(C2xQ8):C12 - GroupNames

G = (C₂×Q₈)⋊C₁₂ order 192 = 2⁶·3

The semidirect product of C₂×Q₈ and C₁₂ acting via C₁₂/C₂=C₆

Aliases: (C₂×Q₈)⋊C₁₂, (C₄×Q₈)⋊₁C₆, C₂²⋊C₄.₁A₄, C₂².₅(C₄×A₄), Q₈.₂(C₂×C₁₂), C₂.₁(D₄.A₄), C₂³.₂₅(C₂×A₄), (C₂²×Q₈).₂C₆, C₂³.₃₂C₂³⋊C₃, (C₄×SL₂(𝔽₃))⋊₃C₂, (C₂×SL₂(𝔽₃))⋊₁C₄, C₂².₁₅(C₂²×A₄), SL₂(𝔽₃).₈(C₂×C₄), (C₂²×SL₂(𝔽₃)).₁C₂, (C₂×SL₂(𝔽₃)).₂₅C₂², C₂.₇(C₂×C₄×A₄), (C₂×C₄).₂(C₂×A₄), (C₂×Q₈).₃₆(C₂×C₆), SmallGroup(192,998)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — Q₈ — (C₂×Q₈)⋊C₁₂

Chief series

C₁ — C₂ — Q₈ — C₂×Q₈ — C₂×SL₂(𝔽₃) — C₂²×SL₂(𝔽₃) — (C₂×Q₈)⋊C₁₂

Lower central

Q₈ — (C₂×Q₈)⋊C₁₂

Upper central

C₁ — C₂² — C₂²⋊C₄

Generators and relations for (C₂×Q₈)⋊C₁₂
G = < a,b,c,d | a²=b⁴=d¹²=1, c²=b², ab=ba, ac=ca, dad^-1=ab², cbc^-1=b^-1, dbd^-1=b²c, dcd^-1=abc >

Subgroups: 245 in 86 conjugacy classes, 27 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², C₆, C₂×C₄, C₂×C₄, Q₈, Q₈, C₂³, C₁₂, C₂×C₆, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, SL₂(𝔽₃), C₂×C₁₂, C₂²×C₆, C₄²⋊C₂, C₄×Q₈, C₄×Q₈, C₂²×Q₈, C₃×C₂²⋊C₄, C₂×SL₂(𝔽₃), C₂×SL₂(𝔽₃), C₂³.₃₂C₂³, C₄×SL₂(𝔽₃), C₂²×SL₂(𝔽₃), (C₂×Q₈)⋊C₁₂
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, C₁₂, A₄, C₂×C₆, C₂×C₁₂, C₂×A₄, C₄×A₄, C₂²×A₄, C₂×C₄×A₄, D₄.A₄, (C₂×Q₈)⋊C₁₂

Smallest permutation representation of (C₂×Q₈)⋊C₁₂
►On 32 points

Generators in S₃₂

(1 7)(2 4)(3 5)(6 8)(9 15)(10 30)(11 17)(12 32)(13 19)(14 22)(16 24)(18 26)(20 28)(21 27)(23 29)(25 31)

(1 12 5 26)(2 23 6 9)(3 18 7 32)(4 29 8 15)(10 20 24 22)(11 13 25 27)(14 30 28 16)(17 19 31 21)

(1 20 5 22)(2 31 6 17)(3 14 7 28)(4 25 8 11)(9 21 23 19)(10 26 24 12)(13 15 27 29)(16 32 30 18)

(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)

G:=sub<Sym(32)| (1,7)(2,4)(3,5)(6,8)(9,15)(10,30)(11,17)(12,32)(13,19)(14,22)(16,24)(18,26)(20,28)(21,27)(23,29)(25,31), (1,12,5,26)(2,23,6,9)(3,18,7,32)(4,29,8,15)(10,20,24,22)(11,13,25,27)(14,30,28,16)(17,19,31,21), (1,20,5,22)(2,31,6,17)(3,14,7,28)(4,25,8,11)(9,21,23,19)(10,26,24,12)(13,15,27,29)(16,32,30,18), (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)>;

