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G = C2×C32⋊D8  order 288 = 25·32

Direct product of C2 and C32⋊D8

direct product, non-abelian, soluble, monomial

Aliases: C2×C32⋊D8, C62.11D4, (C3×C6)⋊D8, C32⋊2(C2×D8), C22.14S3≀C2, C3⋊Dic3.29D4, D6⋊S3⋊9C22, C32⋊2C8⋊4C22, C3⋊Dic3.7C23, C2.16(C2×S3≀C2), (C3×C6).16(C2×D4), (C2×C32⋊2C8)⋊4C2, (C2×D6⋊S3)⋊11C2, (C2×C3⋊Dic3).94C22, SmallGroup(288,883)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3⋊Dic3 — C2×C32⋊D8
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — D6⋊S3 — C32⋊D8 — C2×C32⋊D8
Lower central
C32 — C3×C6 — C3⋊Dic3 — C2×C32⋊D8
Upper central
C1 — C22

Generators and relations for C2×C32⋊D8
 G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

Subgroups: 688 in 130 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, D4, C23, C32, Dic3, D6, C2×C6, C2×C8, D8, C2×D4, C3×S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C2×D8, C3⋊Dic3, S3×C6, C62, C2×C3⋊D4, C32⋊2C8, D6⋊S3, D6⋊S3, C2×C3⋊Dic3, S3×C2×C6, C32⋊D8, C2×C32⋊2C8, C2×D6⋊S3, C2×C32⋊D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C2×D8, S3≀C2, C32⋊D8, C2×S3≀C2, C2×C32⋊D8

Character table of C2×C32⋊D8

 class 12A2B2C2D2E2F2G3A3B4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M6N8A8B8C8D
 size 111112121212441818444444121212121212121218181818
ρ1111111111111111111111111111111    trivial
ρ211111-1-111111111111-1-1-111-111-1-1-1-1    linear of order 2
ρ311-1-1-1-11111-11-1-111-1-1-11-1111-1-1-1-111    linear of order 2
ρ411-1-1-11-1111-11-1-111-1-11-1111-1-1-111-1-1    linear of order 2
ρ511-1-111-1-111-11-1-111-1-11-11-1-1-111-1-111    linear of order 2
ρ611-1-11-11-111-11-1-111-1-1-11-1-1-111111-1-1    linear of order 2
ρ71111-1-1-1-11111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ81111-111-11111111111111-1-11-1-1-1-1-1-1    linear of order 2
ρ922-2-20000222-2-2-222-2-2000000000000    orthogonal lifted from D4
ρ102222000022-2-2222222000000000000    orthogonal lifted from D4
ρ112-22-200002200-22-2-22-200000000√2-√2√2-√2    orthogonal lifted from D8
ρ122-2-22000022002-2-2-2-2200000000√2-√2-√2√2    orthogonal lifted from D8
ρ132-2-22000022002-2-2-2-2200000000-√2√2√2-√2    orthogonal lifted from D8
ρ142-22-200002200-22-2-22-200000000-√2√2-√2√2    orthogonal lifted from D8
ρ154444-200-2-2100-211-2-21000110110000    orthogonal lifted from S3≀C2
ρ1644442002-2100-211-2-21000-1-10-1-10000    orthogonal lifted from S3≀C2
ρ1744440-2-201-2001-2-211-2111001000000    orthogonal lifted from S3≀C2
ρ1844-4-402-201-200-12-21-12-11-1001000000    orthogonal lifted from C2×S3≀C2
ρ1944-4-4200-2-21002-11-22-1000110-1-10000    orthogonal lifted from C2×S3≀C2
ρ2044-4-4-2002-21002-11-22-1000-1-10110000    orthogonal lifted from C2×S3≀C2
ρ2144-4-40-2201-200-12-21-121-1100-1000000    orthogonal lifted from C2×S3≀C2
ρ22444402201-2001-2-211-2-1-1-100-1000000    orthogonal lifted from S3≀C2
ρ234-4-440000-2100-2-1-1221000√-3-√-30-√-3√-30000    complex lifted from C32⋊D8
ρ244-44-40000-210021-12-2-1000√-3-√-30√-3-√-30000    complex lifted from C32⋊D8
ρ254-4-4400001-200122-1-1-2√-3√-3-√-300-√-3000000    complex lifted from C32⋊D8
ρ264-44-400001-200-1-22-112√-3-√-3-√-300√-3000000    complex lifted from C32⋊D8
ρ274-44-400001-200-1-22-112-√-3√-3√-300-√-3000000    complex lifted from C32⋊D8
ρ284-4-4400001-200122-1-1-2-√-3-√-3√-300√-3000000    complex lifted from C32⋊D8
ρ294-4-440000-2100-2-1-1221000-√-3√-30√-3-√-30000    complex lifted from C32⋊D8
ρ304-44-40000-210021-12-2-1000-√-3√-30-√-3√-30000    complex lifted from C32⋊D8

Smallest permutation representation of C2×C32⋊D8
►On 48 points
Generators in S48
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 43)

(1 35 22)(3 24 37)(5 39 18)(7 20 33)(9 46 26)(11 28 48)(13 42 30)(15 32 44)

(2 23 36)(4 38 17)(6 19 40)(8 34 21)(10 27 47)(12 41 29)(14 31 43)(16 45 25)

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

Matrix representation of C2×C32⋊D8 ►in GL6(𝔽73)

7200000
0720000
0072000
0007200
0000720
0000072
,
100000
010000
0072100
0072000
000010
000001
,
100000
010000
001000
000100
0000721
0000720
,
33500000
22400000
00003013
00006043
000100
001000
,
100000
60720000
000100
001000
00003013
00006043

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[33,22,0,0,0,0,50,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,30,60,0,0,0,0,13,43,0,0],[1,60,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,30,60,0,0,0,0,13,43] >;
 

C2×C32⋊D8 in GAP, Magma, Sage, TeX 

C_2\times C_3^2\rtimes D_8
 
% in TeX
 
G:=Group("C2xC3^2:D8");
 
// GroupNames label
 
G:=SmallGroup(288,883);
 
// by ID
 
G=gap.SmallGroup(288,883);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,675,346,80,2693,2028,362,797,1203]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

Export

Character table of C2×C32⋊D8 in TeX

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