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G = C24.63D4  order 128 = 27

18th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.63D4, C42⋊6(C2×C4), (C2×C4).19C42, C4.15(C2×C42), C42⋊C2⋊14C4, C42⋊6C4⋊10C2, (C22×C4).35Q8, C23.22(C4⋊C4), M4(2)⋊19(C2×C4), (C2×M4(2))⋊13C4, (C22×C4).254D4, C23.537(C2×D4), (C23×C4).217C22, (C2×C42).231C22, C23.190(C22⋊C4), C4.16(C2.C42), (C22×C4).1299C23, C2.4(C42⋊C22), (C22×M4(2)).11C2, C42⋊C2.257C22, (C2×M4(2)).296C22, C22.9(C2.C42), (C2×C4⋊C4)⋊21C4, C4.25(C2×C4⋊C4), C22.9(C2×C4⋊C4), C4⋊C4.185(C2×C4), (C2×C4).38(C4⋊C4), (C2×C4).178(C2×Q8), (C2×C4).1492(C2×D4), C4.103(C2×C22⋊C4), (C2×C4).516(C22×C4), (C22×C4).253(C2×C4), (C2×C42⋊C2).6C2, (C2×C4).251(C22⋊C4), C22.104(C2×C22⋊C4), C2.10(C2×C2.C42), SmallGroup(128,465)

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

Derived series
C1 — C4 — C24.63D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C24.63D4
Lower central
C1 — C2 — C4 — C24.63D4
Upper central
C1 — C2×C4 — C23×C4 — C24.63D4
Jennings
C1 — C2 — C2 — C22×C4 — C24.63D4

Generators and relations for C24.63D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, eae-1=ad=da, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=bcde3 >

Subgroups: 340 in 202 conjugacy classes, 108 normal (32 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C42⋊6C4, C2×C42⋊C2, C22×M4(2), C24.63D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C2.C42, C42⋊C22, C24.63D4

Smallest permutation representation of C24.63D4
►On 32 points
Generators in S32
(1 19)(2 24)(3 21)(4 18)(5 23)(6 20)(7 17)(8 22)(9 26)(10 31)(11 28)(12 25)(13 30)(14 27)(15 32)(16 29)

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

(1 5)(3 7)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J4K···4Z8A···8H
order12222···244444···44···48···8
size11112···211112···24···44···4

44 irreducible representations

dim11111112224
type+++++-+
imageC1C2C2C2C4C4C4D4Q8D4C42⋊C22
kernelC24.63D4C42⋊6C4C2×C42⋊C2C22×M4(2)C2×C4⋊C4C42⋊C2C2×M4(2)C22×C4C22×C4C24C2
# reps142141285214

Matrix representation of C24.63D4 ►in GL6(𝔽17)

1600000
0160000
00161500
000100
000101
0001610
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
001010
0016001
,
100000
010000
0016000
0001600
0000160
0000016
,
410000
2130000
0016002
0010116
00713016
006001
,
13160000
040000
00160150
000011
000010
0001160

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,15,1,1,16,0,0,0,0,0,1,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,1,16,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[4,2,0,0,0,0,1,13,0,0,0,0,0,0,16,1,7,6,0,0,0,0,13,0,0,0,0,1,0,0,0,0,2,16,16,1],[13,0,0,0,0,0,16,4,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,1,1,16,0,0,0,1,0,0] >;
 

C24.63D4 in GAP, Magma, Sage, TeX 

C_2^4._{63}D_4
 
% in TeX
 
G:=Group("C2^4.63D4");
 
// GroupNames label
 
G:=SmallGroup(128,465);
 
// by ID
 
G=gap.SmallGroup(128,465);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,2019,248,4037]);
 
// 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,a*c=c*a,e*a*e^-1=a*d=d*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*c*d*e^3>;
 
// generators/relations
 

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