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G = C2×C22⋊SD16order 128 = 27

Direct product of C2 and C22⋊SD16

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

Aliases: C2×C22⋊SD16, C236SD16, C24.174D4, C4⋊C42C23, (C2×C8)⋊8C23, D4.37(C2×D4), (C2×Q8)⋊1C23, C4.34C22≀C2, (C2×D4).292D4, C223(C2×SD16), C4.34(C22×D4), C22⋊C865C22, (C2×C4).216C24, (C22×C8)⋊34C22, (D4×C23).17C2, C23.847(C2×D4), (C22×C4).416D4, C22⋊Q859C22, D4⋊C469C22, C2.5(C22×SD16), (C2×SD16)⋊65C22, (C22×SD16)⋊15C2, (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×C4⋊C4)⋊45C22, (C2×C22⋊C8)⋊33C2, (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

C1C2×C4 — C2×C22⋊SD16
C1C2C22C2×C4C22×C4C23×C4D4×C23 — C2×C22⋊SD16
C1C2C2×C4 — C2×C22⋊SD16
C1C23C23×C4 — C2×C22⋊SD16
C1C2C2C2×C4 — C2×C22⋊SD16

Generators and relations for C2×C22⋊SD16
 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 >

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

Smallest permutation representation of C2×C22⋊SD16
On 32 points
Generators in S32
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)
(2 30)(4 32)(6 26)(8 28)(9 22)(11 24)(13 18)(15 20)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(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 27)(2 30)(3 25)(4 28)(5 31)(6 26)(7 29)(8 32)(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)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (2,30)(4,32)(6,26)(8,28)(9,22)(11,24)(13,18)(15,20), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(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), (1,27)(2,30)(3,25)(4,28)(5,31)(6,26)(7,29)(8,32)(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)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (2,30)(4,32)(6,26)(8,28)(9,22)(11,24)(13,18)(15,20), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(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), (1,27)(2,30)(3,25)(4,28)(5,31)(6,26)(7,29)(8,32)(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),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28)], [(2,30),(4,32),(6,26),(8,28),(9,22),(11,24),(13,18),(15,20)], [(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(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)], [(1,27),(2,30),(3,25),(4,28),(5,31),(6,26),(7,29),(8,32),(9,22),(10,17),(11,20),(12,23),(13,18),(14,21),(15,24),(16,19)])

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

Matrix representation of C2×C22⋊SD16 in 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] >;

C2×C22⋊SD16 in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes {\rm 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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