D12.33C2^3 - GroupNames

G = D₁₂.₃₃C₂³ order 192 = 2⁶·3

14^th non-split extension by D₁₂ of C₂³ acting via C₂³/C₂²=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₂.₃₈C₂⁴, D₁₂.₃₃C₂³, 2₊¹⁺⁴:₉S₃, Dic₆.₃₃C₂³, Q₈oD₁₂:₉C₂, C₄oD₄.₃₂D₆, (C₃xD₄).₃₇D₄, C₃:₆(D₄oSD₁₆), C₃:C₈.₁₇C₂³, (C₃xQ₈).₃₇D₄, D₄:S₃:₂₁C₂², (C₂xD₄).₁₁₈D₆, Q₈.₁₃D₆:₉C₂, C₁₂.₂₇₀(C₂xD₄), C₄.₃₈(S₃xC₂³), Q₈.₁₄D₆:₁₁C₂, D₁₂:₆C₂²:₁₂C₂, C₄oD₁₂:₁₁C₂², D₄.₁₉(C₃:D₄), D₄.Dic₃:₁₁C₂, D₄.S₃:₂₁C₂², Q₈.₂₆(C₃:D₄), (C₃xD₄).₂₆C₂³, D₄.₂₆(C₂²xS₃), C₃:Q₁₆:₁₈C₂², C₆.₁₇₂(C₂²xD₄), (C₃xQ₈).₂₆C₂³, Q₈.₃₆(C₂²xS₃), (C₂xC₁₂).₁₁₉C₂³, Q₈:₂S₃:₂₂C₂², (C₂xDic₆):₄₃C₂², (C₆xD₄).₁₆₉C₂², C₄.Dic₃:₁₇C₂², (C₃x2₊¹⁺⁴):₃C₂, (C₂xC₃:C₈):₂₅C₂², (C₂xC₆).₈₆(C₂xD₄), C₄.₇₆(C₂xC₃:D₄), (C₂xD₄.S₃):₃₂C₂, C₂².₇(C₂xC₃:D₄), C₂.₄₅(C₂²xC₃:D₄), (C₂xC₄).₁₀₃(C₂²xS₃), (C₃xC₄oD₄).₂₉C₂², SmallGroup(192,1395)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — D₁₂.₃₃C₂³

Chief series

C₁ — C₃ — C₆ — C₁₂ — D₁₂ — C₄oD₁₂ — Q₈oD₁₂ — D₁₂.₃₃C₂³

Lower central

C₃ — C₆ — C₁₂ — D₁₂.₃₃C₂³

Upper central

C₁ — C₂ — C₄oD₄ — 2₊¹⁺⁴

Generators and relations for D₁₂.₃₃C₂³
G = < a,b,c,d,e | a¹²=b²=1, c²=d²=e²=a⁶, bab=a^-1, ac=ca, ad=da, eae^-1=a⁷, bc=cb, bd=db, ebe^-1=a³b, dcd^-1=a⁶c, ce=ec, de=ed >

Subgroups: 600 in 258 conjugacy classes, 107 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, Dic₃, C₁₂, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₄oD₄, C₄oD₄, C₃:C₈, C₃:C₈, Dic₆, Dic₆, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xD₄, C₃xQ₈, C₂²xC₆, C₈oD₄, C₂xSD₁₆, C₄oD₈, C₈:C₂², C₈.C₂², 2₊¹⁺⁴, 2_-¹⁺⁴, C₂xC₃:C₈, C₄.Dic₃, D₄:S₃, D₄.S₃, Q₈:₂S₃, C₃:Q₁₆, C₂xDic₆, C₄oD₁₂, D₄:₂S₃, S₃xQ₈, C₆xD₄, C₆xD₄, C₃xC₄oD₄, C₃xC₄oD₄, C₃xC₄oD₄, D₄oSD₁₆, D₁₂:₆C₂², C₂xD₄.S₃, D₄.Dic₃, Q₈.₁₃D₆, Q₈.₁₄D₆, Q₈oD₁₂, C₃x2₊¹⁺⁴, D₁₂.₃₃C₂³
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₂⁴, C₃:D₄, C₂²xS₃, C₂²xD₄, C₂xC₃:D₄, S₃xC₂³, D₄oSD₁₆, C₂²xC₃:D₄, D₁₂.₃₃C₂³

Smallest permutation representation of D₁₂.₃₃C₂³
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4	8
type	+	+	+	+	+	+	+	+	+	+	+	+	+				-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₆	D₆	C₃:D₄	C₃:D₄	D₄oSD₁₆	D₁₂.₃₃C₂³
kernel	D₁₂.₃₃C₂³	D₁₂:₆C₂²	C₂xD₄.S₃	D₄.Dic₃	Q₈.₁₃D₆	Q₈.₁₄D₆	Q₈oD₁₂	C₃x2₊¹⁺⁴	2₊¹⁺⁴	C₃xD₄	C₃xQ₈	C₂xD₄	C₄oD₄	D₄	Q₈	C₃	C₁
# reps	1	3	3	1	3	3	1	1	1	3	1	3	4	6	2	2	1

Matrix representation of D₁₂.₃₃C₂³ ►in GL₆(F₇₃)

64	0	0	0	0	0
71	8	0	0	0	0
0	0	0	1	0	0
0	0	72	0	0	0
0	0	72	1	72	2
0	0	0	1	72	1

55	7	0	0	0	0
6	18	0	0	0	0
0	0	6	67	0	61
0	0	67	6	61	12
0	0	6	6	0	0
0	0	12	67	6	61

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	0	72	0
0	0	1	72	1	71
0	0	1	0	0	0
0	0	0	1	72	1

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	72	0	0
0	0	1	0	0	0
0	0	72	1	72	2
0	0	72	0	72	1

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	0	1	0
0	0	1	72	1	71
0	0	72	0	0	0
0	0	72	1	0	1

G:=sub<GL(6,GF(73))| [64,71,0,0,0,0,0,8,0,0,0,0,0,0,0,72,72,0,0,0,1,0,1,1,0,0,0,0,72,72,0,0,0,0,2,1],[55,6,0,0,0,0,7,18,0,0,0,0,0,0,6,67,6,12,0,0,67,6,6,67,0,0,0,61,0,6,0,0,61,12,0,61],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0,1,0,0,72,1,0,72,0,0,0,71,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,72,72,0,0,72,0,1,0,0,0,0,0,72,72,0,0,0,0,2,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,72,72,0,0,0,72,0,1,0,0,1,1,0,0,0,0,0,71,0,1] >;

D₁₂.₃₃C₂³ in GAP, Magma, Sage, TeX

D_{12}._{33}C_2^3

% in TeX

G:=Group("D12.33C2^3");

// GroupNames label

G:=SmallGroup(192,1395);

// by ID

G=gap.SmallGroup(192,1395);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,675,136,1684,235,102,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^12=b^2=1,c^2=d^2=e^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^7,b*c=c*b,b*d=d*b,e*b*e^-1=a^3*b,d*c*d^-1=a^6*c,c*e=e*c,d*e=e*d>;

// generators/relations

G = D12.33C23 order 192 = 26·3

14th non-split extension by D12 of C23 acting via C23/C22=C2

G = D₁₂.₃₃C₂³ order 192 = 2⁶·3

14^th non-split extension by D₁₂ of C₂³ acting via C₂³/C₂²=C₂