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G = C2×C22⋊SD16  order 128 = 27

Direct product of C2 and C22⋊SD16

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

Aliases: C2×C22⋊SD16, C23⋊6SD16, C24.174D4, C4⋊C4⋊2C23, (C2×C8)⋊8C23, D4.37(C2×D4), (C2×Q8)⋊1C23, C4.34C22≀C2, (C2×D4).292D4, C22⋊3(C2×SD16), C4.34(C22×D4), C22⋊C8⋊65C22, (C2×C4).216C24, (C22×C8)⋊34C22, (D4×C23).17C2, (C22×C4).416D4, C23.847(C2×D4), C22⋊Q8⋊59C22, D4⋊C4⋊69C22, C2.5(C22×SD16), (C22×SD16)⋊15C2, (C2×SD16)⋊65C22, (C2×D4).378C23, (C22×Q8)⋊12C22, C22.113C22≀C2, (C22×C4).954C23, (C23×C4).536C22, C22.476(C22×D4), C22.110(C8⋊C22), (C22×D4).560C22, C2.8(C2×C8⋊C22), (C2×C22⋊C8)⋊33C2, (C2×C4⋊C4)⋊45C22, (C2×C22⋊Q8)⋊51C2, (C2×D4⋊C4)⋊35C2, C2.34(C2×C22≀C2), (C2×C4).1090(C2×D4), SmallGroup(128,1729)

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

Derived series
C1 — C2×C4 — C2×C22⋊SD16
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — D4×C23 — C2×C22⋊SD16
Lower central
C1 — C2 — C2×C4 — C2×C22⋊SD16
Upper central
C1 — C23 — C23×C4 — C2×C22⋊SD16
Jennings
C1 — C2 — C2 — C2×C4 — C2×C22⋊SD16

Subgroups: 1148 in 498 conjugacy classes, 124 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×12], C4 [×4], C4 [×6], C22, C22 [×10], C22 [×76], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×18], D4 [×8], D4 [×28], Q8 [×6], C23, C23 [×6], C23 [×88], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], C2×C8 [×4], SD16 [×16], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×12], C2×D4 [×50], C2×Q8 [×2], C2×Q8 [×5], C24, C24 [×22], C22⋊C8 [×4], D4⋊C4 [×8], C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C2×SD16 [×8], C2×SD16 [×8], C23×C4, C22×D4 [×6], C22×D4 [×11], C22×Q8, C25, C2×C22⋊C8, C2×D4⋊C4 [×2], C22⋊SD16 [×8], C2×C22⋊Q8, C22×SD16 [×2], D4×C23, C2×C22⋊SD16

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], SD16 [×4], C2×D4 [×18], C24, C22≀C2 [×4], C2×SD16 [×6], C8⋊C22 [×2], C22×D4 [×3], C22⋊SD16 [×4], C2×C22≀C2, C22×SD16, C2×C8⋊C22, C2×C22⋊SD16

Generators and relations
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d3 >

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

(2 28)(4 30)(6 32)(8 26)(9 22)(11 24)(13 18)(15 20)

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

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

Matrix representation ►G ⊆ GL5(𝔽17)

160000
016000
001600
000160
000016
,
160000
01000
00100
00010
000716
,
10000
01000
00100
000160
000016
,
10000
051200
05500
000715
000710
,
160000
00100
01000
000160
000016

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,7,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,5,5,0,0,0,12,5,0,0,0,0,0,7,7,0,0,0,15,10],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16] >;
 

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S4A4B4C4D4E4F4G4H4I4J8A···8H
order12···222222···244444444448···8
size11···122224···422224488884···4

38 irreducible representations

dim111111122224
type+++++++++++
imageC1C2C2C2C2C2C2D4D4D4SD16C8⋊C22
kernelC2×C22⋊SD16C2×C22⋊C8C2×D4⋊C4C22⋊SD16C2×C22⋊Q8C22×SD16D4×C23C22×C4C2×D4C24C23C22
# reps112812138182

In GAP, Magma, Sage, TeX 

C_2\times C_2^2\rtimes SD_{16}
 
% in TeX
 
G:=Group("C2xC2^2:SD16");
 
// GroupNames label
 
G:=SmallGroup(128,1729);
 
// by ID
 
G=gap.SmallGroup(128,1729);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
 
// generators/relations
 

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