C2^3:C4:5S3 - GroupNames

G = C₂³⋊C₄⋊₅S₃ order 192 = 2⁶·3

The semidirect product of C₂³⋊C₄ and S₃ acting through Inn(C₂³⋊C₄)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³⋊C₄⋊₅S₃, (C₂×Dic₆)⋊₄C₄, (C₂×D₄).₁₁₉D₆, C₂³.₁₀(C₄×S₃), C₂²⋊C₄.₄₃D₆, (C₆×D₄).₇C₂², C₂².₂₃(S₃×D₄), (C₂²×Dic₃)⋊₄C₄, C₂³.₆D₆⋊₃C₂, C₂³.₇D₆⋊₃C₂, (C₂²×C₆).₁C₂³, (C₂×Dic₃).₁₃₂D₄, C₂³.₁₁(C₂²×S₃), C₂³.₁₆D₆⋊₂₃C₂, Dic₃.₃(C₂²⋊C₄), C₃⋊₁(C₂³.C₂³), C₆.D₄.₁C₂², (C₂²×Dic₃).₂₅C₂², (S₃×C₂×C₄)⋊₁C₄, (C₂×C₃⋊D₄)⋊₁C₄, (C₂×C₄).₅(C₄×S₃), (C₃×C₂³⋊C₄)⋊₃C₂, (C₂×C₁₂).₅(C₂×C₄), (C₂×C₆).₁₆(C₂×D₄), C₂².₁₂(S₃×C₂×C₄), C₂.₁₁(S₃×C₂²⋊C₄), C₆.₁₀(C₂×C₂²⋊C₄), (C₂²×C₆).₅(C₂×C₄), (C₂×C₆).₆(C₂²×C₄), (C₂×D₄⋊₂S₃).₁C₂, (C₂²×S₃).₁(C₂×C₄), (C₂×Dic₃).₁(C₂×C₄), (C₂×C₃⋊D₄).₁C₂², (C₃×C₂²⋊C₄).₈₂C₂², SmallGroup(192,299)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₂³⋊C₄⋊₅S₃

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×C₆ — C₂²×Dic₃ — C₂×D₄⋊₂S₃ — C₂³⋊C₄⋊₅S₃

Lower central

C₃ — C₆ — C₂×C₆ — C₂³⋊C₄⋊₅S₃

Upper central

C₁ — C₂ — C₂³ — C₂³⋊C₄

Generators and relations for C₂³⋊C₄⋊₅S₃
G = < a,b,c,d,e,f | a²=b²=c²=d⁴=e³=f²=1, ab=ba, faf=ac=ca, dad^-1=abc, ae=ea, dbd^-1=bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=bcd, fef=e^-1 >

Subgroups: 448 in 158 conjugacy classes, 51 normal (31 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, Dic₃, Dic₃, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, Dic₆, C₄×S₃, C₂×Dic₃, C₂×Dic₃, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×S₃, C₂²×C₆, C₂³⋊C₄, C₂³⋊C₄, C₄²⋊C₂, C₂×C₄○D₄, C₄×Dic₃, Dic₃⋊C₄, C₆.D₄, C₃×C₂²⋊C₄, C₂×Dic₆, S₃×C₂×C₄, D₄⋊₂S₃, C₂²×Dic₃, C₂×C₃⋊D₄, C₆×D₄, C₂³.C₂³, C₂³.₆D₆, C₂³.₇D₆, C₃×C₂³⋊C₄, C₂³.₁₆D₆, C₂×D₄⋊₂S₃, C₂³⋊C₄⋊₅S₃
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, C₂³, D₆, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄×S₃, C₂²×S₃, C₂×C₂²⋊C₄, S₃×C₂×C₄, S₃×D₄, C₂³.C₂³, S₃×C₂²⋊C₄, C₂³⋊C₄⋊₅S₃

Smallest permutation representation of C₂³⋊C₄⋊₅S₃
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

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

33 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	3	4A	4B	4C	···	4G	4H	4I	4J	4K	···	4O	6A	6B	6C	6D	6E	12A	···	12E
order	1	2	2	2	2	2	2	3	4	4	4	···	4	4	4	4	4	···	4	6	6	6	6	6	12	···	12
size	1	1	2	2	2	4	12	2	3	3	4	···	4	6	6	6	12	···	12	2	4	4	4	8	8	···	8

33 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4	8
type	+	+	+	+	+	+					+	+	+	+			+		-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	S₃	D₄	D₆	D₆	C₄×S₃	C₄×S₃	S₃×D₄	C₂³.C₂³	C₂³⋊C₄⋊₅S₃
kernel	C₂³⋊C₄⋊₅S₃	C₂³.₆D₆	C₂³.₇D₆	C₃×C₂³⋊C₄	C₂³.₁₆D₆	C₂×D₄⋊₂S₃	C₂×Dic₆	S₃×C₂×C₄	C₂²×Dic₃	C₂×C₃⋊D₄	C₂³⋊C₄	C₂×Dic₃	C₂²⋊C₄	C₂×D₄	C₂×C₄	C₂³	C₂²	C₃	C₁
# reps	1	2	1	1	2	1	2	2	2	2	1	4	2	1	2	2	2	2	1

Matrix representation of C₂³⋊C₄⋊₅S₃ ►in GL₆(𝔽₁₃)

12	0	0	0	0	0
0	12	0	0	0	0
0	0	0	8	5	0
0	0	0	8	0	5
0	0	8	8	0	5
0	0	0	3	0	5

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

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

0	12	0	0	0	0
1	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

0	12	0	0	0	0
12	0	0	0	0	0
0	0	12	0	0	0
0	0	12	0	12	1
0	0	12	12	0	1
0	0	11	0	0	1

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

C₂³⋊C₄⋊₅S₃ in GAP, Magma, Sage, TeX

C_2^3\rtimes C_4\rtimes_5S_3

% in TeX

G:=Group("C2^3:C4:5S3");

// GroupNames label

G:=SmallGroup(192,299);

// by ID

G=gap.SmallGroup(192,299);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,219,58,570,438,6278]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^4=e^3=f^2=1,a*b=b*a,f*a*f=a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=b*c*d,f*e*f=e^-1>;

// generators/relations

G = C23⋊C4⋊5S3 order 192 = 26·3

The semidirect product of C23⋊C4 and S3 acting through Inn(C23⋊C4)

G = C₂³⋊C₄⋊₅S₃ order 192 = 2⁶·3

The semidirect product of C₂³⋊C₄ and S₃ acting through Inn(C₂³⋊C₄)