(C2xDic3):C8 - GroupNames

G = (C₂×Dic₃)⋊C₈ order 192 = 2⁶·3

The semidirect product of C₂×Dic₃ and C₈ acting via C₈/C₂=C₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×Dic₃)⋊C₈, C₂.₆(D₆⋊C₈), C₂²⋊C₈.₂S₃, C₂².₄(S₃×C₈), (C₂×C₄).₁₀₈D₁₂, (C₂×C₁₂).₄₃₉D₄, C₆.₄(C₂²⋊C₈), C₆.₅(C₂³⋊C₄), C₂³.₄₆(C₄×S₃), (C₂²×C₄).₁₈D₆, (C₂×C₆).₂M₄(2), C₂².₄(C₈⋊S₃), C₂².₃₃(D₆⋊C₄), C₆.₂(C₄.₁₀D₄), (C₂²×Dic₃).₂C₄, C₁₂.₅₅D₄.₁₁C₂, C₂.₁(C₁₂.₄₇D₄), C₂.₂(C₂³.₆D₆), (C₂²×C₁₂).₃₂₃C₂², C₃⋊₁(C₂².M₄(2)), (C₂×C₆).₂(C₂×C₈), (C₃×C₂²⋊C₈).₂C₂, (C₂²×C₆).₂₇(C₂×C₄), (C₂×C₄).₂₁₀(C₃⋊D₄), (C₂×C₆).₄₁(C₂²⋊C₄), (C₂×Dic₃⋊C₄).₂₄C₂, SmallGroup(192,28)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — (C₂×Dic₃)⋊C₈

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂×C₁₂ — C₂²×C₁₂ — C₂×Dic₃⋊C₄ — (C₂×Dic₃)⋊C₈

Lower central

C₃ — C₆ — C₂×C₆ — (C₂×Dic₃)⋊C₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²⋊C₈

Generators and relations for (C₂×Dic₃)⋊C₈
G = < a,b,c,d | a²=b⁶=d⁸=1, c²=b³, ab=ba, ac=ca, dad^-1=ab³, cbc^-1=b^-1, bd=db, dcd^-1=ab³c >

Subgroups: 208 in 78 conjugacy classes, 31 normal (29 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, C₂³, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₃⋊C₈, C₂₄, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₂²×C₆, C₂²⋊C₈, C₂²⋊C₈, C₂×C₄⋊C₄, C₂×C₃⋊C₈, Dic₃⋊C₄, C₂×C₂₄, C₂²×Dic₃, C₂²×C₁₂, C₂².M₄(2), C₁₂.₅₅D₄, C₃×C₂²⋊C₈, C₂×Dic₃⋊C₄, (C₂×Dic₃)⋊C₈
Quotients: C₁, C₂, C₄, C₂², S₃, C₈, C₂×C₄, D₄, D₆, C₂²⋊C₄, C₂×C₈, M₄(2), C₄×S₃, D₁₂, C₃⋊D₄, C₂²⋊C₈, C₂³⋊C₄, C₄.₁₀D₄, S₃×C₈, C₈⋊S₃, D₆⋊C₄, C₂².M₄(2), C₂³.₆D₆, D₆⋊C₈, C₁₂.₄₇D₄, (C₂×Dic₃)⋊C₈

Smallest permutation representation of (C₂×Dic₃)⋊C₈
►On 96 points

Generators in S₉₆

(2 55)(4 49)(6 51)(8 53)(9 43)(11 45)(13 47)(15 41)(17 92)(19 94)(21 96)(23 90)(25 35)(27 37)(29 39)(31 33)(57 81)(59 83)(61 85)(63 87)(66 73)(68 75)(70 77)(72 79)

(1 62 78 54 86 71)(2 63 79 55 87 72)(3 64 80 56 88 65)(4 57 73 49 81 66)(5 58 74 50 82 67)(6 59 75 51 83 68)(7 60 76 52 84 69)(8 61 77 53 85 70)(9 17 39 43 92 29)(10 18 40 44 93 30)(11 19 33 45 94 31)(12 20 34 46 95 32)(13 21 35 47 96 25)(14 22 36 48 89 26)(15 23 37 41 90 27)(16 24 38 42 91 28)

