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G = C43⋊C2order 128 = 27

7th semidirect product of C43 and C2 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C437C2, C4238D4, C41C4≀C2, C4⋊Q818C4, C4.15(C4×D4), C41D414C4, C4.4D416C4, C4.50(C41D4), C42.268(C2×C4), C23.569(C2×D4), (C22×C4).761D4, C4⋊M4(2)⋊26C2, C4.90(C4.4D4), C22.16(C4⋊D4), (C22×C4).1406C23, (C2×C42).1074C22, (C2×M4(2)).206C22, C22.26C24.20C2, C2.14(C24.3C22), (C2×C4≀C2)⋊18C2, C2.43(C2×C4≀C2), (C2×C4).740(C2×D4), (C2×Q8).92(C2×C4), (C2×D4).107(C2×C4), (C2×C4).597(C4○D4), (C2×C4).420(C22×C4), (C2×C4○D4).37C22, (C2×C4).200(C22⋊C4), C22.284(C2×C22⋊C4), SmallGroup(128,694)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C43⋊C2
C1C2C4C2×C4C22×C4C2×C42C43 — C43⋊C2
C1C2C2×C4 — C43⋊C2
C1C2×C4C2×C42 — C43⋊C2
C1C2C2C22×C4 — C43⋊C2

Generators and relations for C43⋊C2
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=a-1c-1, bc=cb, dbd=b-1, cd=dc >

Subgroups: 356 in 180 conjugacy classes, 60 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C4 [×12], C22, C22 [×2], C22 [×8], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×22], D4 [×12], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×6], C42 [×10], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C4≀C2 [×8], C4⋊C8 [×2], C2×C42, C2×C42 [×2], C2×C42 [×2], C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×C4○D4 [×2], C43, C2×C4≀C2 [×4], C4⋊M4(2), C22.26C24, C43⋊C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C4≀C2 [×4], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C24.3C22, C2×C4≀C2 [×2], C43⋊C2

Smallest permutation representation of C43⋊C2
On 32 points
Generators in S32
(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 12 14 4)(2 11 13 3)(5 8 15 10)(6 7 16 9)(17 26 24 32)(18 27 21 29)(19 28 22 30)(20 25 23 31)
(1 5 13 16)(2 6 14 15)(3 9 12 8)(4 10 11 7)(17 21 19 23)(18 22 20 24)(25 32 27 30)(26 29 28 31)
(1 26)(2 30)(3 19)(4 24)(5 29)(6 25)(7 20)(8 21)(9 23)(10 18)(11 22)(12 17)(13 28)(14 32)(15 27)(16 31)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4AD4AE4AF8A8B8C8D
order1222222244444···4448888
size1111228811112···2888888

44 irreducible representations

dim111111112222
type+++++++
imageC1C2C2C2C2C4C4C4D4D4C4○D4C4≀C2
kernelC43⋊C2C43C2×C4≀C2C4⋊M4(2)C22.26C24C4.4D4C41D4C4⋊Q8C42C22×C4C2×C4C4
# reps1141142262416

Matrix representation of C43⋊C2 in GL4(𝔽17) generated by

16000
01300
0010
0004
,
16000
01600
0040
00013
,
13000
01300
00130
00013
,
01600
16000
0001
0010
G:=sub<GL(4,GF(17))| [16,0,0,0,0,13,0,0,0,0,1,0,0,0,0,4],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,13],[13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[0,16,0,0,16,0,0,0,0,0,0,1,0,0,1,0] >;

C43⋊C2 in GAP, Magma, Sage, TeX

C_4^3\rtimes C_2
% in TeX

G:=Group("C4^3:C2");
// GroupNames label

G:=SmallGroup(128,694);
// by ID

G=gap.SmallGroup(128,694);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,248,124]);
// Polycyclic

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

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