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G = C42.211D4  order 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⋊C8⋊2C22, D4.4(C2×D4), C4⋊D8⋊17C2, C4⋊4(C8⋊C22), C4⋊Q8⋊53C22, D4.D4⋊1C2, (C2×C8).11C23, C4.67(C22×D4), D4.2D4⋊12C2, C4⋊C4.377C23, C4⋊1D4⋊33C22, 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⋊C4⋊15C22, Q8⋊C4⋊17C22, (C4×D4).310C22, C23.36D4⋊5C2, C4.4D4⋊53C22, (C2×SD16).2C22, C22.32(C4⋊D4), (C2×C42).809C22, C22.26C24⋊3C2, (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

Derived series
C1 — C2×C4 — C42.211D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C4×D4 — C42.211D4
Lower central
C1 — C2 — C2×C4 — C42.211D4
Upper central
C1 — C22 — C2×C42 — C42.211D4
Jennings
C1 — C2 — C2 — C2×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, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, D4⋊C4, Q8⋊C4, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C23×C4, C22×D4, C2×C4○D4, C23.36D4, C4⋊M4(2), C4⋊D8, D4.D4, D4.2D4, C2×C4×D4, C22.26C24, C2×C8⋊C22, C42.211D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C8⋊C22, C22×D4, 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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