C2^2xC3:D4 - GroupNames

G = C₂²×C₃⋊D₄ order 96 = 2⁵·3

Direct product of C₂² and C₃⋊D₄

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂²×C₃⋊D₄, C₂⁴⋊₅S₃, C₂³⋊₅D₆, D₆⋊₃C₂³, C₆.₁₅C₂⁴, Dic₃⋊₂C₂³, (C₂×C₆)⋊₉D₄, C₆⋊₃(C₂×D₄), C₃⋊₃(C₂²×D₄), (C₂×C₆)⋊₃C₂³, (C₂³×C₆)⋊₄C₂, (S₃×C₂³)⋊₅C₂, C₂.₁₅(S₃×C₂³), (C₂²×C₆)⋊₇C₂², C₂²⋊₃(C₂²×S₃), (C₂²×S₃)⋊₇C₂², (C₂²×Dic₃)⋊₉C₂, (C₂×Dic₃)⋊₁₁C₂², SmallGroup(96,219)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₂²×C₃⋊D₄

Chief series

C₁ — C₃ — C₆ — D₆ — C₂²×S₃ — S₃×C₂³ — C₂²×C₃⋊D₄

Lower central

C₃ — C₆ — C₂²×C₃⋊D₄

Upper central

C₁ — C₂³ — C₂⁴

Generators and relations for C₂²×C₃⋊D₄
G = < a,b,c,d,e | a²=b²=c³=d⁴=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd^-1=ece=c^-1, ede=d^-1 >

Subgroups: 498 in 236 conjugacy classes, 105 normal (11 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₆, C₂×C₄, D₄, C₂³, C₂³, C₂³, Dic₃, D₆, D₆, C₂×C₆, C₂×C₆, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂×Dic₃, C₃⋊D₄, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂²×D₄, C₂²×Dic₃, C₂×C₃⋊D₄, S₃×C₂³, C₂³×C₆, C₂²×C₃⋊D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₂⁴, C₃⋊D₄, C₂²×S₃, C₂²×D₄, C₂×C₃⋊D₄, S₃×C₂³, C₂²×C₃⋊D₄

Smallest permutation representation of C₂²×C₃⋊D₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

C₂²×C₃⋊D₄ is a maximal subgroup of
C₂⁴.₂₃D₆ C₂⁴.₂₄D₆ C₂⁴.₆₀D₆ C₂⁴.₂₅D₆ C₂³⋊₃D₁₂ C₂⁴.₂₇D₆ C₂⁴.₇₆D₆ C₂⁴.₃₂D₆ C₂⁴.₃₅D₆ C₂⁴.₃₈D₆ C₂³⋊₄D₁₂ C₂⁴⋊₇D₆ C₂⁴⋊₈D₆ C₂⁴.₄₄D₆ C₂⁴.₄₅D₆ C₂⁴⋊₁₂D₆ C₂²×S₃×D₄
C₂²×C₃⋊D₄ is a maximal quotient of
C₂⁴.₈₃D₆ C₂⁴⋊₁₂D₆ C₂⁴.₅₂D₆ C₂⁴.₅₃D₆ C₆.₄₄2_-¹⁺⁴ C₆.₄₅2_-¹⁺⁴ C₁₂.C₂⁴ C₆.₁₀₄2_-¹⁺⁴ C₆.₁₀₅2_-¹⁺⁴ (C₂×D₄)⋊₄₃D₆ C₆.₁₄₅2₊¹⁺⁴ C₆.₁₄₆2₊¹⁺⁴ C₆.₁₀₇2_-¹⁺⁴ (C₂×C₁₂)⋊₁₇D₄ C₆.₁₀₈2_-¹⁺⁴ C₆.₁₄₈2₊¹⁺⁴ D₁₂.₃₂C₂³ D₁₂.₃₃C₂³ D₁₂.₃₄C₂³ D₁₂.₃₅C₂³

36 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	2N	2O	3	4A	4B	4C	4D	6A	···	6O
order	1	2	···	2	2	2	2	2	2	2	2	2	3	4	4	4	4	6	···	6
size	1	1	···	1	2	2	2	2	6	6	6	6	2	6	6	6	6	2	···	2

36 irreducible representations

dim	1	1	1	1	1	2	2	2	2
type	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	S₃	D₄	D₆	C₃⋊D₄
kernel	C₂²×C₃⋊D₄	C₂²×Dic₃	C₂×C₃⋊D₄	S₃×C₂³	C₂³×C₆	C₂⁴	C₂×C₆	C₂³	C₂²
# reps	1	1	12	1	1	1	4	7	8

Matrix representation of C₂²×C₃⋊D₄ ►in GL₆(𝔽₁₃)

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

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	1	0
0	0	0	0	0	1

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

0	1	0	0	0	0
12	0	0	0	0	0
0	0	0	12	0	0
0	0	1	0	0	0
0	0	0	0	4	11
0	0	0	0	2	9

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

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,4,2,0,0,0,0,11,9],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₂²×C₃⋊D₄ in GAP, Magma, Sage, TeX

C_2^2\times C_3\rtimes D_4

% in TeX

G:=Group("C2^2xC3:D4");

// GroupNames label

G:=SmallGroup(96,219);

// by ID

G=gap.SmallGroup(96,219);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,579,2309]);

// Polycyclic

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

// generators/relations

G = C22×C3⋊D4 order 96 = 25·3

Direct product of C22 and C3⋊D4

G = C₂²×C₃⋊D₄ order 96 = 2⁵·3

Direct product of C₂² and C₃⋊D₄