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G = C32:SD32  order 288 = 25·32

The semidirect product of C32 and SD32 acting via SD32/C4=D4

non-abelian, soluble, monomial

Aliases: C32:SD32, C4.2S3wrC2, (C3xC6).2D8, (C3xC12).6D4, C32:2D8.C2, C2.4(C32:D8), C32:2C16:2C2, C32:2Q16:1C2, C32:4C8.2C22, SmallGroup(288,383)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C32:4C8 — C32:SD32
Chief series
C1 — C32 — C3xC6 — C3xC12 — C32:4C8 — C32:2D8 — C32:SD32
Lower central
C32 — C3xC6 — C3xC12 — C32:4C8 — C32:SD32
Upper central
C1 — C2 — C4

Generators and relations for C32:SD32
 G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=b-1, dad=a-1, cbc-1=a, bd=db, dcd=c7 >

Subgroups: 252 in 46 conjugacy classes, 11 normal (all characteristic)
Quotients: C1, C2, C22, D4, D8, SD32, S3wrC2, C32:D8, C32:SD32
C1
C2
24C2
2C3
2C3
C4
12C22
12C4
2C6
2C6
8S3
24C6
C32
6D4
6Q8
9C8
2C12
2C12
4D6
4Dic3
12C12
12C2xC6
C3xC6
8C3xS3
9Q16
9C16
9D8
2Dic6
2D12
6C3:C8
6C3xQ8
6C3xD4
6C3:C8
C3xC12
4C3xDic3
4S3xC6
9SD32
6C3:Q16
6D4:S3
C32:4C8
2C3xDic6
2C3xD12
C32:2Q16
C32:2D8
C32:2C16
C32:SD32

Character table of C32:SD32

 class 12A2B3A3B4A4B6A6B6C6D8A8B12A12B12C12D16A16B16C16D
 size 112444224442424181888242418181818
ρ1111111111111111111111    trivial
ρ211-1111111-1-1111111-1-1-1-1    linear of order 2
ρ311-1111-111-1-11111-1-11111    linear of order 2
ρ4111111-111111111-1-1-1-1-1-1    linear of order 2
ρ522022202200-2-222000000    orthogonal lifted from D4
ρ622022-20220000-2-200-√2√2-√2√2    orthogonal lifted from D8
ρ722022-20220000-2-200√2-√2√2-√2    orthogonal lifted from D8
ρ82-202200-2-200√2-√20000ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ1615+ζ169    complex lifted from SD32
ρ92-202200-2-200-√2√20000ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ165+ζ163    complex lifted from SD32
ρ102-202200-2-200√2-√20000ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16    complex lifted from SD32
ρ112-202200-2-200-√2√20000ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611    complex lifted from SD32
ρ1244-2-2140-211100-21000000    orthogonal lifted from S3wrC2
ρ134401-2421-200001-2-1-10000    orthogonal lifted from S3wrC2
ρ14442-2140-21-1-100-21000000    orthogonal lifted from S3wrC2
ρ154401-24-21-200001-2110000    orthogonal lifted from S3wrC2
ρ164401-2-401-20000-12√-3-√-30000    complex lifted from C32:D8
ρ174401-2-401-20000-12-√-3√-30000    complex lifted from C32:D8
ρ18440-21-40-21-√-3√-3002-1000000    complex lifted from C32:D8
ρ19440-21-40-21√-3-√-3002-1000000    complex lifted from C32:D8
ρ208-80-42004-2000000000000    orthogonal faithful, Schur index 2
ρ218-802-400-24000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C32:SD32
►On 48 points
Generators in S48
(1 39 32)(3 18 41)(5 43 20)(7 22 45)(9 47 24)(11 26 33)(13 35 28)(15 30 37)

(2 40 17)(4 19 42)(6 44 21)(8 23 46)(10 48 25)(12 27 34)(14 36 29)(16 31 38)

(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)

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

Matrix representation of C32:SD32 ►in GL6(F97)

100000
010000
0096100
0096000
000010
000001
,
100000
010000
001000
000100
0000961
0000960
,
34770000
10540000
000001
000010
00411500
00825600
,
96950000
010000
000100
001000
00004115
00008256

G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0],[34,10,0,0,0,0,77,54,0,0,0,0,0,0,0,0,41,82,0,0,0,0,15,56,0,0,0,1,0,0,0,0,1,0,0,0],[96,0,0,0,0,0,95,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,41,82,0,0,0,0,15,56] >;
 

C32:SD32 in GAP, Magma, Sage, TeX 

C_3^2\rtimes {\rm SD}_{32}
 
% in TeX
 
G:=Group("C3^2:SD32");
 
// GroupNames label
 
G:=SmallGroup(288,383);
 
// by ID
 
G=gap.SmallGroup(288,383);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,120,254,135,142,675,346,80,2693,2028,691,797,2372]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,c*a*c^-1=b^-1,d*a*d=a^-1,c*b*c^-1=a,b*d=d*b,d*c*d=c^7>;
 
// generators/relations
 

Export

Subgroup lattice of C32:SD32 in TeX
Character table of C32:SD32 in TeX

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