D13:A4 - GroupNames

G = D₁₃⋊A₄ order 312 = 2³·3·13

The semidirect product of D₁₃ and A₄ acting via A₄/C₂²=C₃

metabelian, soluble, monomial, A-group

Aliases: D₁₃⋊A₄, C₁₃⋊A₄⋊C₂, C₁₃⋊(C₂×A₄), (C₂×C₂₆)⋊₂C₆, C₂²⋊(C₁₃⋊C₆), (C₂²×D₁₃)⋊₂C₃, SmallGroup(312,51)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₆ — D₁₃⋊A₄

Chief series

C₁ — C₁₃ — C₂×C₂₆ — C₁₃⋊A₄ — D₁₃⋊A₄

Lower central

C₂×C₂₆ — D₁₃⋊A₄

Upper central

C₁

Generators and relations for D₁₃⋊A₄
G = < a,b,c,d,e | a¹³=b²=c²=d²=e³=1, bab=a^-1, ac=ca, ad=da, eae^-1=a⁹, bc=cb, bd=db, ebe^-1=a⁸b, ece^-1=cd=dc, ede^-1=c >

C₁

₃C₂

₁₃C₂

₃₉C₂

₅₂C₃

C₁₃

C₂²

₃₉C₂²

₅₂C₆

D₁₃

₃C₂₆

₃D₁₃

₄C₁₃⋊C₃

₁₃C₂³

₁₃A₄

C₂×C₂₆

₃D₂₆

₄C₁₃⋊C₆

₁₃C₂×A₄

C₂²×D₁₃

C₁₃⋊A₄

D₁₃⋊A₄

Character table of D₁₃⋊A₄

class	1	2A	2B	2C	3A	3B	6A	6B	13A	13B	26A	26B	26C	26D	26E	26F
size	1	3	13	39	52	52	52	52	6	6	6	6	6	6	6	6
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	-1	-1	1	1	1	1	1	1	1	1	linear of order 2
ρ₃	1	1	1	1	ζ₃²	ζ₃	ζ₃	ζ₃²	1	1	1	1	1	1	1	1	linear of order 3
ρ₄	1	1	-1	-1	ζ₃	ζ₃²	ζ₆	ζ₆⁵	1	1	1	1	1	1	1	1	linear of order 6
ρ₅	1	1	-1	-1	ζ₃²	ζ₃	ζ₆⁵	ζ₆	1	1	1	1	1	1	1	1	linear of order 6
ρ₆	1	1	1	1	ζ₃	ζ₃²	ζ₃²	ζ₃	1	1	1	1	1	1	1	1	linear of order 3
ρ₇	3	-1	-3	1	0	0	0	0	3	3	-1	-1	-1	-1	-1	-1	orthogonal lifted from C₂×A₄
ρ₈	3	-1	3	-1	0	0	0	0	3	3	-1	-1	-1	-1	-1	-1	orthogonal lifted from A₄
ρ₉	6	6	0	0	0	0	0	0	^-1+√13/₂	^-1-√13/₂	^-1-√13/₂	^-1-√13/₂	^-1+√13/₂	^-1-√13/₂	^-1+√13/₂	^-1+√13/₂	orthogonal lifted from C₁₃⋊C₆
