C2xC2^4:4S3 - GroupNames

G = C₂×C₂⁴⋊₄S₃ order 192 = 2⁶·3

Direct product of C₂ and C₂⁴⋊₄S₃

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂×C₂⁴⋊₄S₃, C₂⁵⋊₄S₃, C₂⁴⋊₁₆D₆, C₆⋊₃C₂²≀C₂, (C₂⁴×C₆)⋊₃C₂, (C₂²×C₆)⋊₁₈D₄, C₂³⋊₈(C₃⋊D₄), (C₂×C₆).₃₂₂C₂⁴, (C₂²×S₃)⋊₃C₂³, (C₂³×C₆)⋊₁₈C₂², (C₂×Dic₃)⋊₄C₂³, C₆.₁₇₆(C₂²×D₄), (S₃×C₂³)⋊₁₆C₂², C₆.D₄⋊₆₆C₂², C₂².₃₃₀(S₃×C₂³), C₂³.₂₅₁(C₂²×S₃), (C₂²×C₆).₄₂₉C₂³, (C₂²×Dic₃)⋊₃₇C₂², C₃⋊₄(C₂×C₂²≀C₂), (C₂×C₆)⋊₁₆(C₂×D₄), C₂²⋊₅(C₂×C₃⋊D₄), (C₂²×C₃⋊D₄)⋊₁₈C₂, (C₂×C₃⋊D₄)⋊₄₉C₂², C₂.₄₈(C₂²×C₃⋊D₄), (C₂×C₆.D₄)⋊₃₂C₂, SmallGroup(192,1399)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₂×C₂⁴⋊₄S₃

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×S₃ — S₃×C₂³ — C₂²×C₃⋊D₄ — C₂×C₂⁴⋊₄S₃

Lower central

C₃ — C₂×C₆ — C₂×C₂⁴⋊₄S₃

Upper central

C₁ — C₂³ — C₂⁵

Generators and relations for C₂×C₂⁴⋊₄S₃
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f³=g²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, gbg=be=eb, bf=fb, gcg=cd=dc, ce=ec, cf=fc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg=f^-1 >

Subgroups: 1528 in 662 conjugacy classes, 159 normal (10 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₂×C₄, D₄, C₂³, C₂³, C₂³, Dic₃, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂×Dic₃, C₂×Dic₃, C₃⋊D₄, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂×C₂²⋊C₄, C₂²≀C₂, C₂²×D₄, C₂⁵, C₆.D₄, C₂²×Dic₃, C₂×C₃⋊D₄, C₂×C₃⋊D₄, S₃×C₂³, C₂³×C₆, C₂³×C₆, C₂×C₂²≀C₂, C₂×C₆.D₄, C₂⁴⋊₄S₃, C₂²×C₃⋊D₄, C₂⁴×C₆, C₂×C₂⁴⋊₄S₃
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₂⁴, C₃⋊D₄, C₂²×S₃, C₂²≀C₂, C₂²×D₄, C₂×C₃⋊D₄, S₃×C₂³, C₂×C₂²≀C₂, C₂⁴⋊₄S₃, C₂²×C₃⋊D₄, C₂×C₂⁴⋊₄S₃

Smallest permutation representation of C₂×C₂⁴⋊₄S₃
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

60 conjugacy classes

class	1	2A	···	2G	2H	···	2S	2T	2U	3	4A	···	4F	6A	···	6AE
order	1	2	···	2	2	···	2	2	2	3	4	···	4	6	···	6
size	1	1	···	1	2	···	2	12	12	2	12	···	12	2	···	2

60 irreducible representations

dim	1	1	1	1	1	2	2	2	2
type	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	S₃	D₄	D₆	C₃⋊D₄
kernel	C₂×C₂⁴⋊₄S₃	C₂×C₆.D₄	C₂⁴⋊₄S₃	C₂²×C₃⋊D₄	C₂⁴×C₆	C₂⁵	C₂²×C₆	C₂⁴	C₂³
# reps	1	3	8	3	1	1	12	7	24

Matrix representation of C₂×C₂⁴⋊₄S₃ ►in GL₆(𝔽₁₃)

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

1	0	0	0	0	0
0	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

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	1

1	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	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	1	0
0	0	0	0	0	1

3	0	0	0	0	0
0	9	0	0	0	0
0	0	9	0	0	0
0	0	0	3	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

G:=sub<GL(6,GF(13))| [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],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

C₂×C₂⁴⋊₄S₃ in GAP, Magma, Sage, TeX

C_2\times C_2^4\rtimes_4S_3

% in TeX

G:=Group("C2xC2^4:4S3");

// GroupNames label

G:=SmallGroup(192,1399);

// by ID

G=gap.SmallGroup(192,1399);

# by ID

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

// Polycyclic

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

// generators/relations

G = C2×C24⋊4S3 order 192 = 26·3

Direct product of C2 and C24⋊4S3

G = C₂×C₂⁴⋊₄S₃ order 192 = 2⁶·3

Direct product of C₂ and C₂⁴⋊₄S₃