C2xA4:C8 - GroupNames

G = C₂×A₄⋊C₈ order 192 = 2⁶·3

Direct product of C₂ and A₄⋊C₈

direct product, non-abelian, soluble, monomial

Aliases: C₂×A₄⋊C₈, C₂⁴.₁Dic₃, (C₂×A₄)⋊C₈, C₂³⋊(C₃⋊C₈), A₄⋊₂(C₂×C₈), C₄.₃₀(C₂×S₄), (C₄×A₄).₃C₄, (C₂×C₄).₂₃S₄, C₄.₆(A₄⋊C₄), (C₂³×C₄).₁S₃, (C₂²×A₄).₂C₄, (C₂²×C₄).₁₀D₆, C₂².₈(A₄⋊C₄), (C₄×A₄).₁₄C₂², C₂³.₁(C₂×Dic₃), (C₂²×C₄).₂Dic₃, C₂²⋊(C₂×C₃⋊C₈), (C₂×C₄×A₄).₅C₂, C₂.₁(C₂×A₄⋊C₄), (C₂×A₄).₆(C₂×C₄), SmallGroup(192,967)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — A₄ — C₂×A₄⋊C₈

Chief series

C₁ — C₂² — A₄ — C₂×A₄ — C₄×A₄ — A₄⋊C₈ — C₂×A₄⋊C₈

Lower central

A₄ — C₂×A₄⋊C₈

Upper central

C₁ — C₂×C₄

Generators and relations for C₂×A₄⋊C₈
G = < a,b,c,d,e | a²=b²=c²=d³=e⁸=1, ab=ba, ac=ca, ad=da, ae=ea, dbd^-1=ebe^-1=bc=cb, dcd^-1=b, ce=ec, ede^-1=d^-1 >

Subgroups: 254 in 89 conjugacy classes, 27 normal (19 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₁₂, A₄, C₂×C₆, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂⁴, C₃⋊C₈, C₂×C₁₂, C₂×A₄, C₂×A₄, C₂²⋊C₈, C₂²×C₈, C₂³×C₄, C₂×C₃⋊C₈, C₄×A₄, C₂²×A₄, C₂×C₂²⋊C₈, A₄⋊C₈, C₂×C₄×A₄, C₂×A₄⋊C₈
Quotients: C₁, C₂, C₄, C₂², S₃, C₈, C₂×C₄, Dic₃, D₆, C₂×C₈, C₃⋊C₈, C₂×Dic₃, S₄, C₂×C₃⋊C₈, A₄⋊C₄, C₂×S₄, A₄⋊C₈, C₂×A₄⋊C₄, C₂×A₄⋊C₈

Smallest permutation representation of C₂×A₄⋊C₈
►On 48 points

Generators in S₄₈

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

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

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

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

(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)

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

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

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

40 conjugacy classes

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

40 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	3	3	3	3	3
type	+	+	+				+	-	+	-		+		+
image	C₁	C₂	C₂	C₄	C₄	C₈	S₃	Dic₃	D₆	Dic₃	C₃⋊C₈	S₄	A₄⋊C₄	C₂×S₄	A₄⋊C₄	A₄⋊C₈
kernel	C₂×A₄⋊C₈	A₄⋊C₈	C₂×C₄×A₄	C₄×A₄	C₂²×A₄	C₂×A₄	C₂³×C₄	C₂²×C₄	C₂²×C₄	C₂⁴	C₂³	C₂×C₄	C₄	C₄	C₂²	C₂
# reps	1	2	1	2	2	8	1	1	1	1	4	2	2	2	2	8

Matrix representation of C₂×A₄⋊C₈ ►in GL₅(𝔽₇₃)

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

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

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

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

29	42	0	0	0
71	44	0	0	0
0	0	0	0	72
0	0	0	72	0
0	0	72	0	0

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72],[0,1,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[29,71,0,0,0,42,44,0,0,0,0,0,0,0,72,0,0,0,72,0,0,0,72,0,0] >;

C₂×A₄⋊C₈ in GAP, Magma, Sage, TeX

C_2\times A_4\rtimes C_8

% in TeX

G:=Group("C2xA4:C8");

// GroupNames label

G:=SmallGroup(192,967);

// by ID

G=gap.SmallGroup(192,967);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,28,58,1124,4037,285,2358,475]);

// Polycyclic

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

// generators/relations

G = C2×A4⋊C8 order 192 = 26·3

Direct product of C2 and A4⋊C8

G = C₂×A₄⋊C₈ order 192 = 2⁶·3

Direct product of C₂ and A₄⋊C₈