C3xC2^3:2Q8 - GroupNames

G = C₃×C₂³⋊₂Q₈ order 192 = 2⁶·3

Direct product of C₃ and C₂³⋊₂Q₈

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

Aliases: C₃×C₂³⋊₂Q₈, C₆.₁₅₈2₊¹⁺⁴, C₂³⋊₂(C₃×Q₈), (C₂²×C₆)⋊₂Q₈, C₂²⋊Q₈⋊₁₀C₆, C₂².₄(C₆×Q₈), C₂⁴.₁₉(C₂×C₆), (C₆×Q₈)⋊₂₉C₂², C₆.₆₁(C₂²×Q₈), (C₂×C₆).₃₆₃C₂⁴, (C₂×C₁₂).₆₇₂C₂³, C₂³.₄₃(C₂²×C₆), C₂².₃₇(C₂³×C₆), (C₂³×C₆).₁₈C₂², (C₂²×C₆).₂₆₂C₂³, C₂.₁₀(C₃×2₊¹⁺⁴), (C₂²×C₁₂).₄₅₁C₂², C₄⋊C₄⋊₄(C₂×C₆), C₂.₇(Q₈×C₂×C₆), (C₂×Q₈)⋊₅(C₂×C₆), (C₂×C₆).₁₇(C₂×Q₈), (C₃×C₄⋊C₄)⋊₃₈C₂², (C₃×C₂²⋊Q₈)⋊₃₇C₂, C₂²⋊C₄.₁₇(C₂×C₆), (C₂×C₂²⋊C₄).₁₃C₆, (C₆×C₂²⋊C₄).₃₃C₂, (C₂×C₄).₃₀(C₂²×C₆), (C₂²×C₄).₆₈(C₂×C₆), (C₃×C₂²⋊C₄).₁₅₁C₂², SmallGroup(192,1432)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃×C₂³⋊₂Q₈

Chief series

C₁ — C₂ — C₂² — C₂×C₆ — C₂×C₁₂ — C₃×C₂²⋊C₄ — C₃×C₂²⋊Q₈ — C₃×C₂³⋊₂Q₈

Lower central

C₁ — C₂² — C₃×C₂³⋊₂Q₈

Upper central

C₁ — C₂×C₆ — C₃×C₂³⋊₂Q₈

Generators and relations for C₃×C₂³⋊₂Q₈
G = < a,b,c,d,e,f | a³=b²=c²=d²=e⁴=1, f²=e², ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe^-1=fbf^-1=bd=db, fcf^-1=cd=dc, ce=ec, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 386 in 242 conjugacy classes, 162 normal (10 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², C₆, C₆, C₆, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂×C₁₂, C₂×C₁₂, C₃×Q₈, C₂²×C₆, C₂²×C₆, C₂×C₂²⋊C₄, C₂²⋊Q₈, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₂²×C₁₂, C₆×Q₈, C₂³×C₆, C₂³⋊₂Q₈, C₆×C₂²⋊C₄, C₃×C₂²⋊Q₈, C₃×C₂³⋊₂Q₈
Quotients: C₁, C₂, C₃, C₂², C₆, Q₈, C₂³, C₂×C₆, C₂×Q₈, C₂⁴, C₃×Q₈, C₂²×C₆, C₂²×Q₈, 2₊¹⁺⁴, C₆×Q₈, C₂³×C₆, C₂³⋊₂Q₈, Q₈×C₂×C₆, C₃×2₊¹⁺⁴, C₃×C₂³⋊₂Q₈

Smallest permutation representation of C₃×C₂³⋊₂Q₈
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

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

66 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	3A	3B	4A	···	4L	6A	···	6F	6G	···	6R	12A	···	12X
order	1	2	2	2	2	···	2	3	3	4	···	4	6	···	6	6	···	6	12	···	12
size	1	1	1	1	2	···	2	1	1	4	···	4	1	···	1	2	···	2	4	···	4

66 irreducible representations

dim	1	1	1	1	1	1	2	2	4	4
type	+	+	+				-		+
image	C₁	C₂	C₂	C₃	C₆	C₆	Q₈	C₃×Q₈	2₊¹⁺⁴	C₃×2₊¹⁺⁴
kernel	C₃×C₂³⋊₂Q₈	C₆×C₂²⋊C₄	C₃×C₂²⋊Q₈	C₂³⋊₂Q₈	C₂×C₂²⋊C₄	C₂²⋊Q₈	C₂²×C₆	C₂³	C₆	C₂
# reps	1	3	12	2	6	24	4	8	2	4

Matrix representation of C₃×C₂³⋊₂Q₈ ►in GL₆(𝔽₁₃)

9	0	0	0	0	0
0	9	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	3	0
0	0	0	0	0	3

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

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

6	5	0	0	0	0
3	7	0	0	0	0
0	0	1	0	11	0
0	0	0	0	1	1
0	0	0	0	12	0
0	0	0	1	1	0

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,1,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,1,12,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[12,1,0,0,0,0,11,1,0,0,0,0,0,0,12,0,0,0,0,0,11,1,12,1,0,0,0,0,0,1,0,0,0,0,1,0],[6,3,0,0,0,0,5,7,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,11,1,12,1,0,0,0,1,0,0] >;

C₃×C₂³⋊₂Q₈ in GAP, Magma, Sage, TeX

C_3\times C_2^3\rtimes_2Q_8

% in TeX

G:=Group("C3xC2^3:2Q8");

// GroupNames label

G:=SmallGroup(192,1432);

// by ID

G=gap.SmallGroup(192,1432);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,2102,555,520,1571]);

// Polycyclic

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

// generators/relations

G = C3×C23⋊2Q8 order 192 = 26·3

Direct product of C3 and C23⋊2Q8

G = C₃×C₂³⋊₂Q₈ order 192 = 2⁶·3

Direct product of C₃ and C₂³⋊₂Q₈