C3xC2^3.31D4 - GroupNames

G = C₃×C₂³.₃₁D₄ order 192 = 2⁶·3

Direct product of C₃ and C₂³.₃₁D₄

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₃×C₂³.₃₁D₄, C₄⋊C₄⋊₂C₁₂, (C₆×Q₈)⋊₃C₄, C₆.₂₃C₄≀C₂, (C₂×Q₈)⋊₂C₁₂, C₂²⋊C₈.₂C₆, (C₂×C₆).₁₁Q₁₆, C₂²⋊Q₈.₁C₆, (C₂×C₁₂).₄₄₄D₄, (C₂×C₆).₂₃SD₁₆, C₂³.₃₅(C₃×D₄), C₂².₂(C₃×Q₁₆), C₆.₃₀(C₂³⋊C₄), (C₂²×C₆).₁₅₀D₄, C₂².₂(C₃×SD₁₆), C₆.₁₅(Q₈⋊C₄), C₂.C₄².₈C₆, (C₂²×C₁₂).₃₈₅C₂², (C₃×C₄⋊C₄)⋊₄C₄, C₂.₅(C₃×C₄≀C₂), (C₂×C₄).₈(C₂×C₁₂), (C₂×C₄).₉₆(C₃×D₄), C₂.₅(C₃×C₂³⋊C₄), (C₃×C₂²⋊C₈).₄C₂, C₂.₃(C₃×Q₈⋊C₄), (C₂×C₁₂).₁₇₅(C₂×C₄), (C₂²×C₄).₂₀(C₂×C₆), (C₃×C₂²⋊Q₈).₁₁C₂, C₂².₃₆(C₃×C₂²⋊C₄), (C₂×C₆).₁₂₃(C₂²⋊C₄), (C₃×C₂.C₄²).₂₄C₂, SmallGroup(192,134)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₄ — C₃×C₂³.₃₁D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂²×C₁₂ — C₃×C₂.C₄² — C₃×C₂³.₃₁D₄

Lower central

C₁ — C₂² — C₂×C₄ — C₃×C₂³.₃₁D₄

Upper central

C₁ — C₂×C₆ — C₂²×C₁₂ — C₃×C₂³.₃₁D₄

Generators and relations for C₃×C₂³.₃₁D₄
G = < a,b,c,d,e,f | a³=b²=c²=d²=1, e⁴=d, f²=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe^-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=bcde³ >

Subgroups: 162 in 80 conjugacy classes, 34 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₂, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂₄, C₂×C₁₂, C₂×C₁₂, C₃×Q₈, C₂²×C₆, C₂.C₄², C₂²⋊C₈, C₂²⋊Q₈, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₃×C₄⋊C₄, C₂×C₂₄, C₂²×C₁₂, C₂²×C₁₂, C₆×Q₈, C₂³.₃₁D₄, C₃×C₂.C₄², C₃×C₂²⋊C₈, C₃×C₂²⋊Q₈, C₃×C₂³.₃₁D₄
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, D₄, C₁₂, C₂×C₆, C₂²⋊C₄, SD₁₆, Q₁₆, C₂×C₁₂, C₃×D₄, C₂³⋊C₄, Q₈⋊C₄, C₄≀C₂, C₃×C₂²⋊C₄, C₃×SD₁₆, C₃×Q₁₆, C₂³.₃₁D₄, C₃×C₂³⋊C₄, C₃×Q₈⋊C₄, C₃×C₄≀C₂, C₃×C₂³.₃₁D₄

Smallest permutation representation of C₃×C₂³.₃₁D₄
►On 48 points

Generators in S₄₈

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

(2 38)(4 40)(6 34)(8 36)(10 30)(12 32)(14 26)(16 28)(18 48)(20 42)(22 44)(24 46)

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

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

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

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

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

57 conjugacy classes

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

57 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+									+	+		-							+
image	C₁	C₂	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₆	C₁₂	C₁₂	D₄	D₄	SD₁₆	Q₁₆	C₃×D₄	C₃×D₄	C₄≀C₂	C₃×SD₁₆	C₃×Q₁₆	C₃×C₄≀C₂	C₂³⋊C₄	C₃×C₂³⋊C₄
kernel	C₃×C₂³.₃₁D₄	C₃×C₂.C₄²	C₃×C₂²⋊C₈	C₃×C₂²⋊Q₈	C₂³.₃₁D₄	C₃×C₄⋊C₄	C₆×Q₈	C₂.C₄²	C₂²⋊C₈	C₂²⋊Q₈	C₄⋊C₄	C₂×Q₈	C₂×C₁₂	C₂²×C₆	C₂×C₆	C₂×C₆	C₂×C₄	C₂³	C₆	C₂²	C₂²	C₂	C₆	C₂
# reps	1	1	1	1	2	2	2	2	2	2	4	4	1	1	2	2	2	2	4	4	4	8	1	2

Matrix representation of C₃×C₂³.₃₁D₄ ►in GL₄(𝔽₇₃) generated by

8	0	0	0
0	8	0	0
0	0	64	0
0	0	0	64

1	0	0	0
0	1	0	0
0	0	1	0
0	0	22	72

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

72	0	0	0
0	72	0	0
0	0	72	0
0	0	0	72

6	67	0	0
6	6	0	0
0	0	63	54
0	0	72	10

72	0	0	0
0	1	0	0
0	0	72	0
0	0	57	27

G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,1,22,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[6,6,0,0,67,6,0,0,0,0,63,72,0,0,54,10],[72,0,0,0,0,1,0,0,0,0,72,57,0,0,0,27] >;

C₃×C₂³.₃₁D₄ in GAP, Magma, Sage, TeX

C_3\times C_2^3._{31}D_4

% in TeX

G:=Group("C3xC2^3.31D4");

// GroupNames label

G:=SmallGroup(192,134);

// by ID

G=gap.SmallGroup(192,134);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,1683,1522,248,2951]);

// Polycyclic

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

// generators/relations

G = C3×C23.31D4 order 192 = 26·3

Direct product of C3 and C23.31D4

G = C₃×C₂³.₃₁D₄ order 192 = 2⁶·3

Direct product of C₃ and C₂³.₃₁D₄