C3xC2^3:C8 - GroupNames

G = C₃×C₂³⋊C₈ order 192 = 2⁶·3

Direct product of C₃ and C₂³⋊C₈

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₃×C₂³⋊C₈, C₂³⋊C₂₄, C₂⁴.₂C₁₂, C₂²⋊C₈⋊₁C₆, (C₂²×C₆)⋊₁C₈, (C₂³×C₆).₁C₄, (C₂×C₁₂).₄₄₁D₄, C₂².₂(C₂×C₂₄), (C₂²×C₁₂).₄C₄, (C₂²×C₄).₂C₁₂, C₆.₂₇(C₂³⋊C₄), C₆.₂₀(C₂²⋊C₈), C₂³.₂₆(C₂×C₁₂), (C₂×C₆).₁₄M₄(2), C₆.₁₁(C₄.D₄), (C₂²×C₁₂).₁C₂², C₂².₂(C₃×M₄(2)), (C₃×C₂²⋊C₈)⋊₃C₂, (C₂×C₆).₂₀(C₂×C₈), (C₂×C₄).₉₁(C₃×D₄), C₂.₁(C₃×C₂³⋊C₄), C₂.₃(C₃×C₂²⋊C₈), (C₂×C₂²⋊C₄).₁C₆, (C₆×C₂²⋊C₄).₃C₂, (C₂²×C₄).₆(C₂×C₆), C₂.₁(C₃×C₄.D₄), (C₂²×C₆).₁₀₆(C₂×C₄), C₂².₂₃(C₃×C₂²⋊C₄), (C₂×C₆).₁₁₈(C₂²⋊C₄), SmallGroup(192,129)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃×C₂³⋊C₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂²×C₁₂ — C₃×C₂²⋊C₈ — C₃×C₂³⋊C₈

Lower central

C₁ — C₂ — C₂² — C₃×C₂³⋊C₈

Upper central

C₁ — C₂×C₆ — C₂²×C₁₂ — C₃×C₂³⋊C₈

Generators and relations for C₃×C₂³⋊C₈
G = < a,b,c,d,e | a³=b²=c²=d²=e⁸=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe^-1=bcd, ece^-1=cd=dc, de=ed >

Subgroups: 218 in 98 conjugacy classes, 38 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₁₂, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂×C₈, C₂²×C₄, C₂⁴, C₂₄, C₂×C₁₂, C₂×C₁₂, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂²⋊C₈, C₂×C₂²⋊C₄, C₃×C₂²⋊C₄, C₂×C₂₄, C₂²×C₁₂, C₂³×C₆, C₂³⋊C₈, C₃×C₂²⋊C₈, C₆×C₂²⋊C₄, C₃×C₂³⋊C₈
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₈, C₂×C₄, D₄, C₁₂, C₂×C₆, C₂²⋊C₄, C₂×C₈, M₄(2), C₂₄, C₂×C₁₂, C₃×D₄, C₂²⋊C₈, C₂³⋊C₄, C₄.D₄, C₃×C₂²⋊C₄, C₂×C₂₄, C₃×M₄(2), C₂³⋊C₈, C₃×C₂²⋊C₈, C₃×C₂³⋊C₄, C₃×C₄.D₄, C₃×C₂³⋊C₈

Smallest permutation representation of C₃×C₂³⋊C₈
►On 48 points

Generators in S₄₈

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

(2 20)(3 17)(4 8)(6 24)(7 21)(9 41)(10 14)(12 48)(13 45)(16 44)(18 22)(26 35)(27 40)(28 32)(30 39)(31 36)(33 37)(42 46)

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

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

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

66 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	4B	4C	4D	4E	4F	6A	···	6F	6G	6H	6I	6J	6K	6L	6M	6N	8A	···	8H	12A	···	12H	12I	12J	12K	12L	24A	···	24P
order	1	2	2	2	2	2	2	2	3	3	4	4	4	4	4	4	6	···	6	6	6	6	6	6	6	6	6	8	···	8	12	···	12	12	12	12	12	24	···	24
size	1	1	1	1	2	2	4	4	1	1	2	2	2	2	4	4	1	···	1	2	2	2	2	4	4	4	4	4	···	4	2	···	2	4	4	4	4	4	···	4

66 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	4
type	+	+	+										+				+	+
image	C₁	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₈	C₁₂	C₁₂	C₂₄	D₄	M₄(2)	C₃×D₄	C₃×M₄(2)	C₂³⋊C₄	C₄.D₄	C₃×C₂³⋊C₄	C₃×C₄.D₄
kernel	C₃×C₂³⋊C₈	C₃×C₂²⋊C₈	C₆×C₂²⋊C₄	C₂³⋊C₈	C₂²×C₁₂	C₂³×C₆	C₂²⋊C₈	C₂×C₂²⋊C₄	C₂²×C₆	C₂²×C₄	C₂⁴	C₂³	C₂×C₁₂	C₂×C₆	C₂×C₄	C₂²	C₆	C₆	C₂	C₂
# reps	1	2	1	2	2	2	4	2	8	4	4	16	2	2	4	4	1	1	2	2

Matrix representation of C₃×C₂³⋊C₈ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	64	0	0	0
0	0	0	64	0	0
0	0	0	0	64	0
0	0	0	0	0	64

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

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

0	1	0	0	0	0
27	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	1	0	0	0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,27,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₃×C₂³⋊C₈ in GAP, Magma, Sage, TeX

C_3\times C_2^3\rtimes C_8

% in TeX

G:=Group("C3xC2^3:C8");

// GroupNames label

G:=SmallGroup(192,129);

// by ID

G=gap.SmallGroup(192,129);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,1271,172]);

// Polycyclic

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

// generators/relations

G = C3×C23⋊C8 order 192 = 26·3

Direct product of C3 and C23⋊C8

G = C₃×C₂³⋊C₈ order 192 = 2⁶·3

Direct product of C₃ and C₂³⋊C₈