Copied to
clipboard

G = C42.275C23  order 128 = 27

136th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.275C23, D8:5(C2xC4), (C4xD8):28C2, C8:C22:5C4, C4.77(C4xD4), D8:C4:2C2, C4:C4.398D4, (C4xC8):21C22, SD16:3(C2xC4), C2.3(D4oD8), (C4xD4):1C22, C8.2(C22xC4), (C4xQ8):1C22, (C4xSD16):13C2, M4(2):9(C2xC4), C4.23(C23xC4), D4.6(C22xC4), C22.16(C4xD4), C2.D8:66C22, C4.Q8:46C22, C8:C4:37C22, SD16:C4:1C2, Q8.6(C22xC4), C4:C4.363C23, C8o2M4(2):4C2, (C2xC4).203C24, (C2xC8).414C23, C22:C4.185D4, C2.4(D4oSD16), C23.435(C2xD4), D4:C4:99C22, C22.11C24:7C2, Q8:C4:92C22, (C2xD8).158C22, (C2xD4).372C23, (C2xQ8).345C23, M4(2):C4:10C2, C23.24D4:38C2, (C22xC8).440C22, (C22xC4).924C23, C22.147(C22xD4), C23.33C23:4C2, C42:C2.80C22, (C2xSD16).109C22, (C22xD4).321C22, (C2xM4(2)).260C22, C2.63(C2xC4xD4), C4oD4:4(C2xC4), (C2xD4):26(C2xC4), C4.11(C2xC4oD4), (C2xC4).910(C2xD4), (C2xC8:C22).9C2, (C2xD4:C4):52C2, (C2xC4).70(C22xC4), (C2xC4).474(C4oD4), (C2xC4:C4).574C22, (C2xC4oD4).87C22, SmallGroup(128,1678)

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

Derived series
C1 — C4 — C42.275C23
Chief series
C1 — C2 — C22 — C2xC4 — C22xC4 — C42:C2 — C22.11C24 — C42.275C23
Lower central
C1 — C2 — C4 — C42.275C23
Upper central
C1 — C22 — C42:C2 — C42.275C23
Jennings
C1 — C2 — C2 — C2xC4 — C42.275C23

Generators and relations for C42.275C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2, ab=ba, ac=ca, ad=da, eae=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ce=ec, de=ed >

Subgroups: 476 in 253 conjugacy classes, 140 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, C2xC8, M4(2), D8, SD16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C24, C4xC8, C8:C4, D4:C4, D4:C4, Q8:C4, C4.Q8, C2.D8, C2xC22:C4, C2xC4:C4, C2xC4:C4, C42:C2, C42:C2, C4xD4, C4xD4, C4xQ8, C22xC8, C2xM4(2), C2xD8, C2xSD16, C8:C22, C22xD4, C2xC4oD4, C8o2M4(2), C2xD4:C4, C23.24D4, M4(2):C4, C4xD8, C4xSD16, SD16:C4, D8:C4, C22.11C24, C23.33C23, C2xC8:C22, C42.275C23
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22xC4, C2xD4, C4oD4, C24, C4xD4, C23xC4, C22xD4, C2xC4oD4, C2xC4xD4, D4oD8, D4oSD16, C42.275C23

Smallest permutation representation of C42.275C23
►On 32 points
Generators in S32
(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 19 15 5)(2 20 16 6)(3 17 13 7)(4 18 14 8)(9 22 25 30)(10 23 26 31)(11 24 27 32)(12 21 28 29)

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

(5 19)(6 20)(7 17)(8 18)(9 22)(10 23)(11 24)(12 21)(25 30)(26 31)(27 32)(28 29)

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

44 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A···4L4M···4V8A8B8C8D8E···8J
order1222222···24···44···488888···8
size1111224···42···24···422224···4

44 irreducible representations

dim111111111111122244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C4D4D4C4oD4D4oD8D4oSD16
kernelC42.275C23C8o2M4(2)C2xD4:C4C23.24D4M4(2):C4C4xD8C4xSD16SD16:C4D8:C4C22.11C24C23.33C23C2xC8:C22C8:C22C22:C4C4:C4C2xC4C2C2
# reps1111122221111622422

Matrix representation of C42.275C23 ►in GL6(F17)

400000
040000
000010
000001
0016000
0001600
,
1600000
0160000
000100
0016000
000001
0000160
,
1020000
970000
0012500
005500
0000125
000055
,
100000
7160000
001000
0001600
000010
0000016
,
1600000
0160000
0016000
0001600
000010
000001

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[10,9,0,0,0,0,2,7,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,0,0,0,0,12,5,0,0,0,0,5,5],[1,7,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C42.275C23 in GAP, Magma, Sage, TeX 

C_4^2._{275}C_2^3
 
% in TeX
 
G:=Group("C4^2.275C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1678);
 
// by ID
 
G=gap.SmallGroup(128,1678);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,521,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

׿
x
:
Z
F
o
wr
Q
<