C2^4.49D6 - GroupNames

G = C₂⁴.₄₉D₆ order 192 = 2⁶·3

38^th non-split extension by C₂⁴ of D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₄₉D₆, C₆.₈₇2₊¹⁺⁴, (C₆×D₄)⋊₁₂C₄, D₄⋊₆(C₂×Dic₃), (D₄×Dic₃)⋊₃₇C₂, (C₂×D₄)⋊₁₁Dic₃, (C₂×D₄).₂₅₁D₆, C₆.₄₅(C₂³×C₄), C₂³⋊₄(C₂×Dic₃), C₂.₅(D₄⋊₆D₆), C₁₂.₉₄(C₂²×C₄), (C₂×C₆).₂₉₃C₂⁴, C₄⋊Dic₃⋊₇₆C₂², (C₂²×D₄).₁₃S₃, (C₂²×C₄).₂₈₆D₆, C₂.₇(C₂³×Dic₃), (C₂×C₁₂).₅₄₁C₂³, C₃⋊₄(C₂².₁₁C₂⁴), (C₄×Dic₃)⋊₄₀C₂², (C₆×D₄).₂₇₀C₂², C₂².₄₅(S₃×C₂³), (C₂³×C₆).₇₅C₂², C₄.₁₇(C₂²×Dic₃), C₂³.₂₆D₆⋊₃₂C₂, C₆.D₄⋊₅₉C₂², C₂³.₂₁₄(C₂²×S₃), (C₂²×C₆).₂₂₉C₂³, C₂².₁(C₂²×Dic₃), (C₂²×C₁₂).₂₇₄C₂², (C₂×Dic₃).₂₈₃C₂³, (C₂²×Dic₃)⋊₃₁C₂², (D₄×C₂×C₆).₉C₂, (C₂×C₁₂)⋊₁₅(C₂×C₄), (C₃×D₄)⋊₂₀(C₂×C₄), (C₂×C₄)⋊₄(C₂×Dic₃), (C₂²×C₆)⋊₁₂(C₂×C₄), (C₂×C₆).₂₇(C₂²×C₄), (C₂×C₆.D₄)⋊₂₆C₂, (C₂×C₄).₆₂₄(C₂²×S₃), SmallGroup(192,1357)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₂⁴.₄₉D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂×Dic₃ — C₂²×Dic₃ — D₄×Dic₃ — C₂⁴.₄₉D₆

Lower central

C₃ — C₆ — C₂⁴.₄₉D₆

Upper central

C₁ — C₂² — C₂²×D₄

Generators and relations for C₂⁴.₄₉D₆
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁶=1, f²=c, ab=ba, ac=ca, eae^-1=faf^-1=ad=da, bc=cb, fbf^-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 712 in 338 conjugacy classes, 191 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², C₆, C₆, C₆, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²×D₄, C₄×Dic₃, C₄⋊Dic₃, C₆.D₄, C₂²×Dic₃, C₂²×C₁₂, C₆×D₄, C₂³×C₆, C₂².₁₁C₂⁴, C₂³.₂₆D₆, D₄×Dic₃, C₂×C₆.D₄, D₄×C₂×C₆, C₂⁴.₄₉D₆
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, C₂³, Dic₃, D₆, C₂²×C₄, C₂⁴, C₂×Dic₃, C₂²×S₃, C₂³×C₄, 2₊¹⁺⁴, C₂²×Dic₃, S₃×C₂³, C₂².₁₁C₂⁴, D₄⋊₆D₆, C₂³×Dic₃, C₂⁴.₄₉D₆

Smallest permutation representation of C₂⁴.₄₉D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	2D	···	2M	3	4A	4B	4C	4D	4E	···	4T	6A	···	6G	6H	···	6O	12A	12B	12C	12D
order	1	2	2	2	2	···	2	3	4	4	4	4	4	···	4	6	···	6	6	···	6	12	12	12	12
size	1	1	1	1	2	···	2	2	2	2	2	2	6	···	6	2	···	2	4	···	4	4	4	4	4

54 irreducible representations

dim

type

image

C₁

C₂

C₄

S₃

D₆

Dic₃

D₆

2₊¹⁺⁴

D₄⋊₆D₆

kernel

C₂⁴.₄₉D₆

C₂³.₂₆D₆

D₄×Dic₃

C₂×C₆.D₄

D₄×C₂×C₆

C₆×D₄

C₂²×D₄

C₂²×C₄

C₂×D₄

C₂⁴

C₆

C₂

# reps

Matrix representation of C₂⁴.₄₉D₆ ►in GL₆(𝔽₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	12	0	0	0
0	0	11	1	6	7
0	0	0	0	0	1
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	12	0	6	7
0	0	0	12	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	0	12	0	0
0	0	0	0	12	0
0	0	0	0	0	12

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

3	0	0	0	0	0
11	9	0	0	0	0
0	0	9	4	8	10
0	0	0	4	0	0
0	0	0	0	3	0
0	0	0	0	0	10

10	9	0	0	0	0
9	3	0	0	0	0
0	0	2	11	12	11
0	0	0	0	0	10
0	0	5	4	11	2
0	0	0	9	0	0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,11,0,0,0,0,0,1,0,0,0,0,0,6,0,1,0,0,0,7,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,6,0,1,0,0,0,7,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[3,11,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,4,4,0,0,0,0,8,0,3,0,0,0,10,0,0,10],[10,9,0,0,0,0,9,3,0,0,0,0,0,0,2,0,5,0,0,0,11,0,4,9,0,0,12,0,11,0,0,0,11,10,2,0] >;

C₂⁴.₄₉D₆ in GAP, Magma, Sage, TeX

C_2^4._{49}D_6

% in TeX

G:=Group("C2^4.49D6");

// GroupNames label

G:=SmallGroup(192,1357);

// by ID

G=gap.SmallGroup(192,1357);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,387,1123,6278]);

// Polycyclic

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

// generators/relations

G = C24.49D6 order 192 = 26·3

38th non-split extension by C24 of D6 acting via D6/C3=C22

G = C₂⁴.₄₉D₆ order 192 = 2⁶·3

38^th non-split extension by C₂⁴ of D₆ acting via D₆/C₃=C₂²