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G = C2×C322C8order 144 = 24·32

Direct product of C2 and C322C8

direct product, metabelian, soluble, monomial, A-group

Aliases: C2×C322C8, C62.1C4, (C3×C6)⋊2C8, C325(C2×C8), C3⋊Dic3.5C4, C22.2(C32⋊C4), C3⋊Dic3.9C22, (C3×C6).5(C2×C4), C2.3(C2×C32⋊C4), (C2×C3⋊Dic3).6C2, SmallGroup(144,134)

Series: Derived Chief Lower central Upper central

C1C32 — C2×C322C8
C1C32C3×C6C3⋊Dic3C322C8 — C2×C322C8
C32 — C2×C322C8
C1C22

Generators and relations for C2×C322C8
 G = < a,b,c,d | a2=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

2C3
2C3
9C4
9C4
2C6
2C6
2C6
2C6
2C6
2C6
9C8
9C8
9C2×C4
2C2×C6
2C2×C6
6Dic3
6Dic3
6Dic3
6Dic3
9C2×C8
6C2×Dic3
6C2×Dic3

Character table of C2×C322C8

 class 12A2B2C3A3B4A4B4C4D6A6B6C6D6E6F8A8B8C8D8E8F8G8H
 size 111144999944444499999999
ρ1111111111111111111111111    trivial
ρ21111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31-1-1111-111-1-111-1-1-11111-1-1-1-1    linear of order 2
ρ41-1-1111-111-1-111-1-1-1-1-1-1-11111    linear of order 2
ρ5111111-1-1-1-1111111-i-iii-i-iii    linear of order 4
ρ6111111-1-1-1-1111111ii-i-iii-i-i    linear of order 4
ρ71-1-11111-1-11-111-1-1-1-i-iiiii-i-i    linear of order 4
ρ81-1-11111-1-11-111-1-1-1ii-i-i-i-iii    linear of order 4
ρ91-11-111ii-i-i-1-1-1-111ζ87ζ83ζ85ζ8ζ87ζ83ζ85ζ8    linear of order 8
ρ101-11-111ii-i-i-1-1-1-111ζ83ζ87ζ8ζ85ζ83ζ87ζ8ζ85    linear of order 8
ρ1111-1-111i-ii-i1-1-11-1-1ζ85ζ8ζ87ζ83ζ8ζ85ζ83ζ87    linear of order 8
ρ1211-1-111i-ii-i1-1-11-1-1ζ8ζ85ζ83ζ87ζ85ζ8ζ87ζ83    linear of order 8
ρ131-11-111-i-iii-1-1-1-111ζ85ζ8ζ87ζ83ζ85ζ8ζ87ζ83    linear of order 8
ρ1411-1-111-ii-ii1-1-11-1-1ζ83ζ87ζ8ζ85ζ87ζ83ζ85ζ8    linear of order 8
ρ1511-1-111-ii-ii1-1-11-1-1ζ87ζ83ζ85ζ8ζ83ζ87ζ8ζ85    linear of order 8
ρ161-11-111-i-iii-1-1-1-111ζ8ζ85ζ83ζ87ζ8ζ85ζ83ζ87    linear of order 8
ρ174444-2100001-21-21-200000000    orthogonal lifted from C32⋊C4
ρ184-4-441-2000021-2-12-100000000    orthogonal lifted from C2×C32⋊C4
ρ1944441-20000-21-21-2100000000    orthogonal lifted from C32⋊C4
ρ204-4-44-210000-1-212-1200000000    orthogonal lifted from C2×C32⋊C4
ρ214-44-41-200002-12-1-2100000000    symplectic lifted from C322C8, Schur index 2
ρ2244-4-4-21000012-1-2-1200000000    symplectic lifted from C322C8, Schur index 2
ρ234-44-4-210000-12-121-200000000    symplectic lifted from C322C8, Schur index 2
ρ2444-4-41-20000-2-1212-100000000    symplectic lifted from C322C8, Schur index 2

Smallest permutation representation of C2×C322C8
On 48 points
Generators in S48
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)
(2 32 22)(4 24 26)(6 28 18)(8 20 30)(10 36 42)(12 44 38)(14 40 46)(16 48 34)
(1 31 21)(2 32 22)(3 23 25)(4 24 26)(5 27 17)(6 28 18)(7 19 29)(8 20 30)(9 35 41)(10 36 42)(11 43 37)(12 44 38)(13 39 45)(14 40 46)(15 47 33)(16 48 34)
(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)

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

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

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

C2×C322C8 is a maximal subgroup of
C62.3D4  C62.4D4  C62.6D4  C62.7D4  C62.2Q8  (C3×C12)⋊4C8  C322C8⋊C4  C62.6(C2×C4)  C325(C4⋊C8)  C623C8  C22.F9  C62.13D4  C62.(C2×C4)
C2×C322C8 is a maximal quotient of
C62.4C8  (C3×C12)⋊4C8  C623C8

Matrix representation of C2×C322C8 in GL6(𝔽73)

100000
0720000
001000
000100
000010
000001
,
100000
010000
001000
000100
000001
00007272
,
100000
010000
00727200
001000
000001
00007272
,
5100000
010000
000010
000001
00415100
00103200

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[51,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,41,10,0,0,0,0,51,32,0,0,1,0,0,0,0,0,0,1,0,0] >;

C2×C322C8 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_2C_8
% in TeX

G:=Group("C2xC3^2:2C8");
// GroupNames label

G:=SmallGroup(144,134);
// by ID

G=gap.SmallGroup(144,134);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,50,3364,256,4613,881]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations

Export

Subgroup lattice of C2×C322C8 in TeX
Character table of C2×C322C8 in TeX

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