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## G = C23⋊2D4order 64 = 26

### 1st semidirect product of C23 and D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C232D4, C24.4C22, C23.75C23, (C2×C4)⋊2D4, C2.4C22≀C2, (C22×D4)⋊1C2, C2.3(C41D4), C2.6(C4⋊D4), C22.68(C2×D4), C2.C429C2, (C22×C4).7C22, C22.35(C4○D4), (C2×C22⋊C4)⋊6C2, SmallGroup(64,73)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C23⋊2D4
 Chief series C1 — C2 — C22 — C23 — C24 — C22×D4 — C23⋊2D4
 Lower central C1 — C23 — C23⋊2D4
 Upper central C1 — C23 — C23⋊2D4
 Jennings C1 — C23 — C23⋊2D4

Generators and relations for C232D4
G = < a,b,c,d,e | a2=b2=c2=d4=e2=1, eae=ab=ba, ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 317 in 161 conjugacy classes, 43 normal (7 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×7], C22, C22 [×6], C22 [×30], C2×C4 [×6], C2×C4 [×9], D4 [×24], C23, C23 [×6], C23 [×18], C22⋊C4 [×6], C22×C4, C22×C4 [×3], C2×D4 [×18], C24 [×3], C2.C42, C2×C22⋊C4 [×3], C22×D4 [×3], C232D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C232D4

Character table of C232D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E 4F 4G 4H size 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 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 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 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 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 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 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 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 linear of order 2 ρ9 2 -2 2 2 2 -2 -2 -2 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 2 2 2 -2 -2 -2 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 2 0 -2 0 orthogonal lifted from D4 ρ12 2 -2 -2 -2 2 -2 2 2 0 0 0 0 0 0 0 2 0 -2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 -2 -2 -2 2 -2 0 0 -2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 -2 -2 -2 2 -2 0 0 2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 -2 -2 2 -2 2 2 0 0 0 0 0 0 0 -2 0 2 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 -2 -2 2 2 -2 -2 0 -2 0 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 -2 -2 2 2 -2 -2 0 2 0 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 -2 0 0 2 0 0 orthogonal lifted from D4 ρ20 2 2 -2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 -2 0 2 0 orthogonal lifted from D4 ρ21 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 2i 0 0 0 0 0 0 -2i complex lifted from C4○D4 ρ22 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 -2i 0 0 0 0 0 0 2i complex lifted from C4○D4

Smallest permutation representation of C232D4
On 32 points
Generators in S32
(1 29)(2 12)(3 31)(4 10)(5 20)(6 21)(7 18)(8 23)(9 16)(11 14)(13 32)(15 30)(17 28)(19 26)(22 25)(24 27)
(1 5)(2 6)(3 7)(4 8)(9 22)(10 23)(11 24)(12 21)(13 26)(14 27)(15 28)(16 25)(17 30)(18 31)(19 32)(20 29)
(1 27)(2 28)(3 25)(4 26)(5 14)(6 15)(7 16)(8 13)(9 18)(10 19)(11 20)(12 17)(21 30)(22 31)(23 32)(24 29)
(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)
(1 4)(2 3)(5 8)(6 7)(9 17)(10 20)(11 19)(12 18)(13 14)(15 16)(21 31)(22 30)(23 29)(24 32)(25 28)(26 27)

G:=sub<Sym(32)| (1,29)(2,12)(3,31)(4,10)(5,20)(6,21)(7,18)(8,23)(9,16)(11,14)(13,32)(15,30)(17,28)(19,26)(22,25)(24,27), (1,5)(2,6)(3,7)(4,8)(9,22)(10,23)(11,24)(12,21)(13,26)(14,27)(15,28)(16,25)(17,30)(18,31)(19,32)(20,29), (1,27)(2,28)(3,25)(4,26)(5,14)(6,15)(7,16)(8,13)(9,18)(10,19)(11,20)(12,17)(21,30)(22,31)(23,32)(24,29), (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), (1,4)(2,3)(5,8)(6,7)(9,17)(10,20)(11,19)(12,18)(13,14)(15,16)(21,31)(22,30)(23,29)(24,32)(25,28)(26,27)>;

G:=Group( (1,29)(2,12)(3,31)(4,10)(5,20)(6,21)(7,18)(8,23)(9,16)(11,14)(13,32)(15,30)(17,28)(19,26)(22,25)(24,27), (1,5)(2,6)(3,7)(4,8)(9,22)(10,23)(11,24)(12,21)(13,26)(14,27)(15,28)(16,25)(17,30)(18,31)(19,32)(20,29), (1,27)(2,28)(3,25)(4,26)(5,14)(6,15)(7,16)(8,13)(9,18)(10,19)(11,20)(12,17)(21,30)(22,31)(23,32)(24,29), (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), (1,4)(2,3)(5,8)(6,7)(9,17)(10,20)(11,19)(12,18)(13,14)(15,16)(21,31)(22,30)(23,29)(24,32)(25,28)(26,27) );

G=PermutationGroup([(1,29),(2,12),(3,31),(4,10),(5,20),(6,21),(7,18),(8,23),(9,16),(11,14),(13,32),(15,30),(17,28),(19,26),(22,25),(24,27)], [(1,5),(2,6),(3,7),(4,8),(9,22),(10,23),(11,24),(12,21),(13,26),(14,27),(15,28),(16,25),(17,30),(18,31),(19,32),(20,29)], [(1,27),(2,28),(3,25),(4,26),(5,14),(6,15),(7,16),(8,13),(9,18),(10,19),(11,20),(12,17),(21,30),(22,31),(23,32),(24,29)], [(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)], [(1,4),(2,3),(5,8),(6,7),(9,17),(10,20),(11,19),(12,18),(13,14),(15,16),(21,31),(22,30),(23,29),(24,32),(25,28),(26,27)])

Matrix representation of C232D4 in GL6(𝔽5)

 4 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 4 2 0 0 0 0 0 1
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 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 4 0 0 0 0 0 0 4
,
 0 1 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 4 0 0 0 0 0 0 0 3 4 0 0 0 0 0 2
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 3 4 0 0 0 0 3 2

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,2,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,4,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,3,0,0,0,0,4,2] >;

C232D4 in GAP, Magma, Sage, TeX

C_2^3\rtimes_2D_4
% in TeX

G:=Group("C2^3:2D4");
// GroupNames label

G:=SmallGroup(64,73);
// by ID

G=gap.SmallGroup(64,73);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,121,362,332]);
// Polycyclic

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

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