(C6xQ8):C4 - GroupNames

G = (C₆×Q₈)⋊C₄ order 192 = 2⁶·3

1^st semidirect product of C₆×Q₈ and C₄ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₆×Q₈)⋊₁C₄, C₆.₁₇C₄≀C₂, C₄⋊C₄⋊₂Dic₃, (C₂×C₆).₃Q₁₆, (C₂×Q₈)⋊₃Dic₃, C₂²⋊Q₈.₁S₃, (C₂×C₁₂).₂₃₀D₄, (C₂×C₆).₁₁SD₁₆, (C₂²×C₄).₇₆D₆, (C₂²×C₆).₄₅D₄, C₆.₂₁(C₂³⋊C₄), C₆.₁₁(Q₈⋊C₄), C₂.₃(Q₈⋊₂Dic₃), C₂³.₅₆(C₃⋊D₄), C₃⋊₃(C₂³.₃₁D₄), C₂².₂(C₃⋊Q₁₆), C₂.₅(Q₈⋊₃Dic₃), C₁₂.₅₅D₄.₁₆C₂, C₆.C₄².₃₅C₂, C₂².₅(Q₈⋊₂S₃), C₂.₆(C₂³.₇D₆), (C₂²×C₁₂).₃₇₂C₂², C₂².₃₈(C₆.D₄), (C₃×C₄⋊C₄)⋊₂C₄, (C₂×C₄).₈(C₂×Dic₃), (C₂×C₁₂).₁₆₈(C₂×C₄), (C₃×C₂²⋊Q₈).₁₀C₂, (C₂×C₄).₁₆₄(C₃⋊D₄), (C₂×C₆).₉₆(C₂²⋊C₄), SmallGroup(192,97)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — (C₆×Q₈)⋊C₄

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×C₆ — C₂²×C₁₂ — C₆.C₄² — (C₆×Q₈)⋊C₄

Lower central

C₃ — C₂×C₆ — C₂×C₁₂ — (C₆×Q₈)⋊C₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²⋊Q₈

Generators and relations for (C₆×Q₈)⋊C₄
G = < a,b,c,d | a⁶=b⁴=d⁴=1, c²=b², ab=ba, ac=ca, dad^-1=a^-1, cbc^-1=b^-1, dbd^-1=a³b, dcd^-1=b^-1c >

Subgroups: 224 in 80 conjugacy classes, 31 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₃⋊C₈, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₃×Q₈, C₂²×C₆, C₂.C₄², C₂²⋊C₈, C₂²⋊Q₈, C₂×C₃⋊C₈, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₃×C₄⋊C₄, C₂²×Dic₃, C₂²×C₁₂, C₆×Q₈, C₂³.₃₁D₄, C₁₂.₅₅D₄, C₆.C₄², C₃×C₂²⋊Q₈, (C₆×Q₈)⋊C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, Dic₃, D₆, C₂²⋊C₄, SD₁₆, Q₁₆, C₂×Dic₃, C₃⋊D₄, C₂³⋊C₄, Q₈⋊C₄, C₄≀C₂, Q₈⋊₂S₃, C₃⋊Q₁₆, C₆.D₄, C₂³.₃₁D₄, C₂³.₇D₆, Q₈⋊₂Dic₃, Q₈⋊₃Dic₃, (C₆×Q₈)⋊C₄

Smallest permutation representation of (C₆×Q₈)⋊C₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+			+	+	+	-	+	-		-				+	+	-
image	C₁	C₂	C₂	C₂	C₄	C₄	S₃	D₄	D₄	Dic₃	D₆	Dic₃	SD₁₆	Q₁₆	C₃⋊D₄	C₃⋊D₄	C₄≀C₂	C₂³⋊C₄	Q₈⋊₂S₃	C₃⋊Q₁₆	C₂³.₇D₆	Q₈⋊₃Dic₃
kernel	(C₆×Q₈)⋊C₄	C₁₂.₅₅D₄	C₆.C₄²	C₃×C₂²⋊Q₈	C₃×C₄⋊C₄	C₆×Q₈	C₂²⋊Q₈	C₂×C₁₂	C₂²×C₆	C₄⋊C₄	C₂²×C₄	C₂×Q₈	C₂×C₆	C₂×C₆	C₂×C₄	C₂³	C₆	C₆	C₂²	C₂²	C₂	C₂
# reps	1	1	1	1	2	2	1	1	1	1	1	1	2	2	2	2	4	1	1	1	2	2

Matrix representation of (C₆×Q₈)⋊C₄ ►in GL₆(𝔽₇₃)

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

46	0	0	0	0	0
0	27	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	46	2
0	0	0	0	0	27

0	72	0	0	0	0
1	0	0	0	0	0
0	0	30	60	0	0
0	0	13	43	0	0
0	0	0	0	10	61
0	0	0	0	51	63

72	0	0	0	0	0
0	27	0	0	0	0
0	0	1	0	0	0
0	0	72	72	0	0
0	0	0	0	72	0
0	0	0	0	46	1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[46,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,2,27],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,30,13,0,0,0,0,60,43,0,0,0,0,0,0,10,51,0,0,0,0,61,63],[72,0,0,0,0,0,0,27,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,46,0,0,0,0,0,1] >;

(C₆×Q₈)⋊C₄ in GAP, Magma, Sage, TeX

(C_6\times Q_8)\rtimes C_4

% in TeX

G:=Group("(C6xQ8):C4");

// GroupNames label

G:=SmallGroup(192,97);

// by ID

G=gap.SmallGroup(192,97);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,219,1571,570,136,6278]);

// Polycyclic

G:=Group<a,b,c,d|a^6=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^3*b,d*c*d^-1=b^-1*c>;

// generators/relations

G = (C6×Q8)⋊C4 order 192 = 26·3

1st semidirect product of C6×Q8 and C4 acting faithfully

G = (C₆×Q₈)⋊C₄ order 192 = 2⁶·3

1^st semidirect product of C₆×Q₈ and C₄ acting faithfully