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G = C42.63D4  order 128 = 27

45th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.63D4, (C4×D4)⋊8C4, (C4×Q8)⋊8C4, C42.C2⋊6C4, C42⋊5C4⋊2C2, C4.4D4⋊10C4, C42.80(C2×C4), C23.511(C2×D4), (C22×C4).222D4, C22.19(C4○D8), C23.31D4⋊3C2, C22.SD16.1C2, C42.12C4⋊18C2, C4⋊D4.143C22, C22⋊C8.169C22, (C2×C42).183C22, (C22×C4).643C23, C22⋊Q8.148C22, C2.26(C42⋊C22), C2.11(C23.24D4), C2.C42.10C22, C23.36C23.8C2, C2.23(C23.C23), C4⋊C4.21(C2×C4), (C2×D4).16(C2×C4), (C2×Q8).16(C2×C4), (C2×C4).1167(C2×D4), (C2×C4).133(C22×C4), (C2×C4).319(C22⋊C4), C22.197(C2×C22⋊C4), SmallGroup(128,253)

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

Derived series
C1 — C2×C4 — C42.63D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.36C23 — C42.63D4
Lower central
C1 — C22 — C2×C4 — C42.63D4
Upper central
C1 — C22 — C2×C42 — C42.63D4
Jennings
C1 — C2 — C22 — C22×C4 — C42.63D4

Generators and relations for C42.63D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 236 in 109 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2.C42, C2.C42, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22.SD16, C23.31D4, C42⋊5C4, C42.12C4, C23.36C23, C42.63D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C4○D8, C23.C23, C23.24D4, C42⋊C22, C42.63D4

Smallest permutation representation of C42.63D4
►On 32 points
Generators in S32
(1 16 31 22)(2 23 32 9)(3 10 25 24)(4 17 26 11)(5 12 27 18)(6 19 28 13)(7 14 29 20)(8 21 30 15)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A···4H4I4J4K···4Q8A···8H
order12222224···4444···48···8
size11112282···2448···84···4

32 irreducible representations

dim111111111122244
type++++++++
imageC1C2C2C2C2C2C4C4C4C4D4D4C4○D8C23.C23C42⋊C22
kernelC42.63D4C22.SD16C23.31D4C42⋊5C4C42.12C4C23.36C23C4×D4C4×Q8C4.4D4C42.C2C42C22×C4C22C2C2
# reps122111222222822

Matrix representation of C42.63D4 ►in GL6(𝔽17)

040000
1300000
000100
0016000
000001
0000160
,
0160000
100000
004000
000400
000040
000004
,
400000
0130000
001000
0001600
0000130
000004
,
5120000
550000
0000130
000004
001000
0001600

G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,4],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,13,0,0,0,0,0,0,4,0,0] >;
 

C42.63D4 in GAP, Magma, Sage, TeX 

C_4^2._{63}D_4
 
% in TeX
 
G:=Group("C4^2.63D4");
 
// GroupNames label
 
G:=SmallGroup(128,253);
 
// by ID
 
G=gap.SmallGroup(128,253);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,387,520,1123,1018,248,1971]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^-1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
 
// generators/relations
 

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