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G = C24⋊3- 1+2  order 432 = 24·33

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

metabelian, soluble, monomial

Aliases: C24⋊13- 1+2, C3.4A42, (C3×A4).A4, C3.A4⋊1A4, C22⋊3(C9⋊A4), C24⋊C9⋊3C3, (C23×C6).4C32, C22⋊1(C32.A4), (A4×C2×C6).2C3, (C2×C6).4(C3×A4), (C22×C3.A4)⋊3C3, SmallGroup(432,527)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23×C6 — C24⋊3- 1+2
Chief series
C1 — C22 — C24 — C23×C6 — A4×C2×C6 — C24⋊3- 1+2
Lower central
C24 — C23×C6 — C24⋊3- 1+2
Upper central
C1 — C3

Generators and relations for C24⋊3- 1+2
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e9=f3=1, ebe-1=ab=ba, ac=ca, ad=da, eae-1=b, af=fa, bc=cb, bd=db, bf=fb, fcf-1=ede-1=cd=dc, ece-1=d, fdf-1=c, fef-1=e4 >

Subgroups: 358 in 66 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C3, C3, C22, C22, C6, C23, C9, C32, A4, C2×C6, C2×C6, C24, C18, C3×C6, C2×A4, C22×C6, 3- 1+2, C3.A4, C3.A4, C2×C18, C3×A4, C62, C22×A4, C23×C6, C2×C3.A4, C6×A4, C9⋊A4, C32.A4, C22×C3.A4, C24⋊C9, A4×C2×C6, C24⋊3- 1+2
Quotients: C1, C3, C32, A4, 3- 1+2, C3×A4, C9⋊A4, C32.A4, A42, C24⋊3- 1+2

Smallest permutation representation of C24⋊3- 1+2
►On 54 points
Generators in S54
(1 53)(3 46)(4 47)(6 49)(7 50)(9 52)(10 21)(11 22)(13 24)(14 25)(16 27)(17 19)(28 38)(30 40)(31 41)(33 43)(34 44)(36 37)

(2 54)(3 46)(5 48)(6 49)(8 51)(9 52)(10 21)(12 23)(13 24)(15 26)(16 27)(18 20)(29 39)(30 40)(32 42)(33 43)(35 45)(36 37)

(2 54)(3 46)(5 48)(6 49)(8 51)(9 52)(10 21)(11 22)(13 24)(14 25)(16 27)(17 19)(28 38)(29 39)(31 41)(32 42)(34 44)(35 45)

(1 53)(2 54)(4 47)(5 48)(7 50)(8 51)(10 21)(12 23)(13 24)(15 26)(16 27)(18 20)(28 38)(30 40)(31 41)(33 43)(34 44)(36 37)

(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 49 50 51 52 53 54)

(1 22 38)(2 20 42)(3 27 37)(4 25 41)(5 23 45)(6 21 40)(7 19 44)(8 26 39)(9 24 43)(10 30 49)(11 28 53)(12 35 48)(13 33 52)(14 31 47)(15 29 51)(16 36 46)(17 34 50)(18 32 54)
 
G:=sub<Sym(54)| (1,53)(3,46)(4,47)(6,49)(7,50)(9,52)(10,21)(11,22)(13,24)(14,25)(16,27)(17,19)(28,38)(30,40)(31,41)(33,43)(34,44)(36,37), (2,54)(3,46)(5,48)(6,49)(8,51)(9,52)(10,21)(12,23)(13,24)(15,26)(16,27)(18,20)(29,39)(30,40)(32,42)(33,43)(35,45)(36,37), (2,54)(3,46)(5,48)(6,49)(8,51)(9,52)(10,21)(11,22)(13,24)(14,25)(16,27)(17,19)(28,38)(29,39)(31,41)(32,42)(34,44)(35,45), (1,53)(2,54)(4,47)(5,48)(7,50)(8,51)(10,21)(12,23)(13,24)(15,26)(16,27)(18,20)(28,38)(30,40)(31,41)(33,43)(34,44)(36,37), (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)(46,47,48,49,50,51,52,53,54), (1,22,38)(2,20,42)(3,27,37)(4,25,41)(5,23,45)(6,21,40)(7,19,44)(8,26,39)(9,24,43)(10,30,49)(11,28,53)(12,35,48)(13,33,52)(14,31,47)(15,29,51)(16,36,46)(17,34,50)(18,32,54)>;
 
