Copied to
clipboard

G = C42.397D4  order 128 = 27

30th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.397D4, C42.598C23, D4:C8:1C2, C4oD4:2C8, D4:5(C2xC8), Q8:C8:1C2, Q8:5(C2xC8), (C4xD4).13C4, (C4xC8).1C22, C4.4(C22xC8), (C4xQ8).13C4, C42.51(C2xC4), C4.4(C2xM4(2)), C4:C8.245C22, C4.23(C22:C8), (C22xC4).654D4, (C2xC4).15M4(2), C4.129(C8:C22), C42:C2.16C4, C42.12C4:6C2, (C4xD4).262C22, C22.3(C22:C8), (C4xQ8).249C22, C4.123(C8.C22), C23.93(C22:C4), (C2xC42).154C22, C2.1(C42:C22), C2.1(C23.36D4), (C2xC4:C8):2C2, (C2xC4).16(C2xC8), (C4xC4oD4).2C2, C4:C4.176(C2xC4), (C2xC4oD4).14C4, C2.13(C2xC22:C8), (C2xD4).190(C2xC4), (C2xC4).1441(C2xD4), (C2xQ8).173(C2xC4), (C22xC4).176(C2xC4), (C2xC4).303(C22xC4), C22.97(C2xC22:C4), (C2xC4).351(C22:C4), SmallGroup(128,209)

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

Derived series
C1 — C4 — C42.397D4
Chief series
C1 — C2 — C22 — C2xC4 — C42 — C2xC42 — C4xC4oD4 — C42.397D4
Lower central
C1 — C2 — C4 — C42.397D4
Upper central
C1 — C2xC4 — C2xC42 — C42.397D4
Jennings
C1 — C22 — C22 — C42 — C42.397D4

Generators and relations for C42.397D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 236 in 132 conjugacy classes, 62 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C4:C4, C4:C4, C2xC8, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C4xC8, C22:C8, C4:C8, C4:C8, C2xC42, C2xC42, C42:C2, C42:C2, C4xD4, C4xD4, C4xQ8, C22xC8, C2xC4oD4, D4:C8, Q8:C8, C2xC4:C8, C42.12C4, C4xC4oD4, C42.397D4
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, C23, C22:C4, C2xC8, M4(2), C22xC4, C2xD4, C22:C8, C2xC22:C4, C22xC8, C2xM4(2), C8:C22, C8.C22, C2xC22:C8, C23.36D4, C42:C22, C42.397D4

Smallest permutation representation of C42.397D4
►On 64 points
Generators in S64
(1 13 55 27)(2 28 56 14)(3 15 49 29)(4 30 50 16)(5 9 51 31)(6 32 52 10)(7 11 53 25)(8 26 54 12)(17 45 61 37)(18 38 62 46)(19 47 63 39)(20 40 64 48)(21 41 57 33)(22 34 58 42)(23 43 59 35)(24 36 60 44)

(1 21 51 61)(2 22 52 62)(3 23 53 63)(4 24 54 64)(5 17 55 57)(6 18 56 58)(7 19 49 59)(8 20 50 60)(9 45 27 33)(10 46 28 34)(11 47 29 35)(12 48 30 36)(13 41 31 37)(14 42 32 38)(15 43 25 39)(16 44 26 40)

(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(1 20 21 50 51 60 61 8)(2 49 22 59 52 7 62 19)(3 58 23 6 53 18 63 56)(4 5 24 17 54 55 64 57)(9 44 45 26 27 40 33 16)(10 25 46 39 28 15 34 43)(11 38 47 14 29 42 35 32)(12 13 48 41 30 31 36 37)
 
