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G = C42.269D4order 128 = 27

251st non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.269D4, C42.730C23, C4.1032+ 1+4, C4⋊C867C22, (C4×C8)⋊16C22, Q8⋊D431C2, C22⋊D8.2C2, D4.2D42C2, C8.12D43C2, (C4×D4)⋊12C22, C22⋊Q166C2, Q8.D42C2, (C2×Q16)⋊5C22, (C4×Q8)⋊12C22, C22⋊SD1631C2, C4⋊C4.150C23, (C2×C8).327C23, (C2×C4).409C24, Q8⋊C45C22, (C2×D8).24C22, C23.692(C2×D4), (C22×C4).172D4, (C2×SD16)⋊43C22, (C2×D4).158C23, C22.33(C4○D8), C4.4D460C22, (C2×Q8).146C23, C42.C237C22, C42.12C435C2, C4⋊D4.189C22, C22⋊C8.179C22, (C2×C42).876C22, C22.669(C22×D4), C22⋊Q8.194C22, D4⋊C4.107C22, C2.54(D8⋊C22), (C22×C4).1080C23, C42.78C227C2, C23.36C239C2, (C22×D4).388C22, (C22×Q8).321C22, C2.80(C22.29C24), C2.43(C2×C4○D8), (C2×C4).539(C2×D4), (C2×C4.4D4)⋊43C2, SmallGroup(128,1943)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.269D4
C1C2C4C2×C4C22×C4C22×D4C2×C4.4D4 — C42.269D4
C1C2C2×C4 — C42.269D4
C1C22C2×C42 — C42.269D4
C1C2C2C2×C4 — C42.269D4

Generators and relations for C42.269D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=b2c3 >

Subgroups: 452 in 205 conjugacy classes, 86 normal (42 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×15], C8 [×4], C2×C4 [×6], C2×C4 [×14], D4 [×10], Q8 [×8], C23, C23 [×9], C42 [×4], C42, C22⋊C4 [×13], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C2×Q8 [×2], C2×Q8 [×3], C24, C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2×C42, C2×C22⋊C4 [×2], C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4 [×4], C4.4D4 [×2], C42.C2, C422C2, C2×D8 [×2], C2×SD16 [×4], C2×Q16 [×2], C22×D4, C22×Q8, C42.12C4, C22⋊D8, Q8⋊D4, C22⋊SD16, C22⋊Q16, D4.2D4 [×2], Q8.D4 [×2], C42.78C22 [×2], C8.12D4 [×2], C2×C4.4D4, C23.36C23, C42.269D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C22×D4, 2+ 1+4 [×2], C22.29C24, C2×C4○D8, D8⋊C22, C42.269D4

Smallest permutation representation of C42.269D4
On 32 points
Generators in S32
(1 14 29 24)(2 15 30 17)(3 16 31 18)(4 9 32 19)(5 10 25 20)(6 11 26 21)(7 12 27 22)(8 13 28 23)
(1 7 5 3)(2 28 6 32)(4 30 8 26)(9 17 13 21)(10 16 14 12)(11 19 15 23)(18 24 22 20)(25 31 29 27)
(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 2)(3 8)(4 7)(5 6)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(25 26)(27 32)(28 31)(29 30)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4H4I4J4K···4O8A···8H
order1222222224···4444···48···8
size1111228882···2448···84···4

32 irreducible representations

dim11111111111122244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D82+ 1+4D8⋊C22
kernelC42.269D4C42.12C4C22⋊D8Q8⋊D4C22⋊SD16C22⋊Q16D4.2D4Q8.D4C42.78C22C8.12D4C2×C4.4D4C23.36C23C42C22×C4C22C4C2
# reps11111122221122822

Matrix representation of C42.269D4 in GL6(𝔽17)

0130000
400000
00131500
000400
000042
0000013
,
0160000
100000
0016000
0001600
000010
000001
,
3140000
330000
0000160
000041
001000
00131600
,
14140000
1430000
000010
000001
001000
000100

G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,13,0,0,0,0,0,0,0,13,0,0,0,0,0,15,4,0,0,0,0,0,0,4,0,0,0,0,0,2,13],[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],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,0,0,1,13,0,0,0,0,0,16,0,0,16,4,0,0,0,0,0,1,0,0],[14,14,0,0,0,0,14,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C42.269D4 in GAP, Magma, Sage, TeX

C_4^2._{269}D_4
% in TeX

G:=Group("C4^2.269D4");
// GroupNames label

G:=SmallGroup(128,1943);
// by ID

G=gap.SmallGroup(128,1943);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,675,4037,1027,124]);
// Polycyclic

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

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