 in GL5(𝔽157)

 0 1 0 0 0 156 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 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 0 135 46 1 0 0 1 0 0 0 0 0 1 0
,
 144 0 0 0 0 0 144 0 0 0 0 0 1 0 0 0 0 110 134 46 0 0 97 47 22

G:=sub<GL(5,GF(157))| [0,156,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,135,1,0,0,0,46,0,1,0,0,1,0,0],[144,0,0,0,0,0,144,0,0,0,0,0,1,110,97,0,0,0,134,47,0,0,0,46,22] >;

D4×C13⋊C3 in GAP, Magma, Sage, TeX

D_4\times C_{13}\rtimes C_3
% in TeX

G:=Group("D4xC13:C3");
// GroupNames label

G:=SmallGroup(312,23);
// by ID

G=gap.SmallGroup(312,23);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-13,141,464]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^13=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^9>;
// generators/relations

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