C2^4.13D6 - GroupNames

G = C₂⁴.₁₃D₆ order 192 = 2⁶·3

2^nd non-split extension by C₂⁴ of D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₁₃D₆, C₂³.₃₇D₁₂, C₂³.₄Dic₆, (C₂×C₆).₁C₄², C₆.D₄⋊₆C₄, (C₂²×C₆).₇Q₈, C₂³.₄₇(C₄×S₃), C₂²⋊C₄⋊₃Dic₃, C₆.₁₅(C₂³⋊C₄), (C₂²×Dic₃)⋊₃C₄, (C₂²×C₆).₁₇₇D₄, C₃⋊₂(C₂³.₉D₄), C₂².₁(C₄×Dic₃), C₂².₃₇(D₆⋊C₄), C₂³.₂₂(C₃⋊D₄), (C₂³×C₆).₂₃C₂², C₂².₁(C₄⋊Dic₃), C₂³.₁₀(C₂×Dic₃), C₆.₄(C₂.C₄²), C₂.₅(C₆.C₄²), C₂².₂(Dic₃⋊C₄), C₂.₃(C₂³.₆D₆), C₂².₉(C₆.D₄), (C₂×C₆).₂(C₄⋊C₄), (C₃×C₂²⋊C₄)⋊₅C₄, (C₆×C₂²⋊C₄).₂C₂, (C₂×C₂²⋊C₄).₃S₃, (C₂²×C₆).₂₉(C₂×C₄), (C₂×C₆).₅₀(C₂²⋊C₄), (C₂×C₆.D₄).₂C₂, SmallGroup(192,86)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₂⁴.₁₃D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×C₆ — C₂³×C₆ — C₂×C₆.D₄ — C₂⁴.₁₃D₆

Lower central

C₃ — C₆ — C₂×C₆ — C₂⁴.₁₃D₆

Upper central

C₁ — C₂² — C₂⁴ — C₂×C₂²⋊C₄

Generators and relations for C₂⁴.₁₃D₆
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁶=b, f²=abcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf^-1=bd=db, be=eb, ece^-1=fcf^-1=cd=dc, de=ed, df=fd, fef^-1=cde⁵ >

Subgroups: 408 in 142 conjugacy classes, 51 normal (39 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₆, C₆, C₂×C₄, C₂³, C₂³, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂²⋊C₄, C₂²×C₄, C₂⁴, C₂×Dic₃, C₂×C₁₂, C₂²×C₆, C₂²×C₆, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₆.D₄, C₆.D₄, C₃×C₂²⋊C₄, C₃×C₂²⋊C₄, C₂²×Dic₃, C₂²×Dic₃, C₂²×C₁₂, C₂³×C₆, C₂³.₉D₄, C₂×C₆.D₄, C₆×C₂²⋊C₄, C₂⁴.₁₃D₆
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, Q₈, Dic₃, D₆, C₄², C₂²⋊C₄, C₄⋊C₄, Dic₆, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂.C₄², C₂³⋊C₄, C₄×Dic₃, Dic₃⋊C₄, C₄⋊Dic₃, D₆⋊C₄, C₆.D₄, C₂³.₉D₄, C₂³.₆D₆, C₆.C₄², C₂⁴.₁₃D₆

Smallest permutation representation of C₂⁴.₁₃D₆
►On 48 points

Generators in S₄₈

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

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

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

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

(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)

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

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	3	4A	4B	4C	4D	4E	···	4L	6A	···	6G	6H	6I	6J	6K	12A	···	12H
order	1	2	2	2	2	···	2	3	4	4	4	4	4	···	4	6	···	6	6	6	6	6	12	···	12
size	1	1	1	1	2	···	2	2	4	4	4	4	12	···	12	2	···	2	4	4	4	4	4	···	4

42 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+				+	+	-	-	+	-		+		+
image	C₁	C₂	C₂	C₄	C₄	C₄	S₃	D₄	Q₈	Dic₃	D₆	Dic₆	C₄×S₃	D₁₂	C₃⋊D₄	C₂³⋊C₄	C₂³.₆D₆
kernel	C₂⁴.₁₃D₆	C₂×C₆.D₄	C₆×C₂²⋊C₄	C₆.D₄	C₃×C₂²⋊C₄	C₂²×Dic₃	C₂×C₂²⋊C₄	C₂²×C₆	C₂²×C₆	C₂²⋊C₄	C₂⁴	C₂³	C₂³	C₂³	C₂³	C₆	C₂
# reps	1	2	1	4	4	4	1	3	1	2	1	2	4	2	4	2	4

Matrix representation of C₂⁴.₁₃D₆ ►in GL₆(𝔽₁₃)

12	0	0	0	0	0
0	12	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

12	0	0	0	0	0
0	12	0	0	0	0
0	0	12	2	0	0
0	0	0	1	0	0
0	0	0	5	12	5
0	0	0	0	0	1

12	0	0	0	0	0
0	12	0	0	0	0
0	0	1	11	0	0
0	0	0	12	0	0
0	0	5	8	12	5
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	12	0
0	0	0	0	0	12

10	3	0	0	0	0
10	7	0	0	0	0
0	0	12	0	3	0
0	0	0	0	0	1
0	0	8	5	1	0
0	0	0	1	0	0

5	5	0	0	0	0
0	8	0	0	0	0
0	0	12	0	0	0
0	0	12	1	0	0
0	0	0	0	12	5
0	0	1	0	10	1

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,2,1,5,0,0,0,0,0,12,0,0,0,0,0,5,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,5,0,0,0,11,12,8,0,0,0,0,0,12,0,0,0,0,0,5,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[10,10,0,0,0,0,3,7,0,0,0,0,0,0,12,0,8,0,0,0,0,0,5,1,0,0,3,0,1,0,0,0,0,1,0,0],[5,0,0,0,0,0,5,8,0,0,0,0,0,0,12,12,0,1,0,0,0,1,0,0,0,0,0,0,12,10,0,0,0,0,5,1] >;

C₂⁴.₁₃D₆ in GAP, Magma, Sage, TeX

C_2^4._{13}D_6

% in TeX

G:=Group("C2^4.13D6");

// GroupNames label

G:=SmallGroup(192,86);

// by ID

G=gap.SmallGroup(192,86);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,1123,851,6278]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=b,f^2=a*b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^5>;

// generators/relations

G = C24.13D6 order 192 = 26·3

2nd non-split extension by C24 of D6 acting via D6/C3=C22

G = C₂⁴.₁₃D₆ order 192 = 2⁶·3

2^nd non-split extension by C₂⁴ of D₆ acting via D₆/C₃=C₂²