C6^2.A4 - GroupNames

G = C₆².A₄ order 432 = 2⁴·3³

9^th non-split extension by C₆² of A₄ acting via A₄/C₂²=C₃

Aliases: C₆².₉A₄, C₂⁴⋊₆3_-¹⁺², C₂⁴⋊C₉⋊₆C₃, C₃².(C₂²⋊A₄), (C₂²×C₆²).₄C₃, C₂²⋊₂(C₃².A₄), (C₂³×C₆).₁₄C₃², (C₂×C₆).₁₅(C₃×A₄), C₃.₄(C₃×C₂²⋊A₄), SmallGroup(432,554)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³×C₆ — C₆².A₄

Chief series

C₁ — C₂² — C₂⁴ — C₂³×C₆ — C₂⁴⋊C₉ — C₆².A₄

Lower central

C₂⁴ — C₂³×C₆ — C₆².A₄

Upper central

C₁ — C₃ — C₃²

Generators and relations for C₆².A₄
G = < a,b,c,d,e | a⁶=b⁶=c²=d²=1, e³=b², ab=ba, ac=ca, ad=da, eae^-1=ab^-1, bc=cb, bd=db, ebe^-1=a³b⁴, ece^-1=cd=dc, ede^-1=c >

Subgroups: 550 in 176 conjugacy classes, 25 normal (7 characteristic)
C₁, C₂, C₃, C₃, C₂², C₂², C₆, C₂³, C₉, C₃², C₂×C₆, C₂×C₆, C₂⁴, C₃×C₆, C₂²×C₆, 3_-¹⁺², C₃.A₄, C₆², C₆², C₂³×C₆, C₂³×C₆, C₂×C₆², C₃².A₄, C₂⁴⋊C₉, C₂²×C₆², C₆².A₄
Quotients: C₁, C₃, C₃², A₄, 3_-¹⁺², C₃×A₄, C₂²⋊A₄, C₃².A₄, C₃×C₂²⋊A₄, C₆².A₄

Smallest permutation representation of C₆².A₄
►On 36 points

Generators in S₃₆

(1 20)(2 5 8)(3 19 6 22 9 25)(4 23)(7 26)(10 32)(11 14 17)(12 31 15 34 18 28)(13 35)(16 29)(21 24 27)(30 33 36)

(1 26 4 20 7 23)(2 27 5 21 8 24)(3 9 6)(10 29 13 32 16 35)(11 30 14 33 17 36)(12 18 15)(19 25 22)(28 34 31)

(1 10)(2 11)(4 13)(5 14)(7 16)(8 17)(20 32)(21 33)(23 35)(24 36)(26 29)(27 30)

(2 11)(3 12)(5 14)(6 15)(8 17)(9 18)(19 31)(21 33)(22 34)(24 36)(25 28)(27 30)

(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)

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

G:=Group( (1,20)(2,5,8)(3,19,6,22,9,25)(4,23)(7,26)(10,32)(11,14,17)(12,31,15,34,18,28)(13,35)(16,29)(21,24,27)(30,33,36), (1,26,4,20,7,23)(2,27,5,21,8,24)(3,9,6)(10,29,13,32,16,35)(11,30,14,33,17,36)(12,18,15)(19,25,22)(28,34,31), (1,10)(2,11)(4,13)(5,14)(7,16)(8,17)(20,32)(21,33)(23,35)(24,36)(26,29)(27,30), (2,11)(3,12)(5,14)(6,15)(8,17)(9,18)(19,31)(21,33)(22,34)(24,36)(25,28)(27,30), (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) );

G=PermutationGroup([[(1,20),(2,5,8),(3,19,6,22,9,25),(4,23),(7,26),(10,32),(11,14,17),(12,31,15,34,18,28),(13,35),(16,29),(21,24,27),(30,33,36)], [(1,26,4,20,7,23),(2,27,5,21,8,24),(3,9,6),(10,29,13,32,16,35),(11,30,14,33,17,36),(12,18,15),(19,25,22),(28,34,31)], [(1,10),(2,11),(4,13),(5,14),(7,16),(8,17),(20,32),(21,33),(23,35),(24,36),(26,29),(27,30)], [(2,11),(3,12),(5,14),(6,15),(8,17),(9,18),(19,31),(21,33),(22,34),(24,36),(25,28),(27,30)], [(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)]])

56 conjugacy classes

class	1	2A	···	2E	3A	3B	3C	3D	6A	···	6AN	9A	···	9F
order	1	2	···	2	3	3	3	3	6	···	6	9	···	9
size	1	3	···	3	1	1	3	3	3	···	3	48	···	48

56 irreducible representations

dim	1	1	1	3	3	3	3
type	+			+
image	C₁	C₃	C₃	A₄	3_-¹⁺²	C₃×A₄	C₃².A₄
kernel	C₆².A₄	C₂⁴⋊C₉	C₂²×C₆²	C₆²	C₂⁴	C₂×C₆	C₂²
# reps	1	6	2	5	2	10	30

Matrix representation of C₆².A₄ ►in GL₆(𝔽₁₉)

18	0	0	0	0	0
0	8	0	0	0	0
0	0	7	0	0	0
0	0	0	18	0	0
0	0	0	0	12	0
0	0	0	0	0	11

8	0	0	0	0	0
0	11	0	0	0	0
0	0	8	0	0	0
0	0	0	12	0	0
0	0	0	0	7	0
0	0	0	0	0	12

18	0	0	0	0	0
0	1	0	0	0	0
0	0	18	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	18	0	0	0	0
0	0	18	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
0	0	1	0	0	0
7	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	11	0	0

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,8,0,0,0,0,0,0,7,0,0,0,0,0,0,18,0,0,0,0,0,0,12,0,0,0,0,0,0,11],[8,0,0,0,0,0,0,11,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,7,0,0,0,0,0,0,12],[18,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,7,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,1,0,0,0,0,0,0,1,0] >;

C₆².A₄ in GAP, Magma, Sage, TeX

C_6^2.A_4

% in TeX

G:=Group("C6^2.A4");

// GroupNames label

G:=SmallGroup(432,554);

// by ID

G=gap.SmallGroup(432,554);

# by ID

G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,63,169,1515,2839,9077,15882]);

// Polycyclic

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

// generators/relations

G = C62.A4 order 432 = 24·33

9th non-split extension by C62 of A4 acting via A4/C22=C3

G = C₆².A₄ order 432 = 2⁴·3³

9^th non-split extension by C₆² of A₄ acting via A₄/C₂²=C₃