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G = C427D4order 128 = 27

1st semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C427D4, C24.77D4, C4⋊D415C4, C4.117(C4×D4), C22⋊Q815C4, C426C426C2, C23.567(C2×D4), (C22×C4).292D4, C4.189(C4⋊D4), C24.4C428C2, C22.28C22≀C2, C23.83(C22⋊C4), C22.19C24.8C2, (C23×C4).261C22, (C2×C42).288C22, (C22×C4).1373C23, C42⋊C2.26C22, C2.42(C42⋊C22), C2.36(C23.23D4), (C2×M4(2)).186C22, C22.26(C22.D4), (C2×C4≀C2)⋊16C2, C4⋊C4.75(C2×C4), (C2×D4).83(C2×C4), (C2×Q8).71(C2×C4), (C2×C42⋊C2)⋊2C2, (C2×C4).1335(C2×D4), (C2×C4).570(C4○D4), (C2×C4).391(C22×C4), (C22×C4).282(C2×C4), (C2×C4○D4).22C22, (C2×C4).134(C22⋊C4), C22.272(C2×C22⋊C4), SmallGroup(128,629)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C427D4
C1C2C4C2×C4C22×C4C23×C4C2×C42⋊C2 — C427D4
C1C2C2×C4 — C427D4
C1C2×C4C23×C4 — C427D4
C1C2C2C22×C4 — C427D4

Generators and relations for C427D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b, dad=a-1b-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 372 in 179 conjugacy classes, 56 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×4], C4 [×9], C22 [×3], C22 [×15], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×23], D4 [×8], Q8 [×2], C23, C23 [×2], C23 [×5], C42 [×4], C42 [×3], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×2], M4(2) [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×7], C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×4], C24, C22⋊C8 [×2], C4≀C2 [×4], C2×C42 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2 [×4], C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C2×M4(2) [×2], C23×C4, C2×C4○D4, C426C4 [×2], C24.4C4, C2×C4≀C2 [×2], C2×C42⋊C2, C22.19C24, C427D4
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], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C42⋊C22 [×2], C427D4

Smallest permutation representation of C427D4
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 8 3 12)(2 7 4 11)(5 9 15 14)(6 10 16 13)(17 25 19 27)(18 26 20 28)(21 31 23 29)(22 32 24 30)
(1 30 6 26)(2 21 5 19)(3 32 16 28)(4 23 15 17)(7 31 9 27)(8 22 10 20)(11 29 14 25)(12 24 13 18)
(1 27)(2 18)(3 25)(4 20)(5 24)(6 31)(7 26)(8 17)(9 30)(10 23)(11 28)(12 19)(13 21)(14 32)(15 22)(16 29)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G···4P4Q4R4S8A8B8C8D
order1222222224444444···44448888
size1111224481111224···48888888

32 irreducible representations

dim1111111122224
type+++++++++
imageC1C2C2C2C2C2C4C4D4D4D4C4○D4C42⋊C22
kernelC427D4C426C4C24.4C4C2×C4≀C2C2×C42⋊C2C22.19C24C4⋊D4C22⋊Q8C42C22×C4C24C2×C4C2
# reps1212114443144

Matrix representation of C427D4 in GL6(𝔽17)

400000
13130000
0001600
001000
0000013
000040
,
100000
010000
004000
000400
000040
000004
,
16150000
110000
000010
0000016
0016000
000100
,
120000
0160000
0000016
000010
000100
0016000

G:=sub<GL(6,GF(17))| [4,13,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,16,0,0],[1,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16,0,0,0] >;

C427D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_7D_4
% in TeX

G:=Group("C4^2:7D4");
// GroupNames label

G:=SmallGroup(128,629);
// by ID

G=gap.SmallGroup(128,629);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,2019,248,124]);
// Polycyclic

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

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