(1 38 54 28)(2 29 55 39)(3 30 56 40)(4 33 49 31)(5 34 50 32)(6 25 51 35)(7 26 52 36)(8 37 53 27)(9 79 43 72)(10 80 44 65)(11 66 45 73)(12 67 46 74)(13 75 47 68)(14 76 48 69)(15 70 41 77)(16 71 42 78)(17 63 92 87)(18 64 93 88)(19 81 94 57)(20 82 95 58)(21 59 96 83)(22 60 89 84)(23 85 90 61)(24 86 91 62)

(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)

G:=sub<Sym(96)| (2,55)(4,49)(6,51)(8,53)(9,43)(11,45)(13,47)(15,41)(17,92)(19,94)(21,96)(23,90)(25,35)(27,37)(29,39)(31,33)(57,81)(59,83)(61,85)(63,87)(66,73)(68,75)(70,77)(72,79), (1,62,78,54,86,71)(2,63,79,55,87,72)(3,64,80,56,88,65)(4,57,73,49,81,66)(5,58,74,50,82,67)(6,59,75,51,83,68)(7,60,76,52,84,69)(8,61,77,53,85,70)(9,17,39,43,92,29)(10,18,40,44,93,30)(11,19,33,45,94,31)(12,20,34,46,95,32)(13,21,35,47,96,25)(14,22,36,48,89,26)(15,23,37,41,90,27)(16,24,38,42,91,28), (1,38,54,28)(2,29,55,39)(3,30,56,40)(4,33,49,31)(5,34,50,32)(6,25,51,35)(7,26,52,36)(8,37,53,27)(9,79,43,72)(10,80,44,65)(11,66,45,73)(12,67,46,74)(13,75,47,68)(14,76,48,69)(15,70,41,77)(16,71,42,78)(17,63,92,87)(18,64,93,88)(19,81,94,57)(20,82,95,58)(21,59,96,83)(22,60,89,84)(23,85,90,61)(24,86,91,62), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)>;

G:=Group( (2,55)(4,49)(6,51)(8,53)(9,43)(11,45)(13,47)(15,41)(17,92)(19,94)(21,96)(23,90)(25,35)(27,37)(29,39)(31,33)(57,81)(59,83)(61,85)(63,87)(66,73)(68,75)(70,77)(72,79), (1,62,78,54,86,71)(2,63,79,55,87,72)(3,64,80,56,88,65)(4,57,73,49,81,66)(5,58,74,50,82,67)(6,59,75,51,83,68)(7,60,76,52,84,69)(8,61,77,53,85,70)(9,17,39,43,92,29)(10,18,40,44,93,30)(11,19,33,45,94,31)(12,20,34,46,95,32)(13,21,35,47,96,25)(14,22,36,48,89,26)(15,23,37,41,90,27)(16,24,38,42,91,28), (1,38,54,28)(2,29,55,39)(3,30,56,40)(4,33,49,31)(5,34,50,32)(6,25,51,35)(7,26,52,36)(8,37,53,27)(9,79,43,72)(10,80,44,65)(11,66,45,73)(12,67,46,74)(13,75,47,68)(14,76,48,69)(15,70,41,77)(16,71,42,78)(17,63,92,87)(18,64,93,88)(19,81,94,57)(20,82,95,58)(21,59,96,83)(22,60,89,84)(23,85,90,61)(24,86,91,62), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96) );

