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## G = C2×C32⋊2Q8order 144 = 24·32

### Direct product of C2 and C32⋊2Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C32⋊2Q8
 Chief series C1 — C3 — C32 — C3×C6 — C3×Dic3 — C32⋊2Q8 — C2×C32⋊2Q8
 Lower central C32 — C3×C6 — C2×C32⋊2Q8
 Upper central C1 — C22

Generators and relations for C2×C322Q8
G = < a,b,c,d,e | a2=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 224 in 84 conjugacy classes, 40 normal (8 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×6], C22, C6 [×6], C6 [×3], C2×C4 [×3], Q8 [×4], C32, Dic3 [×4], Dic3 [×6], C12 [×4], C2×C6 [×2], C2×C6, C2×Q8, C3×C6, C3×C6 [×2], Dic6 [×8], C2×Dic3 [×2], C2×Dic3 [×3], C2×C12 [×2], C3×Dic3 [×4], C3⋊Dic3 [×2], C62, C2×Dic6 [×2], C322Q8 [×4], C6×Dic3 [×2], C2×C3⋊Dic3, C2×C322Q8
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], Q8 [×2], C23, D6 [×6], C2×Q8, Dic6 [×4], C22×S3 [×2], S32, C2×Dic6 [×2], C322Q8 [×2], C2×S32, C2×C322Q8

Character table of C2×C322Q8

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 1 1 2 2 4 6 6 6 6 18 18 2 2 2 2 2 2 4 4 4 6 6 6 6 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ3 1 -1 1 -1 1 1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 -1 1 1 -1 1 1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 -1 1 -1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 linear of order 2 ρ6 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ8 1 -1 1 -1 1 1 1 -1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 -1 -1 1 1 linear of order 2 ρ9 2 -2 2 -2 -1 2 -1 0 2 0 -2 0 0 1 -2 -1 1 -2 2 1 -1 1 1 0 0 1 0 0 -1 -1 orthogonal lifted from D6 ρ10 2 2 2 2 -1 2 -1 0 2 0 2 0 0 -1 2 -1 -1 2 2 -1 -1 -1 -1 0 0 -1 0 0 -1 -1 orthogonal lifted from S3 ρ11 2 -2 2 -2 2 -1 -1 -2 0 2 0 0 0 -2 1 2 -2 1 -1 1 -1 1 0 -1 -1 0 1 1 0 0 orthogonal lifted from D6 ρ12 2 -2 2 -2 2 -1 -1 2 0 -2 0 0 0 -2 1 2 -2 1 -1 1 -1 1 0 1 1 0 -1 -1 0 0 orthogonal lifted from D6 ρ13 2 2 2 2 2 -1 -1 -2 0 -2 0 0 0 2 -1 2 2 -1 -1 -1 -1 -1 0 1 1 0 1 1 0 0 orthogonal lifted from D6 ρ14 2 2 2 2 2 -1 -1 2 0 2 0 0 0 2 -1 2 2 -1 -1 -1 -1 -1 0 -1 -1 0 -1 -1 0 0 orthogonal lifted from S3 ρ15 2 -2 2 -2 -1 2 -1 0 -2 0 2 0 0 1 -2 -1 1 -2 2 1 -1 1 -1 0 0 -1 0 0 1 1 orthogonal lifted from D6 ρ16 2 2 2 2 -1 2 -1 0 -2 0 -2 0 0 -1 2 -1 -1 2 2 -1 -1 -1 1 0 0 1 0 0 1 1 orthogonal lifted from D6 ρ17 2 -2 -2 2 2 2 2 0 0 0 0 0 0 2 2 -2 -2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ18 2 2 -2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ19 2 2 -2 -2 2 -1 -1 0 0 0 0 0 0 -2 1 -2 2 -1 1 1 1 -1 0 √3 -√3 0 √3 -√3 0 0 symplectic lifted from Dic6, Schur index 2 ρ20 2 -2 -2 2 2 -1 -1 0 0 0 0 0 0 2 -1 -2 -2 1 1 -1 1 1 0 -√3 √3 0 √3 -√3 0 0 symplectic lifted from Dic6, Schur index 2 ρ21 2 -2 -2 2 -1 2 -1 0 0 0 0 0 0 -1 2 1 1 -2 -2 -1 1 1 -√3 0 0 √3 0 0 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ22 2 2 -2 -2 -1 2 -1 0 0 0 0 0 0 1 -2 1 -1 2 -2 1 1 -1 √3 0 0 -√3 0 0 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ23 2 -2 -2 2 2 -1 -1 0 0 0 0 0 0 2 -1 -2 -2 1 1 -1 1 1 0 √3 -√3 0 -√3 √3 0 0 symplectic lifted from Dic6, Schur index 2 ρ24 2 2 -2 -2 -1 2 -1 0 0 0 0 0 0 1 -2 1 -1 2 -2 1 1 -1 -√3 0 0 √3 0 0 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ25 2 -2 -2 2 -1 2 -1 0 0 0 0 0 0 -1 2 1 1 -2 -2 -1 1 1 √3 0 0 -√3 0 0 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ26 2 2 -2 -2 2 -1 -1 0 0 0 0 0 0 -2 1 -2 2 -1 1 1 1 -1 0 -√3 √3 0 -√3 √3 0 0 symplectic lifted from Dic6, Schur index 2 ρ27 4 4 4 4 -2 -2 1 0 0 0 0 0 0 -2 -2 -2 -2 -2 -2 1 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ28 4 -4 4 -4 -2 -2 1 0 0 0 0 0 0 2 2 -2 2 2 -2 -1 1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×S32 ρ29 4 4 -4 -4 -2 -2 1 0 0 0 0 0 0 2 2 2 -2 -2 2 -1 -1 1 0 0 0 0 0 0 0 0 symplectic lifted from C32⋊2Q8, Schur index 2 ρ30 4 -4 -4 4 -2 -2 1 0 0 0 0 0 0 -2 -2 2 2 2 2 1 -1 -1 0 0 0 0 0 0 0 0 symplectic lifted from C32⋊2Q8, Schur index 2