G:=Group( (1,53)(3,46)(4,47)(6,49)(7,50)(9,52)(10,21)(11,22)(13,24)(14,25)(16,27)(17,19)(28,38)(30,40)(31,41)(33,43)(34,44)(36,37), (2,54)(3,46)(5,48)(6,49)(8,51)(9,52)(10,21)(12,23)(13,24)(15,26)(16,27)(18,20)(29,39)(30,40)(32,42)(33,43)(35,45)(36,37), (2,54)(3,46)(5,48)(6,49)(8,51)(9,52)(10,21)(11,22)(13,24)(14,25)(16,27)(17,19)(28,38)(29,39)(31,41)(32,42)(34,44)(35,45), (1,53)(2,54)(4,47)(5,48)(7,50)(8,51)(10,21)(12,23)(13,24)(15,26)(16,27)(18,20)(28,38)(30,40)(31,41)(33,43)(34,44)(36,37), (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)(46,47,48,49,50,51,52,53,54), (1,22,38)(2,20,42)(3,27,37)(4,25,41)(5,23,45)(6,21,40)(7,19,44)(8,26,39)(9,24,43)(10,30,49)(11,28,53)(12,35,48)(13,33,52)(14,31,47)(15,29,51)(16,36,46)(17,34,50)(18,32,54) );
 
G=PermutationGroup([[(1,53),(3,46),(4,47),(6,49),(7,50),(9,52),(10,21),(11,22),(13,24),(14,25),(16,27),(17,19),(28,38),(30,40),(31,41),(33,43),(34,44),(36,37)], [(2,54),(3,46),(5,48),(6,49),(8,51),(9,52),(10,21),(12,23),(13,24),(15,26),(16,27),(18,20),(29,39),(30,40),(32,42),(33,43),(35,45),(36,37)], [(2,54),(3,46),(5,48),(6,49),(8,51),(9,52),(10,21),(11,22),(13,24),(14,25),(16,27),(17,19),(28,38),(29,39),(31,41),(32,42),(34,44),(35,45)], [(1,53),(2,54),(4,47),(5,48),(7,50),(8,51),(10,21),(12,23),(13,24),(15,26),(16,27),(18,20),(28,38),(30,40),(31,41),(33,43),(34,44),(36,37)], [(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),(46,47,48,49,50,51,52,53,54)], [(1,22,38),(2,20,42),(3,27,37),(4,25,41),(5,23,45),(6,21,40),(7,19,44),(8,26,39),(9,24,43),(10,30,49),(11,28,53),(12,35,48),(13,33,52),(14,31,47),(15,29,51),(16,36,46),(17,34,50),(18,32,54)]])
 

32 conjugacy classes

class 1 2A2B2C3A3B3C3D6A6B6C6D6E6F6G···6L9A9B9C9D9E9F18A···18F
order122233336666666···699999918···18
size133911121233339912···1212124848484812···12

32 irreducible representations

dim111133333399
type++++
imageC1C3C3C3A4A43- 1+2C3×A4C9⋊A4C32.A4A42C24⋊3- 1+2
kernelC24⋊3- 1+2C22×C3.A4C24⋊C9A4×C2×C6C3.A4C3×A4C24C2×C6C22C22C3C1
# reps124211246612

Matrix representation of C24⋊3- 1+2 ►in GL9(𝔽19)

100000000
010000000
001000000
0001800000
0001801000
0001810000
000000100
000000010
000000001
,
100000000
010000000
001000000
0000181000
0000180000
0001180000
000000100
000000010
000000001
,
100000000
010000000
001000000
000100000
000010000
000001000
0000001800
0000001801
0000001810
,
100000000
010000000
001000000
000100000
000010000
000001000
0000000181
0000000180
0000001180
,
91117000000
12120000000
14717000000
000010000
000001000
000100000
000000010
000000001
000000100
,
51014000000
1513000000
10713000000
0001100000
0000110000
0000011000
000000001
000000100
000000010

G:=sub<GL(9,GF(19))| [1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,18,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[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],[9,12,14,0,0,0,0,0,0,11,12,7,0,0,0,0,0,0,17,0,17,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[5,15,10,0,0,0,0,0,0,10,1,7,0,0,0,0,0,0,14,3,13,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C24⋊3- 1+2 in GAP, Magma, Sage, TeX 

C_2^4\rtimes 3_-^{1+2}
 
% in TeX
 
G:=Group("C2^4:ES-(3,1)");
 
// GroupNames label
 
G:=SmallGroup(432,527);
 
// by ID
 
G=gap.SmallGroup(432,527);
 
# by ID
 
G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,169,50,766,326,13613,5298]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^9=f^3=1,e*b*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=b,a*f=f*a,b*c=c*b,b*d=d*b,b*f=f*b,f*c*f^-1=e*d*e^-1=c*d=d*c,e*c*e^-1=d,f*d*f^-1=c,f*e*f^-1=e^4>;
 
// generators/relations
 

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