D12.34C2^3 - GroupNames

G = D₁₂.₃₄C₂³ order 192 = 2⁶·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₂.₃₉C₂⁴, D₁₂.₃₄C₂³, 2_-¹⁺⁴⋊₅S₃, Dic₆.₃₄C₂³, C₄○D₄⋊₉D₆, (C₂×Q₈)⋊₁₆D₆, D₄○D₁₂⋊₁₀C₂, (C₃×D₄).₃₈D₄, C₃⋊₇(D₄○SD₁₆), C₃⋊C₈.₁₈C₂³, (C₃×Q₈).₃₈D₄, D₄⋊D₆⋊₁₂C₂, D₄⋊S₃⋊₂₂C₂², C₁₂.₂₇₁(C₂×D₄), C₄.₃₉(S₃×C₂³), (C₆×Q₈)⋊₂₃C₂², Q₈.₁₃D₆⋊₁₀C₂, D₄.₂₀(C₃⋊D₄), D₄.Dic₃⋊₁₂C₂, D₄.S₃⋊₂₂C₂², Q₈.₂₇(C₃⋊D₄), (C₃×D₄).₂₇C₂³, D₄.₂₇(C₂²×S₃), C₃⋊Q₁₆⋊₁₉C₂², C₆.₁₇₃(C₂²×D₄), (C₃×Q₈).₂₇C₂³, Q₈.₃₇(C₂²×S₃), Q₈.₁₁D₆⋊₁₁C₂, (C₂×C₁₂).₁₂₀C₂³, Q₈⋊₂S₃⋊₂₀C₂², C₄○D₁₂.₃₃C₂², C₄.Dic₃⋊₁₈C₂², (C₃×2_-¹⁺⁴)⋊₂C₂, (C₂×D₁₂).₁₈₆C₂², (C₂×C₃⋊C₈)⋊₂₆C₂², (C₂×C₆).₈₇(C₂×D₄), C₄.₇₇(C₂×C₃⋊D₄), (C₃×C₄○D₄)⋊₉C₂², C₂².₈(C₂×C₃⋊D₄), (C₂×Q₈⋊₂S₃)⋊₃₂C₂, C₂.₄₆(C₂²×C₃⋊D₄), (C₂×C₄).₁₀₄(C₂²×S₃), SmallGroup(192,1396)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — D₁₂.₃₄C₂³

Chief series

C₁ — C₃ — C₆ — C₁₂ — D₁₂ — C₂×D₁₂ — D₄○D₁₂ — D₁₂.₃₄C₂³

Lower central

C₃ — C₆ — C₁₂ — D₁₂.₃₄C₂³

Upper central

C₁ — C₂ — C₄○D₄ — 2_-¹⁺⁴

Generators and relations for D₁₂.₃₄C₂³
G = < a,b,c,d,e | a¹²=b²=c²=d²=1, e²=a⁶, bab=dad=a^-1, ac=ca, eae^-1=a⁷, cbc=a⁶b, dbd=a¹⁰b, ebe^-1=a³b, cd=dc, ce=ec, ede^-1=a⁹d >

Subgroups: 664 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, D₄, Q₈, Q₈, Q₈, C₂³, Dic₃, C₁₂, C₁₂, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₄○D₄, C₃⋊C₈, C₃⋊C₈, Dic₆, C₄×S₃, D₁₂, D₁₂, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×D₄, C₃×Q₈, C₃×Q₈, C₃×Q₈, C₂²×S₃, 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×D₁₂, C₄○D₁₂, S₃×D₄, Q₈⋊₃S₃, C₆×Q₈, C₆×Q₈, C₃×C₄○D₄, C₃×C₄○D₄, C₃×C₄○D₄, D₄○SD₁₆, C₂×Q₈⋊₂S₃, Q₈.₁₁D₆, D₄.Dic₃, D₄⋊D₆, Q₈.₁₃D₆, D₄○D₁₂, C₃×2_-¹⁺⁴, D₁₂.₃₄C₂³
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, D₁₂.₃₄C₂³

Smallest permutation representation of D₁₂.₃₄C₂³
►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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 15)(16 24)(17 23)(18 22)(19 21)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 45)(38 44)(39 43)(40 42)(46 48)

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

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

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

C₃⋊D₄

D₄○SD₁₆

D₁₂.₃₄C₂³

kernel

D₁₂.₃₄C₂³

C₂×Q₈⋊₂S₃

Q₈.₁₁D₆

D₄.Dic₃

D₄⋊D₆

Q₈.₁₃D₆

D₄○D₁₂

C₃×2_-¹⁺⁴

2_-¹⁺⁴

C₃×D₄

C₃×Q₈

C₂×Q₈

C₄○D₄

D₄

Q₈

C₃

C₁

# reps

Matrix representation of D₁₂.₃₄C₂³ ►in GL₆(𝔽₇₃)

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

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	6	6	67	67
0	0	6	6	6	61
0	0	12	0	0	67
0	0	12	12	0	61

0	1	0	0	0	0
1	0	0	0	0	0
0	0	67	67	67	12
0	0	67	67	6	6
0	0	0	61	0	6
0	0	61	61	0	12

43	13	0	0	0	0
60	30	0	0	0	0
0	0	0	1	1	72
0	0	0	1	0	72
0	0	72	1	0	0
0	0	0	2	0	72

G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,72,72,72,71,0,0,0,1,1,1],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,72,72,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6,12,12,0,0,6,6,0,12,0,0,67,6,0,0,0,0,67,61,67,61],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,67,67,0,61,0,0,67,67,61,61,0,0,67,6,0,0,0,0,12,6,6,12],[43,60,0,0,0,0,13,30,0,0,0,0,0,0,0,0,72,0,0,0,1,1,1,2,0,0,1,0,0,0,0,0,72,72,0,72] >;

D₁₂.₃₄C₂³ in GAP, Magma, Sage, TeX

D_{12}._{34}C_2^3

% in TeX

G:=Group("D12.34C2^3");

// GroupNames label

G:=SmallGroup(192,1396);

// by ID

G=gap.SmallGroup(192,1396);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e|a^12=b^2=c^2=d^2=1,e^2=a^6,b*a*b=d*a*d=a^-1,a*c=c*a,e*a*e^-1=a^7,c*b*c=a^6*b,d*b*d=a^10*b,e*b*e^-1=a^3*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^9*d>;

// generators/relations

G = D12.34C23 order 192 = 26·3

15th non-split extension by D12 of C23 acting via C23/C22=C2

G = D₁₂.₃₄C₂³ order 192 = 2⁶·3

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