ρ₁₀	6	-2	0	0	0	0	0	0	^-1+√13/₂	^-1-√13/₂	-ζ₁₃¹¹+ζ₁₃⁸-ζ₁₃⁷-ζ₁₃⁶+ζ₁₃⁵-ζ₁₃²	-ζ₁₃¹¹-ζ₁₃⁸+ζ₁₃⁷+ζ₁₃⁶-ζ₁₃⁵-ζ₁₃²	ζ₁₃¹²-ζ₁₃¹⁰-ζ₁₃⁹-ζ₁₃⁴-ζ₁₃³+ζ₁₃	ζ₁₃¹¹-ζ₁₃⁸-ζ₁₃⁷-ζ₁₃⁶-ζ₁₃⁵+ζ₁₃²	-ζ₁₃¹²+ζ₁₃¹⁰-ζ₁₃⁹-ζ₁₃⁴+ζ₁₃³-ζ₁₃	-ζ₁₃¹²-ζ₁₃¹⁰+ζ₁₃⁹+ζ₁₃⁴-ζ₁₃³-ζ₁₃	orthogonal faithful
ρ₁₁	6	-2	0	0	0	0	0	0	^-1-√13/₂	^-1+√13/₂	-ζ₁₃¹²+ζ₁₃¹⁰-ζ₁₃⁹-ζ₁₃⁴+ζ₁₃³-ζ₁₃	ζ₁₃¹²-ζ₁₃¹⁰-ζ₁₃⁹-ζ₁₃⁴-ζ₁₃³+ζ₁₃	ζ₁₃¹¹-ζ₁₃⁸-ζ₁₃⁷-ζ₁₃⁶-ζ₁₃⁵+ζ₁₃²	-ζ₁₃¹²-ζ₁₃¹⁰+ζ₁₃⁹+ζ₁₃⁴-ζ₁₃³-ζ₁₃	-ζ₁₃¹¹-ζ₁₃⁸+ζ₁₃⁷+ζ₁₃⁶-ζ₁₃⁵-ζ₁₃²	-ζ₁₃¹¹+ζ₁₃⁸-ζ₁₃⁷-ζ₁₃⁶+ζ₁₃⁵-ζ₁₃²	orthogonal faithful
ρ₁₂	6	-2	0	0	0	0	0	0	^-1-√13/₂	^-1+√13/₂	-ζ₁₃¹²-ζ₁₃¹⁰+ζ₁₃⁹+ζ₁₃⁴-ζ₁₃³-ζ₁₃	-ζ₁₃¹²+ζ₁₃¹⁰-ζ₁₃⁹-ζ₁₃⁴+ζ₁₃³-ζ₁₃	-ζ₁₃¹¹-ζ₁₃⁸+ζ₁₃⁷+ζ₁₃⁶-ζ₁₃⁵-ζ₁₃²	ζ₁₃¹²-ζ₁₃¹⁰-ζ₁₃⁹-ζ₁₃⁴-ζ₁₃³+ζ₁₃	-ζ₁₃¹¹+ζ₁₃⁸-ζ₁₃⁷-ζ₁₃⁶+ζ₁₃⁵-ζ₁₃²	ζ₁₃¹¹-ζ₁₃⁸-ζ₁₃⁷-ζ₁₃⁶-ζ₁₃⁵+ζ₁₃²	orthogonal faithful
ρ₁₃	6	-2	0	0	0	0	0	0	^-1+√13/₂	^-1-√13/₂	-ζ₁₃¹¹-ζ₁₃⁸+ζ₁₃⁷+ζ₁₃⁶-ζ₁₃⁵-ζ₁₃²	ζ₁₃¹¹-ζ₁₃⁸-ζ₁₃⁷-ζ₁₃⁶-ζ₁₃⁵+ζ₁₃²	-ζ₁₃¹²-ζ₁₃¹⁰+ζ₁₃⁹+ζ₁₃⁴-ζ₁₃³-ζ₁₃	-ζ₁₃¹¹+ζ₁₃⁸-ζ₁₃⁷-ζ₁₃⁶+ζ₁₃⁵-ζ₁₃²	ζ₁₃¹²-ζ₁₃¹⁰-ζ₁₃⁹-ζ₁₃⁴-ζ₁₃³+ζ₁₃	-ζ₁₃¹²+ζ₁₃¹⁰-ζ₁₃⁹-ζ₁₃⁴+ζ₁₃³-ζ₁₃	orthogonal faithful
ρ₁₄	6	-2	0	0	0	0	0	0	^-1+√13/₂	^-1-√13/₂	ζ₁₃¹¹-ζ₁₃⁸-ζ₁₃⁷-ζ₁₃⁶-ζ₁₃⁵+ζ₁₃²	-ζ₁₃¹¹+ζ₁₃⁸-ζ₁₃⁷-ζ₁₃⁶+ζ₁₃⁵-ζ₁₃²	-ζ₁₃¹²+ζ₁₃¹⁰-ζ₁₃⁹-ζ₁₃⁴+ζ₁₃³-ζ₁₃	-ζ₁₃¹¹-ζ₁₃⁸+ζ₁₃⁷+ζ₁₃⁶-ζ₁₃⁵-ζ₁₃²	-ζ₁₃¹²-ζ₁₃¹⁰+ζ₁₃⁹+ζ₁₃⁴-ζ₁₃³-ζ₁₃	ζ₁₃¹²-ζ₁₃¹⁰-ζ₁₃⁹-ζ₁₃⁴-ζ₁₃³+ζ₁₃	orthogonal faithful
ρ₁₅	6	-2	0	0	0	0	0	0	^-1-√13/₂	^-1+√13/₂	ζ₁₃¹²-ζ₁₃¹⁰-ζ₁₃⁹-ζ₁₃⁴-ζ₁₃³+ζ₁₃	-ζ₁₃¹²-ζ₁₃¹⁰+ζ₁₃⁹+ζ₁₃⁴-ζ₁₃³-ζ₁₃	-ζ₁₃¹¹+ζ₁₃⁸-ζ₁₃⁷-ζ₁₃⁶+ζ₁₃⁵-ζ₁₃²	-ζ₁₃¹²+ζ₁₃¹⁰-ζ₁₃⁹-ζ₁₃⁴+ζ₁₃³-ζ₁₃	ζ₁₃¹¹-ζ₁₃⁸-ζ₁₃⁷-ζ₁₃⁶-ζ₁₃⁵+ζ₁₃²	-ζ₁₃¹¹-ζ₁₃⁸+ζ₁₃⁷+ζ₁₃⁶-ζ₁₃⁵-ζ₁₃²	orthogonal faithful
ρ₁₆	6	6	0	0	0	0	0	0	^-1-√13/₂	^-1+√13/₂	^-1+√13/₂	^-1+√13/₂	^-1-√13/₂	^-1+√13/₂	^-1-√13/₂	^-1-√13/₂	orthogonal lifted from C₁₃⋊C₆

