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G = D812D4order 128 = 27

1st semidirect product of D8 and D4 acting through Inn(D8)

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

Aliases: D812D4, C42.447C23, C4.1342+ 1+4, C2.64D42, C22⋊C42D8, (C4×D8)⋊24C2, (C8×D4)⋊10C2, C8.79(C2×D4), D45D46C2, D4⋊D45C2, C87D421C2, C4⋊C868C22, (C4×C8)⋊18C22, C4⋊C4.253D4, D4.26(C2×D4), D4.2D45C2, (C2×D4).227D4, C2.42(D4○D8), C8.18D49C2, C8.12D47C2, (C4×D4)⋊22C22, (C22×D8)⋊13C2, C222(C4○D8), C4.94(C22×D4), C2.D860C22, C22⋊SD1632C2, C4⋊C4.219C23, C4⋊D413C22, C22⋊C861C22, (C2×C8).343C23, (C2×C4).478C24, (C22×C8)⋊18C22, C22⋊C4.192D4, Q8⋊C48C22, (C2×Q16)⋊47C22, C23.464(C2×D4), D4⋊C482C22, (C2×SD16)⋊48C22, (C2×D8).174C22, (C2×D4).417C23, C4.4D415C22, (C2×Q8).199C23, C22⋊Q8.64C22, C22.738(C22×D4), (C22×C4).1122C23, (C22×D4).402C22, C22⋊C4(C2×D8), (C2×C4○D8)⋊9C2, C2.52(C2×C4○D8), (C2×C4).917(C2×D4), (C2×C4○D4)⋊17C22, SmallGroup(128,2012)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D812D4
C1C2C22C2×C4C2×D4C22×D4C22×D8 — D812D4
C1C2C2×C4 — D812D4
C1C22C4×D4 — D812D4
C1C2C2C2×C4 — D812D4

Generators and relations for D812D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 616 in 263 conjugacy classes, 96 normal (44 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×27], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×14], D4 [×4], D4 [×18], Q8 [×4], C23 [×2], C23 [×14], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×4], D8 [×4], D8 [×6], SD16 [×6], Q16 [×2], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×4], C2×D4 [×12], C2×Q8 [×2], C4○D4 [×6], C24 [×2], C4×C8, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×2], C4⋊C8, C2.D8, C2×C22⋊C4 [×2], C4×D4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C22×C8 [×2], C2×D8 [×2], C2×D8 [×2], C2×D8 [×4], C2×SD16 [×4], C2×Q16, C4○D8 [×4], C22×D4 [×2], C2×C4○D4 [×2], C8×D4, C4×D8, D4⋊D4 [×2], C22⋊SD16 [×2], D4.2D4 [×2], C87D4, C8.18D4, C8.12D4, D45D4 [×2], C22×D8, C2×C4○D8, D812D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C4○D8 [×2], C22×D4 [×2], 2+ 1+4, D42, C2×C4○D8, D4○D8, D812D4

Smallest permutation representation of D812D4
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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)
(1 9 24 30)(2 10 17 31)(3 11 18 32)(4 12 19 25)(5 13 20 26)(6 14 21 27)(7 15 22 28)(8 16 23 29)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F···2J2K2L4A···4F4G4H4I4J4K4L8A8B8C8D8E···8J
order1222222···2224···444444488888···8
size1111224···4882···244888822224···4

35 irreducible representations

dim1111111111112222244
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D4C4○D82+ 1+4D4○D8
kernelD812D4C8×D4C4×D8D4⋊D4C22⋊SD16D4.2D4C87D4C8.18D4C8.12D4D45D4C22×D8C2×C4○D8C22⋊C4C4⋊C4D8C2×D4C22C4C2
# reps1112221112112141812

Matrix representation of D812D4 in GL4(𝔽17) generated by

31400
3300
0010
0001
,
31400
141400
00160
00016
,
0400
13000
00016
0010
,
0400
13000
00016
00160
G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,1,0,0,0,0,1],[3,14,0,0,14,14,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,4,0,0,0,0,0,0,1,0,0,16,0],[0,13,0,0,4,0,0,0,0,0,0,16,0,0,16,0] >;

D812D4 in GAP, Magma, Sage, TeX

D_8\rtimes_{12}D_4
% in TeX

G:=Group("D8:12D4");
// GroupNames label

G:=SmallGroup(128,2012);
// by ID

G=gap.SmallGroup(128,2012);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,2019,346,2804,1411,375,172]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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