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G = C42.211D4order 128 = 27

193rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.211D4, C42.321C23, (C2×D4)⋊38D4, C4⋊C82C22, D4.4(C2×D4), C4⋊D817C2, C44(C8⋊C22), C4⋊Q853C22, D4.D41C2, (C2×C8).11C23, C4.67(C22×D4), D4.2D412C2, C4⋊C4.377C23, C41D433C22, C4⋊M4(2)⋊2C2, (C2×C4).240C24, (C2×D8).53C22, (C2×D4).49C23, (C22×C4).793D4, C23.652(C2×D4), (C2×Q8).36C23, C4.168(C4⋊D4), D4⋊C415C22, Q8⋊C417C22, (C4×D4).310C22, C23.36D45C2, C4.4D453C22, (C2×SD16).2C22, C22.32(C4⋊D4), (C2×C42).809C22, C22.26C243C2, (C22×C4).970C23, C22.500(C22×D4), C2.11(D8⋊C22), (C22×D4).567C22, (C2×M4(2)).47C22, (C2×C4×D4)⋊61C2, (C2×C8⋊C22)⋊14C2, C4.150(C2×C4○D4), C2.58(C2×C4⋊D4), C2.15(C2×C8⋊C22), (C2×C4).1419(C2×D4), (C2×C4).271(C4○D4), (C2×C4⋊C4).920C22, (C2×C4○D4).115C22, SmallGroup(128,1768)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.211D4
C1C2C4C2×C4C22×C4C22×D4C2×C4×D4 — C42.211D4
C1C2C2×C4 — C42.211D4
C1C22C2×C42 — C42.211D4
C1C2C2C2×C4 — C42.211D4

Generators and relations for C42.211D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=dad=a-1, cbc-1=b-1, bd=db, dcd=b2c3 >

Subgroups: 572 in 274 conjugacy classes, 102 normal (30 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×4], C4 [×7], C22, C22 [×2], C22 [×24], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×21], D4 [×4], D4 [×18], Q8 [×4], C23, C23 [×12], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×4], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4 [×3], C22×C4 [×11], C2×D4 [×8], C2×D4 [×7], C2×Q8 [×2], C4○D4 [×8], C24, D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4 [×4], C4×D4 [×4], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×D8 [×4], C2×SD16 [×4], C8⋊C22 [×8], C23×C4, C22×D4, C2×C4○D4 [×2], C23.36D4 [×2], C4⋊M4(2), C4⋊D8 [×2], D4.D4 [×2], D4.2D4 [×4], C2×C4×D4, C22.26C24, C2×C8⋊C22 [×2], C42.211D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C8⋊C22 [×2], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, C2×C8⋊C22, D8⋊C22, C42.211D4

Smallest permutation representation of C42.211D4
On 32 points
Generators in S32
(1 27 10 23)(2 24 11 28)(3 29 12 17)(4 18 13 30)(5 31 14 19)(6 20 15 32)(7 25 16 21)(8 22 9 26)
(1 16 5 12)(2 13 6 9)(3 10 7 14)(4 15 8 11)(17 27 21 31)(18 32 22 28)(19 29 23 25)(20 26 24 30)
(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 9)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A···4H4I···4N4O4P8A8B8C8D
order1222222222224···44···4448888
size1111224444882···24···4888888

32 irreducible representations

dim111111111222244
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4C4○D4C8⋊C22D8⋊C22
kernelC42.211D4C23.36D4C4⋊M4(2)C4⋊D8D4.D4D4.2D4C2×C4×D4C22.26C24C2×C8⋊C22C42C22×C4C2×D4C2×C4C4C2
# reps121224112224422

Matrix representation of C42.211D4 in GL6(𝔽17)

0160000
100000
001000
000100
000010
000001
,
100000
010000
000100
0016000
000001
0000160
,
460000
6130000
000010
0000016
0001600
0016000
,
460000
6130000
000010
000001
001000
000100

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

C42.211D4 in GAP, Magma, Sage, TeX

C_4^2._{211}D_4
% in TeX

G:=Group("C4^2.211D4");
// GroupNames label

G:=SmallGroup(128,1768);
// by ID

G=gap.SmallGroup(128,1768);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,2019,4037,1027,124]);
// Polycyclic

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

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