(C2^2xS3):C8 - GroupNames

G = (C₂²×S₃)⋊C₈ order 192 = 2⁶·3

The semidirect product of C₂²×S₃ and C₈ acting via C₈/C₂=C₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂²×S₃)⋊C₈, C₂²⋊C₈⋊₁S₃, C₃⋊₁(C₂³⋊C₈), C₂.₅(D₆⋊C₈), C₂².₃(S₃×C₈), (C₂×C₁₂).₄₃₈D₄, (C₂×C₄).₁₀₇D₁₂, C₆.₃(C₂²⋊C₈), C₆.₄(C₂³⋊C₄), (S₃×C₂³).₁C₄, C₂³.₄₅(C₄×S₃), (C₂²×C₄).₁₇D₆, (C₂×C₆).₁M₄(2), C₆.₂(C₄.D₄), C₁₂.₅₅D₄⋊₂₀C₂, C₂².₃(C₈⋊S₃), C₂².₃₂(D₆⋊C₄), (C₂²×Dic₃).₁C₄, C₂.₁(C₁₂.₄₆D₄), C₂.₁(C₂³.₆D₆), (C₂²×C₁₂).₃₂₂C₂², (C₂×C₆).₁(C₂×C₈), (C₃×C₂²⋊C₈)⋊₁C₂, (C₂×D₆⋊C₄).₂₁C₂, (C₂²×C₆).₂₆(C₂×C₄), (C₂×C₄).₂₀₉(C₃⋊D₄), (C₂×C₆).₄₀(C₂²⋊C₄), SmallGroup(192,27)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — (C₂²×S₃)⋊C₈

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂×C₁₂ — C₂²×C₁₂ — C₂×D₆⋊C₄ — (C₂²×S₃)⋊C₈

Lower central

C₃ — C₆ — C₂×C₆ — (C₂²×S₃)⋊C₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²⋊C₈

Generators and relations for (C₂²×S₃)⋊C₈
G = < a,b,c,d,e | a²=b²=c³=d²=e⁸=1, eae^-1=ab=ba, ac=ca, ede^-1=ad=da, bc=cb, bd=db, be=eb, dcd=c^-1, ce=ec >

Subgroups: 368 in 98 conjugacy classes, 31 normal (29 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂⁴, C₃⋊C₈, C₂₄, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂²⋊C₈, C₂²⋊C₈, C₂×C₂²⋊C₄, C₂×C₃⋊C₈, D₆⋊C₄, C₂×C₂₄, C₂²×Dic₃, C₂²×C₁₂, S₃×C₂³, C₂³⋊C₈, C₁₂.₅₅D₄, C₃×C₂²⋊C₈, C₂×D₆⋊C₄, (C₂²×S₃)⋊C₈
Quotients: C₁, C₂, C₄, C₂², S₃, C₈, C₂×C₄, D₄, D₆, C₂²⋊C₄, C₂×C₈, M₄(2), C₄×S₃, D₁₂, C₃⋊D₄, C₂²⋊C₈, C₂³⋊C₄, C₄.D₄, S₃×C₈, C₈⋊S₃, D₆⋊C₄, C₂³⋊C₈, C₂³.₆D₆, D₆⋊C₈, C₁₂.₄₆D₄, (C₂²×S₃)⋊C₈

Smallest permutation representation of (C₂²×S₃)⋊C₈
►On 48 points

Generators in S₄₈

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

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

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

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

(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)

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3	4A	4B	4C	4D	4E	4F	6A	6B	6C	6D	6E	8A	8B	8C	8D	8E	8F	8G	8H	12A	12B	12C	12D	12E	12F	24A	···	24H
order	1	2	2	2	2	2	2	2	3	4	4	4	4	4	4	6	6	6	6	6	8	8	8	8	8	8	8	8	12	12	12	12	12	12	24	···	24
size	1	1	1	1	2	2	12	12	2	2	2	2	2	12	12	2	2	2	4	4	4	4	4	4	12	12	12	12	2	2	2	2	4	4	4	···	4

42 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+				+	+	+		+					+	+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₈	S₃	D₄	D₆	M₄(2)	D₁₂	C₃⋊D₄	C₄×S₃	S₃×C₈	C₈⋊S₃	C₂³⋊C₄	C₄.D₄	C₂³.₆D₆	C₁₂.₄₆D₄
kernel	(C₂²×S₃)⋊C₈	C₁₂.₅₅D₄	C₃×C₂²⋊C₈	C₂×D₆⋊C₄	C₂²×Dic₃	S₃×C₂³	C₂²×S₃	C₂²⋊C₈	C₂×C₁₂	C₂²×C₄	C₂×C₆	C₂×C₄	C₂×C₄	C₂³	C₂²	C₂²	C₆	C₆	C₂	C₂
# reps	1	1	1	1	2	2	8	1	2	1	2	2	2	2	4	4	1	1	2	2

Matrix representation of (C₂²×S₃)⋊C₈ ►in GL₆(𝔽₇₃)

72	0	0	0	0	0
0	72	0	0	0	0
0	0	72	0	0	0
0	0	0	72	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	72	0	0	0
0	0	0	72	0	0
0	0	0	0	72	0
0	0	0	0	0	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	72	1	0	0
0	0	72	0	0	0
0	0	0	0	72	1
0	0	0	0	72	0

1	0	0	0	0	0
29	72	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

22	1	0	0	0	0
54	51	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	66	14	0	0
0	0	59	7	0	0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,29,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[22,54,0,0,0,0,1,51,0,0,0,0,0,0,0,0,66,59,0,0,0,0,14,7,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C₂²×S₃)⋊C₈ in GAP, Magma, Sage, TeX

(C_2^2\times S_3)\rtimes C_8

% in TeX

G:=Group("(C2^2xS3):C8");

// GroupNames label

G:=SmallGroup(192,27);

// by ID

G=gap.SmallGroup(192,27);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,758,100,570,6278]);

// Polycyclic

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

// generators/relations

G = (C22×S3)⋊C8 order 192 = 26·3

The semidirect product of C22×S3 and C8 acting via C8/C2=C4

G = (C₂²×S₃)⋊C₈ order 192 = 2⁶·3

The semidirect product of C₂²×S₃ and C₈ acting via C₈/C₂=C₄