Copied to
clipboard

G = C42:20D4  order 128 = 27

14th semidirect product of C42 and D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42:20D4, C23.441C24, C24.322C23, C22.2302+ 1+4, C2.29D42, C4:C4:26D4, (C2xD4):35D4, C4:2(C4:D4), C4:2(C4:1D4), C23.48(C2xD4), C23:2D4:19C2, C2.23(Q8:6D4), (C23xC4).394C22, (C2xC42).546C22, (C22xC4).834C23, C22.292(C22xD4), C24.3C22:54C2, (C22xD4).163C22, C2.21(C22.29C24), C2.C42.547C22, (C2xC4xD4):45C2, (C4xC4:C4):86C2, (C2xC4:1D4):6C2, (C2xC4).72(C2xD4), (C2xC4:D4):18C2, C2.11(C2xC4:1D4), C2.34(C2xC4:D4), (C2xC4).893(C4oD4), (C2xC4:C4).866C22, C22.318(C2xC4oD4), (C2xC22:C4).176C22, SmallGroup(128,1273)

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

Derived series
C1 — C23 — C42:20D4
Chief series
C1 — C2 — C22 — C23 — C22xC4 — C23xC4 — C2xC4xD4 — C42:20D4
Lower central
C1 — C23 — C42:20D4
Upper central
C1 — C23 — C42:20D4
Jennings
C1 — C23 — C42:20D4

Generators and relations for C42:20D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=ab2, dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1076 in 474 conjugacy classes, 132 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2xC4, C2xC4, D4, C23, C23, C23, C42, C42, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C24, C2.C42, C2xC42, C2xC42, C2xC22:C4, C2xC4:C4, C2xC4:C4, C4xD4, C4:D4, C4:1D4, C23xC4, C22xD4, C22xD4, C4xC4:C4, C24.3C22, C23:2D4, C2xC4xD4, C2xC4:D4, C2xC4:1D4, C2xC4:1D4, C42:20D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C4:D4, C4:1D4, C22xD4, C2xC4oD4, 2+ 1+4, C2xC4:D4, C2xC4:1D4, C22.29C24, D42, Q8:6D4, C42:20D4

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I···4V
order12···2222222224···44···4
size11···1444488882···24···4

38 irreducible representations

dim111111122224
type+++++++++++
imageC1C2C2C2C2C2C2D4D4D4C4oD42+ 1+4
kernelC42:20D4C4xC4:C4C24.3C22C23:2D4C2xC4xD4C2xC4:D4C2xC4:1D4C42C4:C4C2xD4C2xC4C22
# reps112414348442

Matrix representation of C42:20D4 ►in GL6(F5)

010000
400000
004000
000400
000041
000031
,
400000
040000
001000
000100
000014
000024
,
400000
040000
000100
004000
000020
000043
,
040000
400000
004000
000100
000041
000001

G:=sub<GL(6,GF(5))| [0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,3,0,0,0,0,1,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,4,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,4,0,0,0,0,0,3],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,1,1] >;
 

C42:20D4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{20}D_4
 
% in TeX
 
G:=Group("C4^2:20D4");
 
// GroupNames label
 
G:=SmallGroup(128,1273);
 
// by ID
 
G=gap.SmallGroup(128,1273);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,568,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

׿
x
:
Z
F
o
wr
Q
<