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G = C42.278C23order 128 = 27

139th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.278C23, C22:C4oSD16, (C4xC8):47C22, C4:C4.401D4, C4.102(C4xD4), (C4xQ8):2C22, (C2xSD16):10C4, SD16:11(C2xC4), (C4xSD16):51C2, C8.21(C22xC4), C4.26(C23xC4), D4.9(C22xC4), C22.69(C4xD4), C8:C4:39C22, C2.D8:67C22, C4.Q8:74C22, SD16:C4:3C2, Q8.8(C22xC4), C4:C4.366C23, C8o2M4(2):7C2, (C2xC8).417C23, (C2xC4).206C24, C22:C4.188D4, C2.6(D4oSD16), (C4xD4).58C22, C23.438(C2xD4), Q8:C4:93C22, (C2xD4).375C23, (C2xQ8).347C23, (C22xSD16).4C2, C23.25D4:25C2, C23.38D4:32C2, C22.11C24.7C2, (C22xC8).249C22, (C22xC4).927C23, C22.150(C22xD4), D4:C4.197C22, C23.32C23:6C2, C42:C2.83C22, (C2xSD16).111C22, C23.37D4.10C2, (C22xD4).323C22, (C22xQ8).259C22, (C2xM4(2)).353C22, C2.66(C2xC4xD4), (C2xC8):15(C2xC4), (C2xQ8):21(C2xC4), C4.14(C2xC4oD4), (C2xC4).913(C2xD4), (C2xD4).138(C2xC4), (C2xC4).265(C4oD4), (C2xC4).265(C22xC4), SmallGroup(128,1681)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C42.278C23
C1C2C22C2xC4C22xC4C42:C2C23.32C23 — C42.278C23
C1C2C4 — C42.278C23
C1C22C42:C2 — C42.278C23
C1C2C2C2xC4 — C42.278C23

Generators and relations for C42.278C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=b2, ab=ba, ac=ca, ad=da, eae=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=b2c, de=ed >

Subgroups: 436 in 246 conjugacy classes, 140 normal (26 characteristic)
C1, C2, 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, C2xC8, C2xC8, M4(2), SD16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C24, C4xC8, C8:C4, D4:C4, Q8:C4, C4.Q8, C2.D8, C2xC22:C4, C42:C2, C42:C2, C4xD4, C4xD4, C4xQ8, C4xQ8, C22xC8, C2xM4(2), C2xSD16, C22xD4, C22xQ8, C8o2M4(2), C23.37D4, C23.38D4, C23.25D4, C4xSD16, SD16:C4, C22.11C24, C23.32C23, C22xSD16, C42.278C23
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22xC4, C2xD4, C4oD4, C24, C4xD4, C23xC4, C22xD4, C2xC4oD4, C2xC4xD4, D4oSD16, C42.278C23

Smallest permutation representation of C42.278C23
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 4 3 2)(5 18 7 20)(6 19 8 17)(9 29 11 31)(10 30 12 32)(13 16 15 14)(21 27 23 25)(22 28 24 26)
(1 31 15 23)(2 32 16 24)(3 29 13 21)(4 30 14 22)(5 10 19 26)(6 11 20 27)(7 12 17 28)(8 9 18 25)
(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), (1,4,3,2)(5,18,7,20)(6,19,8,17)(9,29,11,31)(10,30,12,32)(13,16,15,14)(21,27,23,25)(22,28,24,26), (1,31,15,23)(2,32,16,24)(3,29,13,21)(4,30,14,22)(5,10,19,26)(6,11,20,27)(7,12,17,28)(8,9,18,25), (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), (1,4,3,2)(5,18,7,20)(6,19,8,17)(9,29,11,31)(10,30,12,32)(13,16,15,14)(21,27,23,25)(22,28,24,26), (1,31,15,23)(2,32,16,24)(3,29,13,21)(4,30,14,22)(5,10,19,26)(6,11,20,27)(7,12,17,28)(8,9,18,25), (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)], [(1,4,3,2),(5,18,7,20),(6,19,8,17),(9,29,11,31),(10,30,12,32),(13,16,15,14),(21,27,23,25),(22,28,24,26)], [(1,31,15,23),(2,32,16,24),(3,29,13,21),(4,30,14,22),(5,10,19,26),(6,11,20,27),(7,12,17,28),(8,9,18,25)], [(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 2A2B2C2D2E2F2G2H2I4A···4L4M···4X8A8B8C8D8E···8J
order12222222224···44···488888···8
size11112244442···24···422224···4

44 irreducible representations

dim111111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C4D4D4C4oD4D4oSD16
kernelC42.278C23C8o2M4(2)C23.37D4C23.38D4C23.25D4C4xSD16SD16:C4C22.11C24C23.32C23C22xSD16C2xSD16C22:C4C4:C4C2xC4C2
# reps1111144111162244

Matrix representation of C42.278C23 in GL6(F17)

400000
040000
00160150
0000161
000010
000110
,
1600000
0160000
0011500
0011600
000101
00161160
,
1300000
240000
00160150
001601616
000010
0011610
,
16130000
010000
0010700
005700
000555
00125512
,
100000
010000
001000
000100
00160160
00160016

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,16,1,1,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,16,0,0,15,16,1,1,0,0,0,0,0,16,0,0,0,0,1,0],[13,2,0,0,0,0,0,4,0,0,0,0,0,0,16,16,0,1,0,0,0,0,0,16,0,0,15,16,1,1,0,0,0,16,0,0],[16,0,0,0,0,0,13,1,0,0,0,0,0,0,10,5,0,12,0,0,7,7,5,5,0,0,0,0,5,5,0,0,0,0,5,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,16,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;

C42.278C23 in GAP, Magma, Sage, TeX

C_4^2._{278}C_2^3
% in TeX

G:=Group("C4^2.278C2^3");
// GroupNames label

G:=SmallGroup(128,1681);
// by ID

G=gap.SmallGroup(128,1681);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,184,2019,521,2804,1411,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2,d^2=b^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^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e=b^2*c,d*e=e*d>;
// generators/relations

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