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, Q₈○D₁₂⋊₉C₂, C₄○D₄.₃₂D₆, (C₃×D₄).₃₇D₄, C₃⋊₆(D₄○SD₁₆), C₃⋊C₈.₁₇C₂³, (C₃×Q₈).₃₇D₄, D₄⋊S₃⋊₂₁C₂², (C₂×D₄).₁₁₈D₆, Q₈.₁₃D₆⋊₉C₂, C₁₂.₂₇₀(C₂×D₄), C₄.₃₈(S₃×C₂³), Q₈.₁₄D₆⋊₁₁C₂, D₁₂⋊₆C₂²⋊₁₂C₂, C₄○D₁₂⋊₁₁C₂², D₄.₁₉(C₃⋊D₄), D₄.Dic₃⋊₁₁C₂, D₄.S₃⋊₂₁C₂², Q₈.₂₆(C₃⋊D₄), (C₃×D₄).₂₆C₂³, D₄.₂₆(C₂²×S₃), C₃⋊Q₁₆⋊₁₈C₂², C₆.₁₇₂(C₂²×D₄), (C₃×Q₈).₂₆C₂³, Q₈.₃₆(C₂²×S₃), (C₂×C₁₂).₁₁₉C₂³, Q₈⋊₂S₃⋊₂₂C₂², (C₂×Dic₆)⋊₄₃C₂², (C₆×D₄).₁₆₉C₂², C₄.Dic₃⋊₁₇C₂², (C₃×2₊¹⁺⁴)⋊₃C₂, (C₂×C₃⋊C₈)⋊₂₅C₂², (C₂×C₆).₈₆(C₂×D₄), C₄.₇₆(C₂×C₃⋊D₄), (C₂×D₄.S₃)⋊₃₂C₂, C₂².₇(C₂×C₃⋊D₄), C₂.₄₅(C₂²×C₃⋊D₄), (C₂×C₄).₁₀₃(C₂²×S₃), (C₃×C₄○D₄).₂₉C₂², SmallGroup(192,1395)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — D₁₂.₃₃C₂³

Chief series

C₁ — C₃ — C₆ — C₁₂ — D₁₂ — C₄○D₁₂ — Q₈○D₁₂ — D₁₂.₃₃C₂³

Lower central

C₃ — C₆ — C₁₂ — D₁₂.₃₃C₂³

Upper central

C₁ — C₂ — C₄○D₄ — 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, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, Dic₃, C₁₂, C₁₂, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, C₄○D₄, C₃⋊C₈, C₃⋊C₈, Dic₆, Dic₆, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×Q₈, C₂²×C₆, C₈○D₄, C₂×SD₁₆, C₄○D₈, C₈⋊C₂², C₈.C₂², 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×C₃⋊C₈, C₄.Dic₃, D₄⋊S₃, D₄.S₃, Q₈⋊₂S₃, C₃⋊Q₁₆, C₂×Dic₆, C₄○D₁₂, D₄⋊₂S₃, S₃×Q₈, C₆×D₄, C₆×D₄, C₃×C₄○D₄, C₃×C₄○D₄, C₃×C₄○D₄, D₄○SD₁₆, D₁₂⋊₆C₂², C₂×D₄.S₃, D₄.Dic₃, Q₈.₁₃D₆, Q₈.₁₄D₆, Q₈○D₁₂, C₃×2₊¹⁺⁴, D₁₂.₃₃C₂³
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₂⁴, C₃⋊D₄, C₂²×S₃, C₂²×D₄, C₂×C₃⋊D₄, S₃×C₂³, D₄○SD₁₆, C₂²×C₃⋊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	D₄○SD₁₆	D₁₂.₃₃C₂³
kernel	D₁₂.₃₃C₂³	D₁₂⋊₆C₂²	C₂×D₄.S₃	D₄.Dic₃	Q₈.₁₃D₆	Q₈.₁₄D₆	Q₈○D₁₂	C₃×2₊¹⁺⁴	2₊¹⁺⁴	C₃×D₄	C₃×Q₈	C₂×D₄	C₄○D₄	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