C2xQ8xA4 - GroupNames

G = C₂xQ₈xA₄ order 192 = 2⁶·3

Direct product of C₂, Q₈ and A₄

direct product, metabelian, soluble, monomial

Aliases: C₂xQ₈xA₄, C₂²:(C₆xQ₈), C₂³:₃(C₃xQ₈), (Q₈xC₂³):₂C₃, C₄.₇(C₂²xA₄), (C₂³xC₄).₃C₆, C₂.₄(C₂³xA₄), (C₂²xQ₈):₆C₆, C₂⁴.₂₇(C₂xC₆), (C₄xA₄).₂₀C₂², (C₂xA₄).₁₃C₂³, C₂³.₃₀(C₂²xC₆), C₂².₁₉(C₂²xA₄), (C₂²xA₄).₁₇C₂², (C₂xC₄xA₄).₉C₂, (C₂xC₄).₁₁(C₂xA₄), (C₂²xC₄).₄(C₂xC₆), SmallGroup(192,1499)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₂xQ₈xA₄

Chief series

C₁ — C₂² — C₂³ — C₂xA₄ — C₂²xA₄ — C₂xC₄xA₄ — C₂xQ₈xA₄

Lower central

C₂² — C₂³ — C₂xQ₈xA₄

Upper central

C₁ — C₂² — C₂xQ₈

Generators and relations for C₂xQ₈xA₄
G = < a,b,c,d,e,f | a²=b⁴=d²=e²=f³=1, c²=b², ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc^-1=b^-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf^-1=de=ed, fef^-1=d >

Subgroups: 520 in 205 conjugacy classes, 57 normal (12 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₂xC₄, C₂xC₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₁₂, A₄, C₂xC₆, C₂²xC₄, C₂²xC₄, C₂xQ₈, C₂xQ₈, C₂⁴, C₂xC₁₂, C₃xQ₈, C₂xA₄, C₂xA₄, C₂³xC₄, C₂²xQ₈, C₂²xQ₈, C₄xA₄, C₆xQ₈, C₂²xA₄, Q₈xC₂³, C₂xC₄xA₄, Q₈xA₄, C₂xQ₈xA₄
Quotients: C₁, C₂, C₃, C₂², C₆, Q₈, C₂³, A₄, C₂xC₆, C₂xQ₈, C₃xQ₈, C₂xA₄, C₂²xC₆, C₆xQ₈, C₂²xA₄, Q₈xA₄, C₂³xA₄, C₂xQ₈xA₄

Smallest permutation representation of C₂xQ₈xA₄
►On 48 points

Generators in S₄₈

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

(5 25)(6 26)(7 27)(8 28)(13 37)(14 38)(15 39)(16 40)(17 21)(18 22)(19 23)(20 24)(41 47)(42 48)(43 45)(44 46)

(1 11)(2 12)(3 9)(4 10)(17 21)(18 22)(19 23)(20 24)(29 35)(30 36)(31 33)(32 34)(41 47)(42 48)(43 45)(44 46)

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

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

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

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

40 conjugacy classes

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

40 irreducible representations

dim	1	1	1	1	1	1	2	2	3	3	3	6
type	+	+	+				-		+	+	+	-
image	C₁	C₂	C₂	C₃	C₆	C₆	Q₈	C₃xQ₈	A₄	C₂xA₄	C₂xA₄	Q₈xA₄
kernel	C₂xQ₈xA₄	C₂xC₄xA₄	Q₈xA₄	Q₈xC₂³	C₂³xC₄	C₂²xQ₈	C₂xA₄	C₂³	C₂xQ₈	C₂xC₄	Q₈	C₂
# reps	1	3	4	2	6	8	2	4	1	3	4	2

Matrix representation of C₂xQ₈xA₄ ►in GL₅(F₁₃)

12	0	0	0	0
0	12	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

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

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

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

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

9	0	0	0	0
0	9	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	1	0	0

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,1,0,0,0,11,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,5,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[9,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;

C₂xQ₈xA₄ in GAP, Magma, Sage, TeX

C_2\times Q_8\times A_4

% in TeX

G:=Group("C2xQ8xA4");

// GroupNames label

G:=SmallGroup(192,1499);

// by ID

G=gap.SmallGroup(192,1499);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,2,176,303,142,530,909]);

// Polycyclic

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

// generators/relations

G = C2xQ8xA4 order 192 = 26·3

Direct product of C2, Q8 and A4

G = C₂xQ₈xA₄ order 192 = 2⁶·3

Direct product of C₂, Q₈ and A₄