Smallest permutation representation of D₁₃⋊A₄
►On 52 points

Generators in S₅₂

(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)

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

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

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

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

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

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

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

Matrix representation of D₁₃⋊A₄ ►in GL₆(𝔽₇₉)

16	1	0	0	0	0
62	0	1	0	0	0
18	0	0	1	0	0
61	0	0	0	1	0
17	0	0	0	0	1
62	78	78	78	78	78

77	77	78	14	0	13
16	17	16	78	78	15
64	63	63	2	17	65
17	17	17	78	62	0
77	77	77	15	17	14
1	1	1	64	62	64

37	0	77	24	24	77
53	39	34	11	37	58
2	74	26	68	45	24
0	57	7	44	7	57
77	22	43	66	24	72
26	5	63	37	60	65

16	0	11	3	3	11
8	5	50	31	23	53
68	34	60	48	29	78
0	69	74	7	74	69
11	10	40	59	71	45
71	45	15	23	42	76

16	0	0	16	2	0
62	0	0	61	62	1
18	1	0	19	33	0
61	0	0	45	45	0
16	78	78	32	18	78
63	0	1	62	61	0

G:=sub<GL(6,GF(79))| [16,62,18,61,17,62,1,0,0,0,0,78,0,1,0,0,0,78,0,0,1,0,0,78,0,0,0,1,0,78,0,0,0,0,1,78],[77,16,64,17,77,1,77,17,63,17,77,1,78,16,63,17,77,1,14,78,2,78,15,64,0,78,17,62,17,62,13,15,65,0,14,64],[37,53,2,0,77,26,0,39,74,57,22,5,77,34,26,7,43,63,24,11,68,44,66,37,24,37,45,7,24,60,77,58,24,57,72,65],[16,8,68,0,11,71,0,5,34,69,10,45,11,50,60,74,40,15,3,31,48,7,59,23,3,23,29,74,71,42,11,53,78,69,45,76],[16,62,18,61,16,63,0,0,1,0,78,0,0,0,0,0,78,1,16,61,19,45,32,62,2,62,33,45,18,61,0,1,0,0,78,0] >;

D₁₃⋊A₄ in GAP, Magma, Sage, TeX

D_{13}\rtimes A_4

% in TeX

G:=Group("D13:A4");

// GroupNames label

G:=SmallGroup(312,51);

// by ID

G=gap.SmallGroup(312,51);

# by ID

G:=PCGroup([5,-2,-3,-2,2,-13,97,188,7204,909]);

// Polycyclic

G:=Group<a,b,c,d,e|a^13=b^2=c^2=d^2=e^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^9,b*c=c*b,b*d=d*b,e*b*e^-1=a^8*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;

// generators/relations

Export

Subgroup lattice of D₁₃⋊A₄ in TeX
Character table of D₁₃⋊A₄ in TeX

77	77	78	14	0	13
16	17	16	78	78	15
64	63	63	2	17	65
17	17	17	78	62	0
77	77	77	15	17	14
1	1	1	64	62	64

37	0	77	24	24	77
53	39	34	11	37	58
2	74	26	68	45	24
0	57	7	44	7	57
77	22	43	66	24	72
26	5	63	37	60	65

16	0	11	3	3	11
8	5	50	31	23	53
68	34	60	48	29	78
0	69	74	7	74	69
11	10	40	59	71	45
71	45	15	23	42	76

77	77	78	14	0	13
16	17	16	78	78	15
64	63	63	2	17	65
17	17	17	78	62	0
77	77	77	15	17	14
1	1	1	64	62	64

37	0	77	24	24	77
53	39	34	11	37	58
2	74	26	68	45	24
0	57	7	44	7	57
77	22	43	66	24	72
26	5	63	37	60	65

16	0	11	3	3	11
8	5	50	31	23	53
68	34	60	48	29	78
0	69	74	7	74	69
11	10	40	59	71	45
71	45	15	23	42	76

G = D13⋊A4 order 312 = 23·3·13

The semidirect product of D13 and A4 acting via A4/C22=C3

G = D₁₃⋊A₄ order 312 = 2³·3·13

The semidirect product of D₁₃ and A₄ acting via A₄/C₂²=C₃

77	77	78	14	0	13
16	17	16	78	78	15
64	63	63	2	17	65
17	17	17	78	62	0
77	77	77	15	17	14
1	1	1	64	62	64

37	0	77	24	24	77
53	39	34	11	37	58
2	74	26	68	45	24
0	57	7	44	7	57
77	22	43	66	24	72
26	5	63	37	60	65

16	0	11	3	3	11
8	5	50	31	23	53
68	34	60	48	29	78
0	69	74	7	74	69
11	10	40	59	71	45
71	45	15	23	42	76