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G = C2×D4○SD16  order 128 = 27

Direct product of C2 and D4○SD16

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

Aliases: C2×D4○SD16, D8⋊5C23, C8.4C24, C4.9C25, Q16⋊5C23, D4.6C24, Q8.6C24, SD16⋊7C23, M4(2)⋊8C23, 2+ (1+4)⋊9C22, 2- (1+4)⋊7C22, SD16○(C2×D4), D4○(C2×SD16), Q8○(C2×SD16), SD16○(C2×Q8), (C2×C8)⋊7C23, C4○D4.39D4, D4.62(C2×D4), C4○D4⋊2C23, Q8.64(C2×D4), (C2×D4).358D4, C4○D8⋊10C22, C8○D4⋊14C22, (C2×D8)⋊57C22, (C2×Q8).277D4, (C2×Q8)⋊11C23, C2.44(D4×C23), C8⋊C22⋊13C22, (C2×C4).615C24, (C22×C8)⋊29C22, (C2×Q16)⋊60C22, C23.486(C2×D4), C4.126(C22×D4), (C2×SD16)⋊63C22, (C22×SD16)⋊10C2, (C2×D4).346C23, C8.C22⋊13C22, (C22×Q8)⋊48C22, C22.18(C22×D4), (C2×M4(2))⋊60C22, (C2×2+ (1+4))⋊14C2, (C2×2- (1+4))⋊11C2, (C22×C4).1226C23, (C22×D4).443C22, (C2×Q8)○(C2×SD16), (C2×C8○D4)⋊11C2, (C2×C4○D8)⋊31C2, (C2×C8⋊C22)⋊35C2, (C2×C4).1114(C2×D4), (C2×C4○D4)⋊57C22, (C2×C8.C22)⋊34C2, SmallGroup(128,2314)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — C2×D4○SD16
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2+ (1+4) — C2×D4○SD16
Lower central
C1 — C2 — C4 — C2×D4○SD16
Upper central
C1 — C22 — C2×C4○D4 — C2×D4○SD16
Jennings
C1 — C2 — C2 — C4 — C2×D4○SD16

Subgroups: 1172 in 730 conjugacy classes, 428 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×14], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×6], C22 [×34], C8 [×8], C2×C4, C2×C4 [×15], C2×C4 [×40], D4 [×20], D4 [×40], Q8 [×12], Q8 [×16], C23 [×3], C23 [×22], C2×C8, C2×C8 [×15], M4(2) [×12], D8 [×12], SD16 [×40], Q16 [×12], C22×C4 [×3], C22×C4 [×9], C2×D4, C2×D4 [×18], C2×D4 [×39], C2×Q8 [×2], C2×Q8 [×15], C2×Q8 [×17], C4○D4 [×32], C4○D4 [×44], C24 [×3], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×D8 [×3], C2×SD16, C2×SD16 [×33], C2×Q16 [×3], C4○D8 [×24], C8⋊C22 [×24], C8.C22 [×24], C22×D4 [×3], C22×D4 [×3], C22×Q8 [×3], C22×Q8, C2×C4○D4, C2×C4○D4 [×6], C2×C4○D4 [×4], 2+ (1+4) [×8], 2+ (1+4) [×4], 2- (1+4) [×8], 2- (1+4) [×4], C2×C8○D4, C22×SD16 [×3], C2×C4○D8 [×3], C2×C8⋊C22 [×3], C2×C8.C22 [×3], D4○SD16 [×16], C2×2+ (1+4), C2×2- (1+4), C2×D4○SD16

Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], C25, D4○SD16 [×2], D4×C23, C2×D4○SD16

Generators and relations
 G = < a,b,c,d,e | a2=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0001600
001000
000001
0000160
,
100000
010000
000001
0000160
0001600
001000
,
1140000
1260000
0051200
005500
0000512
000055
,
1600000
1410000
000010
0000016
001000
0001600

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[11,12,0,0,0,0,4,6,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,12,5],[16,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,16,0,0] >;
 

44 conjugacy classes

class 1 2A2B2C2D···2I2J···2Q4A···4H4I···4P8A8B8C8D8E···8J
order12222···22···24···44···488888···8
size11112···24···42···24···422224···4

44 irreducible representations

dim1111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4D4○SD16
kernelC2×D4○SD16C2×C8○D4C22×SD16C2×C4○D8C2×C8⋊C22C2×C8.C22D4○SD16C2×2+ (1+4)C2×2- (1+4)C2×D4C2×Q8C4○D4C2
# reps11333316113144

In GAP, Magma, Sage, TeX 

C_2\times D_4\circ SD_{16}
 
% in TeX
 
G:=Group("C2xD4oSD16");
 
// GroupNames label
 
G:=SmallGroup(128,2314);
 
// by ID
 
G=gap.SmallGroup(128,2314);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,-2,448,477,521,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=e^2=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
 
// generators/relations
 

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