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G = D4○SD32order 128 = 27

Central product of D4 and SD32

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D4SD32, Q8SD32, D4.13D8, Q8.13D8, D166C22, C8.17C24, C16.4C23, Q325C22, D8.6C23, SD325C22, Q16.6C23, M4(2).22D4, M5(2)⋊9C22, D4○D86C2, Q8○D85C2, D4○C164C2, C4○D165C2, C4.50(C2×D8), C8.16(C2×D4), C16⋊C226C2, (C2×C16)⋊6C22, C4○D4.36D4, (C2×SD32)⋊6C2, Q32⋊C25C2, C4○D82C22, C22.7(C2×D8), C2.32(C22×D8), C4.23(C22×D4), (C2×C8).295C23, (C2×Q16)⋊34C22, C8○D4.13C22, (C2×D8).98C22, (C2×C4).185(C2×D4), SmallGroup(128,2148)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — D4○SD32
C1C2C4C8C2×C8C8○D4D4○D8 — D4○SD32
C1C2C4C8 — D4○SD32
C1C2C4○D4C8○D4 — D4○SD32
C1C2C2C2C2C4C4C8 — D4○SD32

Generators and relations for D4○SD32
 G = < a,b,c,d | a4=d2=1, b2=c8=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >

Subgroups: 416 in 180 conjugacy classes, 90 normal (15 characteristic)
C1, C2, C2 [×7], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], C8, C8 [×3], C2×C4 [×3], C2×C4 [×9], D4 [×3], D4 [×13], Q8, Q8 [×7], C23 [×3], C16, C16 [×3], C2×C8 [×3], M4(2) [×3], D8, D8 [×3], D8 [×3], SD16 [×6], Q16, Q16 [×3], Q16 [×3], C2×D4 [×6], C2×Q8 [×4], C4○D4, C4○D4 [×10], C2×C16 [×3], M5(2) [×3], D16 [×3], SD32, SD32 [×9], Q32 [×3], C8○D4, C2×D8 [×3], C2×Q16 [×3], C4○D8 [×6], C8⋊C22 [×3], C8.C22 [×3], 2+ 1+4, 2- 1+4, D4○C16, C2×SD32 [×3], C4○D16 [×3], C16⋊C22 [×3], Q32⋊C2 [×3], D4○D8, Q8○D8, D4○SD32
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, C22×D8, D4○SD32

Smallest permutation representation of D4○SD32
On 32 points
Generators in S32
(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)
(1 23 9 31)(2 24 10 32)(3 25 11 17)(4 26 12 18)(5 27 13 19)(6 28 14 20)(7 29 15 21)(8 30 16 22)
(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 24)(2 31)(3 22)(4 29)(5 20)(6 27)(7 18)(8 25)(9 32)(10 23)(11 30)(12 21)(13 28)(14 19)(15 26)(16 17)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H8A8B8C8D8E16A16B16C16D16E···16J
order12222222244444444888881616161616···16
size112228888222288882244422224···4

32 irreducible representations

dim1111111122224
type++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D8D8D4○SD32
kernelD4○SD32D4○C16C2×SD32C4○D16C16⋊C22Q32⋊C2D4○D8Q8○D8M4(2)C4○D4D4Q8C1
# reps1133331131624

Matrix representation of D4○SD32 in GL4(𝔽7) generated by

0651
3056
3361
1631
,
4221
1413
0455
2411
,
4051
1421
1615
5510
,
4566
4322
2110
5656
G:=sub<GL(4,GF(7))| [0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[4,1,0,2,2,4,4,4,2,1,5,1,1,3,5,1],[4,1,1,5,0,4,6,5,5,2,1,1,1,1,5,0],[4,4,2,5,5,3,1,6,6,2,1,5,6,2,0,6] >;

D4○SD32 in GAP, Magma, Sage, TeX

D_4\circ {\rm SD}_{32}
% in TeX

G:=Group("D4oSD32");
// GroupNames label

G:=SmallGroup(128,2148);
// by ID

G=gap.SmallGroup(128,2148);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,448,253,521,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=d^2=1,b^2=c^8=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations

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