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G = C24.150D4  order 128 = 27

5th non-split extension by C24 of D4 acting via D4/C22=C2

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

Aliases: C24.150D4, C4:D4:3C4, C22:Q8:3C4, C22.4C4wrC2, C42:C2:2C4, C23.494(C2xD4), (C22xC4).211D4, C24.4C4:19C2, C22.SD16:17C2, C22.6(C23:C4), C4:D4.132C22, C23.31D4:17C2, C22:C8.127C22, C22.14(C8:C22), (C22xC4).626C23, C22.19C24.2C2, (C23xC4).206C22, C22:Q8.137C22, C23.102(C22:C4), C2.8(C23.36D4), C22.10(C8.C22), C2.C42.503C22, C4:C4:3(C2xC4), (C2xC4oD4):2C4, (C2xD4):2(C2xC4), (C2xQ8):2(C2xC4), C2.21(C2xC4wrC2), C2.15(C2xC23:C4), (C2xC4).1150(C2xD4), (C2xC4).89(C22:C4), (C22xC4).197(C2xC4), (C2xC4).116(C22xC4), (C2xC2.C42):14C2, C22.180(C2xC22:C4), SmallGroup(128,236)

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

Derived series
C1 — C2xC4 — C24.150D4
Chief series
C1 — C2 — C22 — C23 — C22xC4 — C23xC4 — C22.19C24 — C24.150D4
Lower central
C1 — C22 — C2xC4 — C24.150D4
Upper central
C1 — C22 — C23xC4 — C24.150D4
Jennings
C1 — C2 — C22 — C22xC4 — C24.150D4

Generators and relations for C24.150D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, eae-1=ac=ca, ad=da, af=fa, bc=cb, bd=db, ebe-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bde3 >

Subgroups: 388 in 167 conjugacy classes, 48 normal (36 characteristic)
C1, C2, C2, C4, C22, C22, C22, C8, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C22:C4, C4:C4, C4:C4, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C2.C42, C2.C42, C22:C8, C22:C8, C42:C2, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C2xM4(2), C23xC4, C23xC4, C2xC4oD4, C22.SD16, C23.31D4, C2xC2.C42, C24.4C4, C22.19C24, C24.150D4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C23:C4, C4wrC2, C2xC22:C4, C8:C22, C8.C22, C2xC23:C4, C23.36D4, C2xC4wrC2, C24.150D4

Permutation representations of C24.150D4
►On 16 points - transitive group 16T254
Generators in S16
(2 11)(4 13)(6 15)(8 9)

(2 15)(4 9)(6 11)(8 13)

(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)

(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(2 13 15 8)(3 12)(4 6 9 11)(7 16)
 
G:=sub<Sym(16)| (2,11)(4,13)(6,15)(8,9), (2,15)(4,9)(6,11)(8,13), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,13,15,8)(3,12)(4,6,9,11)(7,16)>;
 
G:=Group( (2,11)(4,13)(6,15)(8,9), (2,15)(4,9)(6,11)(8,13), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,13,15,8)(3,12)(4,6,9,11)(7,16) );
 
G=PermutationGroup([[(2,11),(4,13),(6,15),(8,9)], [(2,15),(4,9),(6,11),(8,13)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,13,15,8),(3,12),(4,6,9,11),(7,16)]])
 
G:=TransitiveGroup(16,254);
 

32 conjugacy classes

class 1 2A2B2C2D···2I2J4A4B4C4D4E···4N4O4P4Q8A8B8C8D
order12222···2244444···44448888
size11112···2822224···48888888

32 irreducible representations

dim1111111111222444
type++++++++++-
imageC1C2C2C2C2C2C4C4C4C4D4D4C4wrC2C23:C4C8:C22C8.C22
kernelC24.150D4C22.SD16C23.31D4C2xC2.C42C24.4C4C22.19C24C42:C2C4:D4C22:Q8C2xC4oD4C22xC4C24C22C22C22C22
# reps1221112222318211

Matrix representation of C24.150D4 ►in GL6(F17)

100000
010000
001000
000100
0000160
0000016
,
100000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
1600000
0160000
0016000
0001600
0000160
0000016
,
040000
100000
0000016
0000160
0016000
000100
,
1600000
0130000
0016000
000100
0000016
0000160

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

C24.150D4 in GAP, Magma, Sage, TeX 

C_2^4._{150}D_4
 
% in TeX
 
G:=Group("C2^4.150D4");
 
// GroupNames label
 
G:=SmallGroup(128,236);
 
// by ID
 
G=gap.SmallGroup(128,236);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,387,1123,1018,248,1971]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*d*e^3>;
 
// generators/relations
 

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