G=PermutationGroup([[(2,55),(4,49),(6,51),(8,53),(9,43),(11,45),(13,47),(15,41),(17,92),(19,94),(21,96),(23,90),(25,35),(27,37),(29,39),(31,33),(57,81),(59,83),(61,85),(63,87),(66,73),(68,75),(70,77),(72,79)], [(1,62,78,54,86,71),(2,63,79,55,87,72),(3,64,80,56,88,65),(4,57,73,49,81,66),(5,58,74,50,82,67),(6,59,75,51,83,68),(7,60,76,52,84,69),(8,61,77,53,85,70),(9,17,39,43,92,29),(10,18,40,44,93,30),(11,19,33,45,94,31),(12,20,34,46,95,32),(13,21,35,47,96,25),(14,22,36,48,89,26),(15,23,37,41,90,27),(16,24,38,42,91,28)], [(1,38,54,28),(2,29,55,39),(3,30,56,40),(4,33,49,31),(5,34,50,32),(6,25,51,35),(7,26,52,36),(8,37,53,27),(9,79,43,72),(10,80,44,65),(11,66,45,73),(12,67,46,74),(13,75,47,68),(14,76,48,69),(15,70,41,77),(16,71,42,78),(17,63,92,87),(18,64,93,88),(19,81,94,57),(20,82,95,58),(21,59,96,83),(22,60,89,84),(23,85,90,61),(24,86,91,62)], [(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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)]])

42 conjugacy classes

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	4D	4E	4F	4G	4H	6A	6B	6C	6D	6E	8A	8B	8C	8D	8E	8F	8G	8H	12A	12B	12C	12D	12E	12F	24A	···	24H
order	1	2	2	2	2	2	3	4	4	4	4	4	4	4	4	6	6	6	6	6	8	8	8	8	8	8	8	8	12	12	12	12	12	12	24	···	24
size	1	1	1	1	2	2	2	2	2	2	2	12	12	12	12	2	2	2	4	4	4	4	4	4	12	12	12	12	2	2	2	2	4	4	4	···	4

42 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+			+	+	+		+					+	-		-
image	C₁	C₂	C₂	C₂	C₄	C₈	S₃	D₄	D₆	M₄(2)	D₁₂	C₃⋊D₄	C₄×S₃	S₃×C₈	C₈⋊S₃	C₂³⋊C₄	C₄.₁₀D₄	C₂³.₆D₆	C₁₂.₄₇D₄
kernel	(C₂×Dic₃)⋊C₈	C₁₂.₅₅D₄	C₃×C₂²⋊C₈	C₂×Dic₃⋊C₄	C₂²×Dic₃	C₂×Dic₃	C₂²⋊C₈	C₂×C₁₂	C₂²×C₄	C₂×C₆	C₂×C₄	C₂×C₄	C₂³	C₂²	C₂²	C₆	C₆	C₂	C₂
# reps	1	1	1	1	4	8	1	2	1	2	2	2	2	4	4	1	1	2	2

Matrix representation of (C₂×Dic₃)⋊C₈ ►in GL₆(𝔽₇₃)

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

72	0	0	0	0	0
0	1	0	0	0	0
0	0	19	59	0	0
0	0	5	54	0	0
0	0	0	0	19	59
0	0	0	0	5	54

0	22	0	0	0	0
51	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	7	59	0	0
0	0	14	66	0	0

G:=sub<GL(6,GF(73))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,19,5,0,0,0,0,59,54,0,0,0,0,0,0,19,5,0,0,0,0,59,54],[0,51,0,0,0,0,22,0,0,0,0,0,0,0,0,0,7,14,0,0,0,0,59,66,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C₂×Dic₃)⋊C₈ in GAP, Magma, Sage, TeX

(C_2\times {\rm Dic}_3)\rtimes C_8

% in TeX

G:=Group("(C2xDic3):C8");

// GroupNames label

G:=SmallGroup(192,28);

// by ID

G=gap.SmallGroup(192,28);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,758,100,570,6278]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=d^8=1,c^2=b^3,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^3,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a*b^3*c>;

// generators/relations

G = (C2×Dic3)⋊C8 order 192 = 26·3

The semidirect product of C2×Dic3 and C8 acting via C8/C2=C4

G = (C₂×Dic₃)⋊C₈ order 192 = 2⁶·3

The semidirect product of C₂×Dic₃ and C₈ acting via C₈/C₂=C₄