G:=sub<Sym(64)| (1,13,55,27)(2,28,56,14)(3,15,49,29)(4,30,50,16)(5,9,51,31)(6,32,52,10)(7,11,53,25)(8,26,54,12)(17,45,61,37)(18,38,62,46)(19,47,63,39)(20,40,64,48)(21,41,57,33)(22,34,58,42)(23,43,59,35)(24,36,60,44), (1,21,51,61)(2,22,52,62)(3,23,53,63)(4,24,54,64)(5,17,55,57)(6,18,56,58)(7,19,49,59)(8,20,50,60)(9,45,27,33)(10,46,28,34)(11,47,29,35)(12,48,30,36)(13,41,31,37)(14,42,32,38)(15,43,25,39)(16,44,26,40), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,20,21,50,51,60,61,8)(2,49,22,59,52,7,62,19)(3,58,23,6,53,18,63,56)(4,5,24,17,54,55,64,57)(9,44,45,26,27,40,33,16)(10,25,46,39,28,15,34,43)(11,38,47,14,29,42,35,32)(12,13,48,41,30,31,36,37)>;
 
G:=Group( (1,13,55,27)(2,28,56,14)(3,15,49,29)(4,30,50,16)(5,9,51,31)(6,32,52,10)(7,11,53,25)(8,26,54,12)(17,45,61,37)(18,38,62,46)(19,47,63,39)(20,40,64,48)(21,41,57,33)(22,34,58,42)(23,43,59,35)(24,36,60,44), (1,21,51,61)(2,22,52,62)(3,23,53,63)(4,24,54,64)(5,17,55,57)(6,18,56,58)(7,19,49,59)(8,20,50,60)(9,45,27,33)(10,46,28,34)(11,47,29,35)(12,48,30,36)(13,41,31,37)(14,42,32,38)(15,43,25,39)(16,44,26,40), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,20,21,50,51,60,61,8)(2,49,22,59,52,7,62,19)(3,58,23,6,53,18,63,56)(4,5,24,17,54,55,64,57)(9,44,45,26,27,40,33,16)(10,25,46,39,28,15,34,43)(11,38,47,14,29,42,35,32)(12,13,48,41,30,31,36,37) );
 
G=PermutationGroup([[(1,13,55,27),(2,28,56,14),(3,15,49,29),(4,30,50,16),(5,9,51,31),(6,32,52,10),(7,11,53,25),(8,26,54,12),(17,45,61,37),(18,38,62,46),(19,47,63,39),(20,40,64,48),(21,41,57,33),(22,34,58,42),(23,43,59,35),(24,36,60,44)], [(1,21,51,61),(2,22,52,62),(3,23,53,63),(4,24,54,64),(5,17,55,57),(6,18,56,58),(7,19,49,59),(8,20,50,60),(9,45,27,33),(10,46,28,34),(11,47,29,35),(12,48,30,36),(13,41,31,37),(14,42,32,38),(15,43,25,39),(16,44,26,40)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,20,21,50,51,60,61,8),(2,49,22,59,52,7,62,19),(3,58,23,6,53,18,63,56),(4,5,24,17,54,55,64,57),(9,44,45,26,27,40,33,16),(10,25,46,39,28,15,34,43),(11,38,47,14,29,42,35,32),(12,13,48,41,30,31,36,37)]])
 

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O···4T8A···8P
order1222222244444···44···48···8
size1111224411112···24···44···4

44 irreducible representations

dim11111111111222444
type+++++++++-
imageC1C2C2C2C2C2C4C4C4C4C8D4D4M4(2)C8:C22C8.C22C42:C22
kernelC42.397D4D4:C8Q8:C8C2xC4:C8C42.12C4C4xC4oD4C42:C2C4xD4C4xQ8C2xC4oD4C4oD4C42C22xC4C2xC4C4C4C2
# reps122111222216224112

Matrix representation of C42.397D4 ►in GL6(F17)

100000
010000
00111615
0000160
000100
00101616
,
400000
040000
0016000
0001600
0000160
0000016
,
010000
1300000
0001304
000044
00441313
0001344
,
010000
400000
0001304
00001313
00131344
00401313

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

C42.397D4 in GAP, Magma, Sage, TeX 

C_4^2._{397}D_4
 
% in TeX
 
G:=Group("C4^2.397D4");
 
// GroupNames label
 
G:=SmallGroup(128,209);
 
// by ID
 
G=gap.SmallGroup(128,209);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,387,1123,570,136,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
 
// generators/relations
 

׿
x
:
Z
F
o
wr
Q
<