D6.C2^4 - GroupNames

G = D₆.C₂⁴ order 192 = 2⁶·3

9^th non-split extension by D₆ of C₂⁴ acting via C₂⁴/C₂³=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₆.₁₆C₂⁵, D₆.₉C₂⁴, C₁₂.₅₁C₂⁴, D₁₂.₃₇C₂³, 2₊¹⁺⁴⋊₁₀S₃, Dic₆.₃₈C₂³, Dic₃.₁₁C₂⁴, C₄○D₄⋊₁₄D₆, (C₂×D₄)⋊₃₂D₆, Q₈○D₁₂⋊₁₁C₂, (C₂×C₆).₇C₂⁴, D₄⋊₆D₆⋊₁₀C₂, (S₃×D₄)⋊₁₄C₂², (C₆×D₄)⋊₂₆C₂², C₄.₄₈(S₃×C₂³), C₂.₁₇(S₃×C₂⁴), (S₃×Q₈)⋊₁₆C₂², C₃⋊D₄.₃C₂³, C₄○D₁₂⋊₁₃C₂², (C₄×S₃).₂₀C₂³, C₃⋊₂(C₂.C₂⁵), D₄.₃₁(C₂²×S₃), (C₃×D₄).₃₁C₂³, C₂².₄(S₃×C₂³), (C₃×Q₈).₃₂C₂³, Q₈.₄₂(C₂²×S₃), D₄⋊₂S₃⋊₁₆C₂², (C₂×C₁₂).₁₂₂C₂³, Q₈⋊₃S₃⋊₁₉C₂², (C₂×Dic₆)⋊₄₄C₂², (C₂²×C₆).₇₉C₂³, C₂³.₈₁(C₂²×S₃), (C₃×2₊¹⁺⁴)⋊₅C₂, (C₂²×S₃).₁₄₃C₂³, (C₂×Dic₃).₁₆₉C₂³, (C₂²×Dic₃)⋊₃₉C₂², (S₃×C₄○D₄)⋊₈C₂, (S₃×C₂×C₄)⋊₃₇C₂², (C₂×D₄⋊₂S₃)⋊₃₁C₂, (C₃×C₄○D₄)⋊₁₁C₂², (C₂×C₃⋊D₄)⋊₃₃C₂², (C₂×C₄).₁₀₆(C₂²×S₃), SmallGroup(192,1525)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — D₆.C₂⁴

Chief series

C₁ — C₃ — C₆ — D₆ — C₂²×S₃ — S₃×C₂×C₄ — S₃×C₄○D₄ — D₆.C₂⁴

Lower central

C₃ — C₆ — D₆.C₂⁴

Upper central

C₁ — C₂ — 2₊¹⁺⁴

Generators and relations for D₆.C₂⁴
G = < a,b,c,d,e,f | a⁶=b²=c²=e²=f²=1, d²=a³, bab=eae=a^-1, ac=ca, ad=da, af=fa, cbc=fbf=a³b, bd=db, ebe=a⁴b, cd=dc, ce=ec, cf=fc, de=ed, fdf=a³d, fef=a³e >

Subgroups: 1640 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, C₂³, Dic₃, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, Dic₆, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₂²×S₃, C₂²×C₆, C₂×C₄○D₄, 2₊¹⁺⁴, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×Dic₆, S₃×C₂×C₄, C₄○D₁₂, S₃×D₄, D₄⋊₂S₃, S₃×Q₈, Q₈⋊₃S₃, C₂²×Dic₃, C₂×C₃⋊D₄, C₆×D₄, C₃×C₄○D₄, C₂.C₂⁵, C₂×D₄⋊₂S₃, D₄⋊₆D₆, S₃×C₄○D₄, Q₈○D₁₂, C₃×2₊¹⁺⁴, D₆.C₂⁴
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₂⁴, C₂²×S₃, C₂⁵, S₃×C₂³, C₂.C₂⁵, S₃×C₂⁴, D₆.C₂⁴

Smallest permutation representation of D₆.C₂⁴
►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 33)(2 32)(3 31)(4 36)(5 35)(6 34)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 45)(14 44)(15 43)(16 48)(17 47)(18 46)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)

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

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

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

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

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

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

51 conjugacy classes

class	1	2A	2B	···	2J	2K	···	2P	3	4A	···	4F	4G	4H	4I	···	4Q	6A	6B	···	6J	12A	···	12F
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	D₆.C₂⁴
kernel	D₆.C₂⁴	C₂×D₄⋊₂S₃	D₄⋊₆D₆	S₃×C₄○D₄	Q₈○D₁₂	C₃×2₊¹⁺⁴	2₊¹⁺⁴	C₂×D₄	C₄○D₄	C₃	C₁
# reps	1	9	9	6	6	1	1	9	6	2	1

Matrix representation of D₆.C₂⁴ ►in GL₆(𝔽₁₃)

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

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

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	12	0	0	0
0	0	0	12	0	0

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

1	0	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	0	12
0	0	0	0	12	0

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

D₆.C₂⁴ in GAP, Magma, Sage, TeX

D_6.C_2^4

% in TeX

G:=Group("D6.C2^4");

// GroupNames label

G:=SmallGroup(192,1525);

// by ID

G=gap.SmallGroup(192,1525);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e,f|a^6=b^2=c^2=e^2=f^2=1,d^2=a^3,b*a*b=e*a*e=a^-1,a*c=c*a,a*d=d*a,a*f=f*a,c*b*c=f*b*f=a^3*b,b*d=d*b,e*b*e=a^4*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=a^3*d,f*e*f=a^3*e>;

// generators/relations

G = D6.C24 order 192 = 26·3

9th non-split extension by D6 of C24 acting via C24/C23=C2

G = D₆.C₂⁴ order 192 = 2⁶·3

9^th non-split extension by D₆ of C₂⁴ acting via C₂⁴/C₂³=C₂