G:=Group( (1,7)(2,4)(3,5)(6,8)(9,15)(10,30)(11,17)(12,32)(13,19)(14,22)(16,24)(18,26)(20,28)(21,27)(23,29)(25,31), (1,12,5,26)(2,23,6,9)(3,18,7,32)(4,29,8,15)(10,20,24,22)(11,13,25,27)(14,30,28,16)(17,19,31,21), (1,20,5,22)(2,31,6,17)(3,14,7,28)(4,25,8,11)(9,21,23,19)(10,26,24,12)(13,15,27,29)(16,32,30,18), (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) );

G=PermutationGroup([[(1,7),(2,4),(3,5),(6,8),(9,15),(10,30),(11,17),(12,32),(13,19),(14,22),(16,24),(18,26),(20,28),(21,27),(23,29),(25,31)], [(1,12,5,26),(2,23,6,9),(3,18,7,32),(4,29,8,15),(10,20,24,22),(11,13,25,27),(14,30,28,16),(17,19,31,21)], [(1,20,5,22),(2,31,6,17),(3,14,7,28),(4,25,8,11),(9,21,23,19),(10,26,24,12),(13,15,27,29),(16,32,30,18)], [(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)]])

38 conjugacy classes

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

38 irreducible representations

dim	1	1	1	1	1	1	1	1	3	3	3	3	4	4
type	+	+	+						+	+	+		-
image	C₁	C₂	C₂	C₃	C₄	C₆	C₆	C₁₂	A₄	C₂×A₄	C₂×A₄	C₄×A₄	D₄.A₄	D₄.A₄
kernel	(C₂×Q₈)⋊C₁₂	C₄×SL₂(𝔽₃)	C₂²×SL₂(𝔽₃)	C₂³.₃₂C₂³	C₂×SL₂(𝔽₃)	C₄×Q₈	C₂²×Q₈	C₂×Q₈	C₂²⋊C₄	C₂×C₄	C₂³	C₂²	C₂	C₂
# reps	1	2	1	2	4	4	2	8	1	2	1	4	2	4

Matrix representation of (C₂×Q₈)⋊C₁₂ ►in GL₇(𝔽₁₃)

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	12	0	0	0
0	0	0	0	12	0	0
0	0	0	0	0	1	0
0	0	0	5	8	0	1

12	0	0	0	0	0	0
12	0	1	0	0	0	0
12	1	0	0	0	0	0
0	0	0	0	12	0	0
0	0	0	1	0	0	0
0	0	0	8	5	1	11
0	0	0	0	5	1	12

0	12	1	0	0	0	0
0	12	0	0	0	0	0
1	12	0	0	0	0	0
0	0	0	9	10	0	0
0	0	0	10	4	0	0
0	0	0	11	2	7	7
0	0	0	11	9	4	6

0	8	0	0	0	0	0
0	0	8	0	0	0	0
8	0	0	0	0	0	0
0	0	0	0	0	1	0
0	0	0	11	2	7	7
0	0	0	1	0	0	0
0	0	0	0	10	2	11

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,5,0,0,0,0,12,0,8,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,12,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,12,0,5,5,0,0,0,0,0,1,1,0,0,0,0,0,11,12],[0,0,1,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,10,11,11,0,0,0,10,4,2,9,0,0,0,0,0,7,4,0,0,0,0,0,7,6],[0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,11,1,0,0,0,0,0,2,0,10,0,0,0,1,7,0,2,0,0,0,0,7,0,11] >;

(C₂×Q₈)⋊C₁₂ in GAP, Magma, Sage, TeX

(C_2\times Q_8)\rtimes C_{12}

% in TeX

G:=Group("(C2xQ8):C12");

// GroupNames label

G:=SmallGroup(192,998);

// by ID

G=gap.SmallGroup(192,998);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,1373,92,438,172,775,285,124]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=d^12=1,c^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,d*b*d^-1=b^2*c,d*c*d^-1=a*b*c>;

// generators/relations

G = (C2×Q8)⋊C12 order 192 = 26·3

The semidirect product of C2×Q8 and C12 acting via C12/C2=C6

G = (C₂×Q₈)⋊C₁₂ order 192 = 2⁶·3

The semidirect product of C₂×Q₈ and C₁₂ acting via C₁₂/C₂=C₆