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G = C42.240D4  order 128 = 27

222nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.240D4, C42.707C23, (C4×C8)⋊64C22, C4⋊Q8⋊63C22, C4.4D8⋊40C2, C4⋊C4.91C23, C8⋊C4⋊65C22, C4.20(C8⋊C22), (C4×M4(2))⋊40C2, (C2×C4).336C24, (C2×C8).457C23, (C22×C4).460D4, C23.678(C2×D4), D4⋊C4⋊53C22, C4.77(C4.4D4), (C2×D4).104C23, C42.C2⋊35C22, C23.37D4⋊37C2, C4⋊1D4.147C22, (C2×C42).847C22, C22.596(C22×D4), (C22×C4).1034C23, C22.31(C4.4D4), (C22×D4).369C22, C23.37C23⋊10C2, C42.29C22⋊18C2, C42⋊C2.141C22, (C2×M4(2)).373C22, C4.45(C2×C4○D4), (C2×C4).514(C2×D4), C2.39(C2×C8⋊C22), (C2×C4⋊1D4).24C2, C2.47(C2×C4.4D4), (C2×C4).300(C4○D4), SmallGroup(128,1870)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C42.240D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2) — C4×M4(2) — C42.240D4
Lower central
C1 — C2 — C2×C4 — C42.240D4
Upper central
C1 — C22 — C2×C42 — C42.240D4
Jennings
C1 — C2 — C2 — C2×C4 — C42.240D4

Generators and relations for C42.240D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2c3 >

Subgroups: 580 in 242 conjugacy classes, 96 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C4×C8, C8⋊C4, D4⋊C4, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊1D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C22×D4, C22×D4, C4×M4(2), C23.37D4, C4.4D8, C42.29C22, C2×C4⋊1D4, C23.37C23, C42.240D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C8⋊C22, C22×D4, C2×C4○D4, C2×C4.4D4, C2×C8⋊C22, C42.240D4

Smallest permutation representation of C42.240D4
►On 32 points
Generators in S32
(1 18 25 11)(2 19 26 12)(3 20 27 13)(4 21 28 14)(5 22 29 15)(6 23 30 16)(7 24 31 9)(8 17 32 10)

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111112224
type++++++++++
imageC1C2C2C2C2C2C2D4D4C4○D4C8⋊C22
kernelC42.240D4C4×M4(2)C23.37D4C4.4D8C42.29C22C2×C4⋊1D4C23.37C23C42C22×C4C2×C4C4
# reps11444112284

Matrix representation of C42.240D4 ►in GL6(𝔽17)

1690000
1310000
0001600
001000
0000016
000010
,
1600000
0160000
0001600
001000
000001
0000160
,
1300000
0130000
0000016
000010
0016000
0001600
,
1320000
040000
000010
0000016
001000
0001600

G:=sub<GL(6,GF(17))| [16,13,0,0,0,0,9,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,16,0,0,0],[13,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,16,0,0] >;
 

C42.240D4 in GAP, Magma, Sage, TeX 

C_4^2._{240}D_4
 
% in TeX
 
G:=Group("C4^2.240D4");
 
// GroupNames label
 
G:=SmallGroup(128,1870);
 
// by ID
 
G=gap.SmallGroup(128,1870);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,100,1018,521,2804,172,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*c^3>;
 
// generators/relations
 

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