D12.39C2^3 - GroupNames

G = D₁₂.₃₉C₂³ order 192 = 2⁶·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₆.₁₈C₂⁵, C₁₂.₅₃C₂⁴, D₆.₁₀C₂⁴, D₁₂.₃₉C₂³, 2_-¹⁺⁴⋊₇S₃, Dic₃.₁₃C₂⁴, Dic₆.₄₀C₂³, C₄○D₄⋊₁₆D₆, (C₂×Q₈)⋊₂₈D₆, D₄○D₁₂⋊₁₂C₂, (C₂×C₆).₉C₂⁴, (S₃×D₄)⋊₁₅C₂², C₄.₅₀(S₃×C₂³), C₂.₁₉(S₃×C₂⁴), (C₆×Q₈)⋊₂₅C₂², (S₃×Q₈)⋊₁₈C₂², C₃⋊D₄.₅C₂³, C₄○D₁₂⋊₁₅C₂², (C₂×D₁₂)⋊₄₁C₂², (C₄×S₃).₂₂C₂³, C₃⋊₃(C₂.C₂⁵), Q₈.₁₅D₆⋊₈C₂, (C₃×D₄).₃₃C₂³, D₄.₃₃(C₂²×S₃), C₂².₆(S₃×C₂³), Q₈.₄₄(C₂²×S₃), (C₃×Q₈).₃₄C₂³, D₄⋊₂S₃⋊₁₉C₂², (C₂×C₁₂).₁₂₄C₂³, Q₈⋊₃S₃⋊₁₇C₂², (C₃×2_-¹⁺⁴)⋊₅C₂, (C₂²×S₃).₁₄₄C₂³, (C₂×Dic₃).₃₀₀C₂³, (S₃×C₄○D₄)⋊₁₀C₂, (S₃×C₂×C₄)⋊₃₈C₂², (C₂×Q₈⋊₃S₃)⋊₂₂C₂, (C₃×C₄○D₄)⋊₁₃C₂², (C₂×C₄).₁₀₈(C₂²×S₃), SmallGroup(192,1527)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — D₁₂.₃₉C₂³

Chief series

C₁ — C₃ — C₆ — D₆ — C₂²×S₃ — S₃×C₂×C₄ — S₃×C₄○D₄ — D₁₂.₃₉C₂³

Lower central

C₃ — C₆ — D₁₂.₃₉C₂³

Upper central

C₁ — C₂ — 2_-¹⁺⁴

Generators and relations for D₁₂.₃₉C₂³
G = < a,b,c,d,e | a¹²=b²=c²=d²=e²=1, bab=a^-1, ac=ca, ad=da, eae=a⁵, cbc=a⁶b, bd=db, ebe=a¹⁰b, dcd=ece=a⁶c, de=ed >

Subgroups: 1704 in 810 conjugacy classes, 443 normal (8 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, Dic₃, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, Dic₆, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₂²×S₃, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, 2_-¹⁺⁴, S₃×C₂×C₄, C₂×D₁₂, C₄○D₁₂, S₃×D₄, D₄⋊₂S₃, S₃×Q₈, Q₈⋊₃S₃, C₆×Q₈, C₃×C₄○D₄, C₂.C₂⁵, C₂×Q₈⋊₃S₃, Q₈.₁₅D₆, S₃×C₄○D₄, D₄○D₁₂, C₃×2_-¹⁺⁴, D₁₂.₃₉C₂³
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₂⁴, C₂²×S₃, C₂⁵, S₃×C₂³, C₂.C₂⁵, S₃×C₂⁴, D₁₂.₃₉C₂³

Smallest permutation representation of D₁₂.₃₉C₂³
►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 9)(2 8)(3 7)(4 6)(10 12)(13 23)(14 22)(15 21)(16 20)(17 19)(25 35)(26 34)(27 33)(28 32)(29 31)(37 45)(38 44)(39 43)(40 42)(46 48)

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

(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)

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

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

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

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

51 conjugacy classes

class	1	2A	2B	···	2F	2G	···	2P	3	4A	···	4J	4K	4L	4M	···	4Q	6A	6B	···	6F	12A	···	12J
order	1	2	2	···	2	2	···	2	3	4	···	4	4	4	4	···	4	6	6	···	6	12	···	12
size	1	1	2	···	2	6	···	6	2	2	···	2	3	3	6	···	6	2	4	···	4	4	···	4

51 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4	8
type	+	+	+	+	+	+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₆	D₆	C₂.C₂⁵	D₁₂.₃₉C₂³
kernel	D₁₂.₃₉C₂³	C₂×Q₈⋊₃S₃	Q₈.₁₅D₆	S₃×C₄○D₄	D₄○D₁₂	C₃×2_-¹⁺⁴	2_-¹⁺⁴	C₂×Q₈	C₄○D₄	C₃	C₁
# reps	1	5	5	10	10	1	1	5	10	2	1

Matrix representation of D₁₂.₃₉C₂³ ►in GL₆(𝔽₁₃)

12	12	0	0	0	0
1	0	0	0	0	0
0	0	1	5	0	0
0	0	10	12	0	0
0	0	0	0	1	5
0	0	0	0	10	12

12	12	0	0	0	0
0	1	0	0	0	0
0	0	1	5	0	0
0	0	0	12	0	0
0	0	0	0	12	8
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

12	0	0	0	0	0
0	12	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	12	0
0	0	0	0	0	12

1	0	0	0	0	0
12	12	0	0	0	0
0	0	8	1	0	0
0	0	2	5	0	0
0	0	0	0	5	12
0	0	0	0	11	8

G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,10,0,0,0,0,5,12,0,0,0,0,0,0,1,10,0,0,0,0,5,12],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,5,12,0,0,0,0,0,0,12,0,0,0,0,0,8,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,8,2,0,0,0,0,1,5,0,0,0,0,0,0,5,11,0,0,0,0,12,8] >;

D₁₂.₃₉C₂³ in GAP, Magma, Sage, TeX

D_{12}._{39}C_2^3

% in TeX

G:=Group("D12.39C2^3");

// GroupNames label

G:=SmallGroup(192,1527);

// by ID

G=gap.SmallGroup(192,1527);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,570,1684,438,102,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^12=b^2=c^2=d^2=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^5,c*b*c=a^6*b,b*d=d*b,e*b*e=a^10*b,d*c*d=e*c*e=a^6*c,d*e=e*d>;

// generators/relations

G = D12.39C23 order 192 = 26·3

20th non-split extension by D12 of C23 acting via C23/C22=C2

G = D₁₂.₃₉C₂³ order 192 = 2⁶·3

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