C2^3.35D12 - GroupNames

G = C₂³.₃₅D₁₂ order 192 = 2⁶·3

1^st non-split extension by C₂³ of D₁₂ acting via D₁₂/D₆=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₃₅D₁₂, C₂².₂Dic₁₂, C₆.₈C₄≀C₂, C₄⋊Dic₃⋊₃C₄, (C₂×C₆).₁Q₁₆, (C₂×Dic₆)⋊₂C₄, C₂²⋊C₈.₁S₃, (C₂×C₆).₁SD₁₆, (C₂×C₁₂).₄₃₇D₄, C₆.₃(C₂³⋊C₄), (C₂²×C₄).₇₀D₆, (C₂²×C₆).₃₉D₄, C₆.₆(Q₈⋊C₄), C₂.₆(D₁₂⋊C₄), C₂².₄(C₂₄⋊C₂), C₂².₅₈(D₆⋊C₄), C₁₂.₄₈D₄.₁C₂, C₃⋊₂(C₂³.₃₁D₄), C₂.₃(C₂.Dic₁₂), C₆.C₄².₂₁C₂, (C₂²×C₁₂).₄₀C₂², C₂.₆(C₂³.₆D₆), (C₂×C₄).₁₂(C₄×S₃), (C₂×C₁₂).₂₄(C₂×C₄), (C₃×C₂²⋊C₈).₁C₂, (C₂×C₄).₂₀₈(C₃⋊D₄), (C₂×C₆).₃₉(C₂²⋊C₄), SmallGroup(192,26)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — C₂³.₃₅D₁₂

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂×C₁₂ — C₂²×C₁₂ — C₆.C₄² — C₂³.₃₅D₁₂

Lower central

C₃ — C₂×C₆ — C₂×C₁₂ — C₂³.₃₅D₁₂

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²⋊C₈

Generators and relations for C₂³.₃₅D₁₂
G = < a,b,c,d,e | a²=b²=c²=1, d¹²=cb=bc, e²=b, ab=ba, dad^-1=eae^-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede^-1=acd¹¹ >

Subgroups: 256 in 80 conjugacy classes, 29 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, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂₄, Dic₆, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₂²×C₆, C₂.C₄², C₂²⋊C₈, C₂²⋊Q₈, Dic₃⋊C₄, C₄⋊Dic₃, C₆.D₄, C₂×C₂₄, C₂×Dic₆, C₂²×Dic₃, C₂²×C₁₂, C₂³.₃₁D₄, C₆.C₄², C₃×C₂²⋊C₈, C₁₂.₄₈D₄, C₂³.₃₅D₁₂
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, D₆, C₂²⋊C₄, SD₁₆, Q₁₆, C₄×S₃, D₁₂, C₃⋊D₄, C₂³⋊C₄, Q₈⋊C₄, C₄≀C₂, C₂₄⋊C₂, Dic₁₂, D₆⋊C₄, C₂³.₃₁D₄, C₂³.₆D₆, C₂.Dic₁₂, D₁₂⋊C₄, C₂³.₃₅D₁₂

Smallest permutation representation of C₂³.₃₅D₁₂
►On 48 points

Generators in S₄₈

(2 25)(4 27)(6 29)(8 31)(10 33)(12 35)(14 37)(16 39)(18 41)(20 43)(22 45)(24 47)

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

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

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

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

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

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

39 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	24A	···	24H
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	24	···	24
size	1	1	1	1	2	2	2	2	2	4	12	12	12	12	24	24	2	2	2	4	4	4	4	4	4	2	2	2	2	4	4	4	···	4

39 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+			+	+	+	+		-			+			-	+
image	C₁	C₂	C₂	C₂	C₄	C₄	S₃	D₄	D₄	D₆	SD₁₆	Q₁₆	C₄×S₃	C₃⋊D₄	D₁₂	C₄≀C₂	C₂₄⋊C₂	Dic₁₂	C₂³⋊C₄	C₂³.₆D₆	D₁₂⋊C₄
kernel	C₂³.₃₅D₁₂	C₆.C₄²	C₃×C₂²⋊C₈	C₁₂.₄₈D₄	C₄⋊Dic₃	C₂×Dic₆	C₂²⋊C₈	C₂×C₁₂	C₂²×C₆	C₂²×C₄	C₂×C₆	C₂×C₆	C₂×C₄	C₂×C₄	C₂³	C₆	C₂²	C₂²	C₆	C₂	C₂
# reps	1	1	1	1	2	2	1	1	1	1	2	2	2	2	2	4	4	4	1	2	2

Matrix representation of C₂³.₃₅D₁₂ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	57
0	0	0	0	0	72

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	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	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

26	65	0	0	0	0
18	59	0	0	0	0
0	0	14	66	0	0
0	0	7	7	0	0
0	0	0	0	9	50
0	0	0	0	65	64

59	15	0	0	0	0
55	14	0	0	0	0
0	0	7	7	0	0
0	0	14	66	0	0
0	0	0	0	49	70
0	0	0	0	70	24

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,57,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[26,18,0,0,0,0,65,59,0,0,0,0,0,0,14,7,0,0,0,0,66,7,0,0,0,0,0,0,9,65,0,0,0,0,50,64],[59,55,0,0,0,0,15,14,0,0,0,0,0,0,7,14,0,0,0,0,7,66,0,0,0,0,0,0,49,70,0,0,0,0,70,24] >;

C₂³.₃₅D₁₂ in GAP, Magma, Sage, TeX

C_2^3._{35}D_{12}

% in TeX

G:=Group("C2^3.35D12");

// GroupNames label

G:=SmallGroup(192,26);

// by ID

G=gap.SmallGroup(192,26);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,85,92,422,387,268,570,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c*b=b*c,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*c*d^11>;

// generators/relations

G = C23.35D12 order 192 = 26·3

1st non-split extension by C23 of D12 acting via D12/D6=C2

G = C₂³.₃₅D₁₂ order 192 = 2⁶·3

1^st non-split extension by C₂³ of D₁₂ acting via D₁₂/D₆=C₂