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G = C24⋊23- 1+2  order 432 = 24·33

2nd semidirect product of C24 and 3- 1+2 acting via 3- 1+2/C3=C32

metabelian, soluble, monomial

Aliases: C24⋊23- 1+2, C3.5A42, C3.A4⋊2A4, C22⋊1(C9⋊A4), C24⋊C9⋊4C3, (C23×C6).5C32, (C2×C6).5(C3×A4), (C22×C3.A4)⋊4C3, (C3×C22⋊A4).2C3, SmallGroup(432,528)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23×C6 — C24⋊23- 1+2
Chief series
C1 — C22 — C24 — C23×C6 — C22×C3.A4 — C24⋊23- 1+2
Lower central
C24 — C23×C6 — C24⋊23- 1+2
Upper central
C1 — C3

Generators and relations for C24⋊23- 1+2
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e9=f3=1, eae-1=fbf-1=ab=ba, ac=ca, ad=da, faf-1=b, bc=cb, bd=db, ebe-1=a, fcf-1=cd=dc, ce=ec, de=ed, fdf-1=c, fef-1=e4 >

Subgroups: 394 in 63 conjugacy classes, 15 normal (7 characteristic)
C1, C2, C3, C3, C22, C22, C6, C23, C9, C32, A4, C2×C6, C2×C6, C24, C18, C22×C6, 3- 1+2, C3.A4, C3.A4, C2×C18, C3×A4, C22⋊A4, C23×C6, C2×C3.A4, C9⋊A4, C22×C3.A4, C24⋊C9, C3×C22⋊A4, C24⋊23- 1+2
Quotients: C1, C3, C32, A4, 3- 1+2, C3×A4, C9⋊A4, A42, C24⋊23- 1+2

Smallest permutation representation of C24⋊23- 1+2
►On 36 points
Generators in S36
(1 34)(2 11)(3 27)(4 28)(5 14)(6 21)(7 31)(8 17)(9 24)(10 25)(12 36)(13 19)(15 30)(16 22)(18 33)(20 29)(23 32)(26 35)

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

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

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

(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)

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

32 conjugacy classes

class 1 2A2B2C3A3B3C3D6A6B6C6D6E6F9A9B9C9D9E9F18A···18L
order1222333366666699999918···18
size133911484833339912121212484812···12

32 irreducible representations

dim1111333399
type+++
imageC1C3C3C3A43- 1+2C3×A4C9⋊A4A42C24⋊23- 1+2
kernelC24⋊23- 1+2C22×C3.A4C24⋊C9C3×C22⋊A4C3.A4C24C2×C6C22C3C1
# reps14222241212

Matrix representation of C24⋊23- 1+2 ►in GL9(𝔽19)

100000000
010000000
001000000
000010000
000100000
000181818000
000000100
000000010
000000001
,
100000000
010000000
001000000
000001000
000181818000
000100000
000000100
000000010
000000001
,
100000000
010000000
001000000
000100000
000010000
000001000
0000001800
0000001801
0000001810
,
100000000
010000000
001000000
000100000
000010000
000001000
0000000181
0000000180
0000001180
,
001000000
1100000000
0110000000
0001100000
000888000
0000110000
000000100
000000010
000000001
,
100000000
0110000000
007000000
000100000
000001000
000181818000
000000001
000000100
000000010

G:=sub<GL(9,GF(19))| [1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,1,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,18,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,18,0,0,0,0,0,0,1,0,0],[0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,11,8,0,0,0,0,0,0,0,0,8,11,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C24⋊23- 1+2 in GAP, Magma, Sage, TeX 

C_2^4\rtimes_23_-^{1+2}
 
% in TeX
 
G:=Group("C2^4:2ES-(3,1)");
 
// GroupNames label
 
G:=SmallGroup(432,528);
 
// by ID
 
G=gap.SmallGroup(432,528);
 
# by ID
 
G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,63,169,50,766,326,13613,5298]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^9=f^3=1,e*a*e^-1=f*b*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f^-1=b,b*c=c*b,b*d=d*b,e*b*e^-1=a,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,f*d*f^-1=c,f*e*f^-1=e^4>;
 
// generators/relations
 

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