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G = C42.266D4order 128 = 27

248th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.266D4, C42.397C23, C4.1002+ 1+4, (C2×C4)⋊7SD16, C85D422C2, C4⋊C876C22, (C4×C8)⋊42C22, C4⋊Q870C22, C4⋊SD1636C2, C4.4D827C2, (C4×Q8)⋊11C22, C4.86(C2×SD16), C22⋊SD1630C2, C4⋊C4.147C23, C4.27(C8⋊C22), (C2×C4).406C24, (C2×C8).325C23, C23.690(C2×D4), (C22×C4).496D4, D4⋊C445C22, (C2×SD16)⋊42C22, (C2×D4).156C23, (C2×Q8).143C23, C2.23(C22×SD16), C22.25(C2×SD16), C42.12C443C2, C41D4.162C22, C22⋊C8.219C22, (C2×C42).873C22, C22.666(C22×D4), C22⋊Q8.191C22, (C22×C4).1077C23, (C22×D4).387C22, C23.37C2316C2, C2.77(C22.29C24), (C2×C4).866(C2×D4), C2.55(C2×C8⋊C22), (C2×C41D4).26C2, SmallGroup(128,1940)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.266D4
C1C2C4C2×C4C22×C4C22×D4C2×C41D4 — C42.266D4
C1C2C2×C4 — C42.266D4
C1C22C2×C42 — C42.266D4
C1C2C2C2×C4 — C42.266D4

Generators and relations for C42.266D4
 G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c3 >

Subgroups: 604 in 244 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×6], C22, C22 [×2], C22 [×22], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×8], D4 [×28], Q8 [×6], C23, C23 [×16], C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], SD16 [×8], C22×C4 [×3], C2×D4 [×4], C2×D4 [×22], C2×Q8 [×2], C2×Q8, C24 [×2], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C2×C42, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C41D4 [×4], C41D4 [×2], C4⋊Q8 [×2], C2×SD16 [×8], C22×D4 [×2], C22×D4 [×2], C42.12C4, C22⋊SD16 [×4], C4⋊SD16 [×4], C4.4D8 [×2], C85D4 [×2], C2×C41D4, C23.37C23, C42.266D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C24, C2×SD16 [×6], C8⋊C22 [×2], C22×D4, 2+ 1+4 [×2], C22.29C24, C22×SD16, C2×C8⋊C22, C42.266D4

Smallest permutation representation of C42.266D4
On 32 points
Generators in S32
(1 12 31 19)(2 13 32 20)(3 14 25 21)(4 15 26 22)(5 16 27 23)(6 9 28 24)(7 10 29 17)(8 11 30 18)
(1 3 5 7)(2 30 6 26)(4 32 8 28)(9 22 13 18)(10 12 14 16)(11 24 15 20)(17 19 21 23)(25 27 29 31)
(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 11 5 15)(2 14 6 10)(3 9 7 13)(4 12 8 16)(17 32 21 28)(18 27 22 31)(19 30 23 26)(20 25 24 29)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I4J4K4L4M4N8A···8H
order12222222224···44444448···8
size11112288882···24488884···4

32 irreducible representations

dim1111111122244
type++++++++++++
imageC1C2C2C2C2C2C2C2D4D4SD16C8⋊C222+ 1+4
kernelC42.266D4C42.12C4C22⋊SD16C4⋊SD16C4.4D8C85D4C2×C41D4C23.37C23C42C22×C4C2×C4C4C4
# reps1144221122822

Matrix representation of C42.266D4 in GL6(𝔽17)

010000
1600000
0001600
001000
000001
0000160
,
0160000
100000
0016000
0001600
000010
000001
,
5120000
550000
000001
000010
0001600
0016000
,
12120000
1250000
000010
000001
001000
000100

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

C42.266D4 in GAP, Magma, Sage, TeX

C_4^2._{266}D_4
% in TeX

G:=Group("C4^2.266D4");
// GroupNames label

G:=SmallGroup(128,1940);
// by ID

G=gap.SmallGroup(128,1940);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,675,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations

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