Smallest permutation representation of C2×C322Q8
On 48 points
Generators in S48
(1 5)(2 6)(3 7)(4 8)(9 34)(10 35)(11 36)(12 33)(13 38)(14 39)(15 40)(16 37)(17 29)(18 30)(19 31)(20 32)(21 27)(22 28)(23 25)(24 26)(41 48)(42 45)(43 46)(44 47)
(1 39 35)(2 36 40)(3 37 33)(4 34 38)(5 14 10)(6 11 15)(7 16 12)(8 9 13)(17 22 41)(18 42 23)(19 24 43)(20 44 21)(25 30 45)(26 46 31)(27 32 47)(28 48 29)
(1 35 39)(2 36 40)(3 33 37)(4 34 38)(5 10 14)(6 11 15)(7 12 16)(8 9 13)(17 41 22)(18 42 23)(19 43 24)(20 44 21)(25 30 45)(26 31 46)(27 32 47)(28 29 48)
(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)
(1 47 3 45)(2 46 4 48)(5 44 7 42)(6 43 8 41)(9 17 11 19)(10 20 12 18)(13 22 15 24)(14 21 16 23)(25 39 27 37)(26 38 28 40)(29 36 31 34)(30 35 32 33)

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

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

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

C2×C322Q8 is a maximal subgroup of
C62.4D4  Dic35Dic6  C62.8C23  C62.9C23  C62.10C23  D6⋊Dic6  Dic3.D12  C62.35C23  D62Dic6  C62.65C23  D64Dic6  C62.85C23  C123Dic6  C12⋊Dic6  C62.95C23  C62.101C23  C623Q8  C624Q8  C62.15D4  C2×S3×Dic6  Dic6.24D6
C2×C322Q8 is a maximal quotient of
C62.39C23  C62.42C23  C123Dic6  C12⋊Dic6  C623Q8  C624Q8

Matrix representation of C2×C322Q8 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 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 1 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 12 12
,
 4 10 0 0 0 0 10 9 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[4,10,0,0,0,0,10,9,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×C322Q8 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_2Q_8
% in TeX

G:=Group("C2xC3^2:2Q8");
// GroupNames label

G:=SmallGroup(144,152);
// by ID

G=gap.SmallGroup(144,152);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,121,55,490,3461]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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