C2^4.52D6 - GroupNames

G = C₂⁴.₅₂D₆ order 192 = 2⁶·3

41^st non-split extension by C₂⁴ of D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₅₂D₆, C₆.₉₀2₊¹⁺⁴, (C₂×C₁₂)⋊₁₄D₄, D₆⋊₃D₄⋊₄₀C₂, C₁₂⋊₃D₄⋊₂₉C₂, (C₂×D₄).₂₃₀D₆, C₁₂.₂₅₂(C₂×D₄), (C₂²×D₄)⋊₁₄S₃, C₂⁴⋊₄S₃⋊₁₃C₂, (C₂×D₁₂)⋊₅₇C₂², (C₂×C₆).₃₀₀C₂⁴, C₄⋊Dic₃⋊₇₈C₂², C₆.₁₄₇(C₂²×D₄), (C₂²×C₄).₂₈₈D₆, C₂³.₁₂D₆⋊₂₈C₂, C₂.₉₃(D₄⋊₆D₆), (C₂×C₁₂).₅₄₅C₂³, C₃⋊₆(C₂².₂₉C₂⁴), (C₄×Dic₃)⋊₄₂C₂², (C₂×Dic₆)⋊₆₈C₂², (C₆×D₄).₂₇₁C₂², (C₂³×C₆).₇₉C₂², C₆.D₄⋊₃₉C₂², C₂³.₂₆D₆⋊₃₃C₂, C₂³.₁₄₆(C₂²×S₃), C₂².₃₁₃(S₃×C₂³), (C₂²×C₆).₂₃₄C₂³, (C₂²×S₃).₁₃₁C₂³, (C₂²×C₁₂).₂₇₇C₂², (C₂×Dic₃).₁₅₅C₂³, (D₄×C₂×C₆)⋊₇C₂, (C₂×C₄)⋊₆(C₃⋊D₄), (S₃×C₂×C₄)⋊₃₁C₂², C₄.₉₇(C₂×C₃⋊D₄), (C₂×C₄○D₁₂)⋊₂₉C₂, (C₂×C₆).₅₈₃(C₂×D₄), (C₂×C₃⋊D₄)⋊₂₈C₂², C₂².₃₆(C₂×C₃⋊D₄), C₂.₂₀(C₂²×C₃⋊D₄), (C₂×C₄).₆₂₈(C₂²×S₃), SmallGroup(192,1364)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₂⁴.₅₂D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×S₃ — S₃×C₂×C₄ — C₂×C₄○D₁₂ — C₂⁴.₅₂D₆

Lower central

C₃ — C₂×C₆ — C₂⁴.₅₂D₆

Upper central

C₁ — C₂² — C₂²×D₄

Generators and relations for C₂⁴.₅₂D₆
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁶=f²=d, ab=ba, ac=ca, eae^-1=ad=da, faf^-1=acd, bc=cb, fbf^-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e⁵ >

Subgroups: 904 in 334 conjugacy classes, 111 normal (21 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, Dic₆, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₄²⋊C₂, C₂²≀C₂, C₄⋊D₄, C₄.₄D₄, C₄⋊₁D₄, C₂²×D₄, C₂×C₄○D₄, C₄×Dic₃, C₄⋊Dic₃, C₆.D₄, C₂×Dic₆, S₃×C₂×C₄, C₂×D₁₂, C₄○D₁₂, C₂×C₃⋊D₄, C₂²×C₁₂, C₆×D₄, C₆×D₄, C₂³×C₆, C₂².₂₉C₂⁴, C₂³.₂₆D₆, C₂³.₁₂D₆, D₆⋊₃D₄, C₁₂⋊₃D₄, C₂⁴⋊₄S₃, C₂×C₄○D₁₂, D₄×C₂×C₆, C₂⁴.₅₂D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₂⁴, C₃⋊D₄, C₂²×S₃, C₂²×D₄, 2₊¹⁺⁴, C₂×C₃⋊D₄, S₃×C₂³, C₂².₂₉C₂⁴, D₄⋊₆D₆, C₂²×C₃⋊D₄, C₂⁴.₅₂D₆

Smallest permutation representation of C₂⁴.₅₂D₆
►On 48 points

Generators in S₄₈

(2 8)(4 10)(6 12)(13 19)(15 21)(17 23)(25 44)(26 39)(27 46)(28 41)(29 48)(30 43)(31 38)(32 45)(33 40)(34 47)(35 42)(36 37)

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

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

C₃⋊D₄

2₊¹⁺⁴

D₄⋊₆D₆

kernel

C₂⁴.₅₂D₆

C₂³.₂₆D₆

C₂³.₁₂D₆

D₆⋊₃D₄

C₁₂⋊₃D₄

C₂⁴⋊₄S₃

C₂×C₄○D₁₂

D₄×C₂×C₆

C₂²×D₄

C₂×C₁₂

C₂²×C₄

C₂×D₄

C₂⁴

C₂×C₄

C₆

C₂

# reps

Matrix representation of C₂⁴.₅₂D₆ ►in GL₆(𝔽₁₃)

12	0	0	0	0	0
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	0	0	1

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	12	0
0	0	0	0	0	12

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

10	0	0	0	0	0
0	4	0	0	0	0
0	0	0	12	0	0
0	0	1	0	0	0
0	0	0	0	0	12
0	0	0	0	1	0

0	4	0	0	0	0
10	0	0	0	0	0
0	0	0	0	0	12
0	0	0	0	1	0
0	0	0	12	0	0
0	0	1	0	0	0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[10,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[0,10,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12,0,0,0] >;

C₂⁴.₅₂D₆ in GAP, Magma, Sage, TeX

C_2^4._{52}D_6

% in TeX

G:=Group("C2^4.52D6");

// GroupNames label

G:=SmallGroup(192,1364);

// by ID

G=gap.SmallGroup(192,1364);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,570,6278]);

// Polycyclic

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

// generators/relations

G = C24.52D6 order 192 = 26·3

41st non-split extension by C24 of D6 acting via D6/C3=C22

G = C₂⁴.₅₂D₆ order 192 = 2⁶·3

41^st non-split extension by C₂⁴ of D₆ acting via D₆/C₃=C₂²