(C3xD4).31D4 - GroupNames

G = (C₃×D₄).₃₁D₄ order 192 = 2⁶·3

1^st non-split extension by C₃×D₄ of D₄ acting via D₄/C₂²=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₃×D₄).₃₁D₄, (C₂×C₆)⋊₁₁SD₁₆, C₆.₇₂C₂²≀C₂, (C₂×D₄).₁₉₉D₆, C₁₂.₂₀₆(C₂×D₄), (C₂×C₁₂).₃₀₁D₄, C₃⋊₅(C₂²⋊SD₁₆), C₆.₆₄(C₂×SD₁₆), (C₂²×D₄).₅S₃, D₄⋊Dic₃⋊₃₉C₂, D₄.₁₃(C₃⋊D₄), C₄⋊Dic₃⋊₂₂C₂², C₂²⋊₄(D₄.S₃), (C₂²×C₆).₁₉₇D₄, (C₂²×C₄).₁₆₆D₆, C₁₂.₅₅D₄⋊₁₆C₂, C₁₂.₄₈D₄⋊₂₆C₂, C₆.₁₀₃(C₈⋊C₂²), (C₂×C₁₂).₄₇₃C₂³, C₂.₅(C₂⁴⋊₄S₃), (C₂×Dic₆)⋊₁₅C₂², (C₆×D₄).₂₄₁C₂², C₂³.₉₃(C₃⋊D₄), C₂.₂₃(D₁₂⋊₆C₂²), (C₂²×C₁₂).₁₉₈C₂², (D₄×C₂×C₆).₄C₂, (C₂×C₃⋊C₈)⋊₁₁C₂², C₄.₅₉(C₂×C₃⋊D₄), (C₂×D₄.S₃)⋊₂₃C₂, (C₂×C₆).₅₅₄(C₂×D₄), C₂.₁₇(C₂×D₄.S₃), (C₂×C₄).₈₄(C₃⋊D₄), (C₂×C₄).₅₅₉(C₂²×S₃), C₂².₂₁₇(C₂×C₃⋊D₄), SmallGroup(192,777)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — (C₃×D₄).₃₁D₄

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂×C₁₂ — C₂×Dic₆ — C₂×D₄.S₃ — (C₃×D₄).₃₁D₄

Lower central

C₃ — C₆ — C₂×C₁₂ — (C₃×D₄).₃₁D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×D₄

Generators and relations for (C₃×D₄).₃₁D₄
G = < a,b,c,d,e | a³=b⁴=c²=1, d⁴=e²=b², ab=ba, ac=ca, dad^-1=eae^-1=a^-1, cbc=ebe^-1=b^-1, bd=db, dcd^-1=bc, ece^-1=b^-1c, ede^-1=d³ >

Subgroups: 488 in 188 conjugacy classes, 51 normal (25 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₂³, Dic₃, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, C₃⋊C₈, Dic₆, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₂²×C₆, C₂²×C₆, C₂²⋊C₈, D₄⋊C₄, C₂²⋊Q₈, C₂×SD₁₆, C₂²×D₄, C₂×C₃⋊C₈, Dic₃⋊C₄, C₄⋊Dic₃, D₄.S₃, C₆.D₄, C₂×Dic₆, C₂²×C₁₂, C₆×D₄, C₆×D₄, C₂³×C₆, C₂²⋊SD₁₆, C₁₂.₅₅D₄, D₄⋊Dic₃, C₁₂.₄₈D₄, C₂×D₄.S₃, D₄×C₂×C₆, (C₃×D₄).₃₁D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, SD₁₆, C₂×D₄, C₃⋊D₄, C₂²×S₃, C₂²≀C₂, C₂×SD₁₆, C₈⋊C₂², D₄.S₃, C₂×C₃⋊D₄, C₂²⋊SD₁₆, D₁₂⋊₆C₂², C₂×D₄.S₃, C₂⁴⋊₄S₃, (C₃×D₄).₃₁D₄

Smallest permutation representation of (C₃×D₄).₃₁D₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+					+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₄	D₆	D₆	SD₁₆	C₃⋊D₄	C₃⋊D₄	C₃⋊D₄	C₈⋊C₂²	D₄.S₃	D₁₂⋊₆C₂²
kernel	(C₃×D₄).₃₁D₄	C₁₂.₅₅D₄	D₄⋊Dic₃	C₁₂.₄₈D₄	C₂×D₄.S₃	D₄×C₂×C₆	C₂²×D₄	C₂×C₁₂	C₃×D₄	C₂²×C₆	C₂²×C₄	C₂×D₄	C₂×C₆	C₂×C₄	D₄	C₂³	C₆	C₂²	C₂
# reps	1	1	2	1	2	1	1	1	4	1	1	2	4	2	8	2	1	2	2

Matrix representation of (C₃×D₄).₃₁D₄ ►in GL₆(𝔽₇₃)

64	0	0	0	0	0
0	8	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

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	0	1
0	0	0	0	72	0

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

0	72	0	0	0	0
1	0	0	0	0	0
0	0	0	72	0	0
0	0	1	0	0	0
0	0	0	0	67	67
0	0	0	0	6	67

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	67	6
0	0	0	0	6	6

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,67,6,0,0,0,0,67,67],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,67,6,0,0,0,0,6,6] >;

(C₃×D₄).₃₁D₄ in GAP, Magma, Sage, TeX

(C_3\times D_4)._{31}D_4

% in TeX

G:=Group("(C3xD4).31D4");

// GroupNames label

G:=SmallGroup(192,777);

// by ID

G=gap.SmallGroup(192,777);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,1684,851,102,6278]);

// Polycyclic

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

// generators/relations

G = (C3×D4).31D4 order 192 = 26·3

1st non-split extension by C3×D4 of D4 acting via D4/C22=C2

G = (C₃×D₄).₃₁D₄ order 192 = 2⁶·3

1^st non-split extension by C₃×D₄ of D₄ acting via D₄/C₂²=C₂