Copied to
clipboard

G = C42.696C23order 128 = 27

111st non-split extension by C42 of C23 acting via C23/C22=C2

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

Aliases: C42.696C23, C4.1202- 1+4, (C8×Q8)⋊34C2, C4⋊Q8.35C4, C84Q841C2, C4.22(C8○D4), C22⋊Q8.27C4, C4⋊C8.238C22, (C2×C8).442C23, (C4×C8).341C22, C42.230(C2×C4), (C2×C4).682C24, C42.C2.20C4, (C4×Q8).283C22, C8⋊C4.101C22, C2.34(Q8○M4(2)), C22⋊C8.238C22, C42.6C4.34C2, C23.107(C22×C4), (C2×C42).789C22, (C22×C4).946C23, C22.205(C23×C4), C42.12C4.47C2, C42⋊C2.88C22, C42.7C22.3C2, C23.37C23.24C2, C2.25(C23.32C23), C2.33(C2×C8○D4), C4⋊C4.122(C2×C4), C22⋊C4.23(C2×C4), (C2×C4).84(C22×C4), (C2×Q8).166(C2×C4), (C22×C4).361(C2×C4), SmallGroup(128,1717)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.696C23
C1C2C4C2×C4C22×C4C2×C42C23.37C23 — C42.696C23
C1C22 — C42.696C23
C1C2×C4 — C42.696C23
C1C2C2C2×C4 — C42.696C23

Generators and relations for C42.696C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=b, d2=a2, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=a2c, ede=b2d >

Subgroups: 204 in 161 conjugacy classes, 126 normal (28 characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×12], C22, C22 [×3], C8 [×8], C2×C4 [×6], C2×C4 [×8], C2×C4 [×5], Q8 [×6], C23, C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×16], C2×C8 [×8], C22×C4 [×3], C2×Q8 [×4], C4×C8 [×2], C4×C8 [×4], C8⋊C4 [×6], C22⋊C8 [×4], C4⋊C8 [×12], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C42.12C4, C42.6C4, C42.7C22 [×4], C8×Q8 [×2], C84Q8 [×6], C23.37C23, C42.696C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C8○D4 [×2], C23×C4, 2- 1+4 [×2], C23.32C23, C2×C8○D4, Q8○M4(2), C42.696C23

Smallest permutation representation of C42.696C23
On 64 points
Generators in S64
(1 31 55 23)(2 32 56 24)(3 25 49 17)(4 26 50 18)(5 27 51 19)(6 28 52 20)(7 29 53 21)(8 30 54 22)(9 58 38 42)(10 59 39 43)(11 60 40 44)(12 61 33 45)(13 62 34 46)(14 63 35 47)(15 64 36 48)(16 57 37 41)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 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 63 55 47)(2 48 56 64)(3 57 49 41)(4 42 50 58)(5 59 51 43)(6 44 52 60)(7 61 53 45)(8 46 54 62)(9 26 38 18)(10 19 39 27)(11 28 40 20)(12 21 33 29)(13 30 34 22)(14 23 35 31)(15 32 36 24)(16 17 37 25)
(2 56)(4 50)(6 52)(8 54)(9 34)(10 14)(11 36)(12 16)(13 38)(15 40)(18 26)(20 28)(22 30)(24 32)(33 37)(35 39)(41 45)(42 62)(43 47)(44 64)(46 58)(48 60)(57 61)(59 63)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I···4S8A···8H8I···8T
order12222444444444···48···88···8
size11114111122224···42···24···4

44 irreducible representations

dim1111111111244
type+++++++-
imageC1C2C2C2C2C2C2C4C4C4C8○D42- 1+4Q8○M4(2)
kernelC42.696C23C42.12C4C42.6C4C42.7C22C8×Q8C84Q8C23.37C23C22⋊Q8C42.C2C4⋊Q8C4C4C2
# reps1114261844822

Matrix representation of C42.696C23 in GL6(𝔽17)

100000
010000
00161600
002100
009701
00122160
,
1300000
0130000
001000
000100
000010
000001
,
800000
080000
001014150
0071322
0058414
0081277
,
010000
100000
000500
0010000
00111555
00108512
,
100000
0160000
001000
000100
001014160
0007016

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,2,9,12,0,0,16,1,7,2,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,10,7,5,8,0,0,14,13,8,12,0,0,15,2,4,7,0,0,0,2,14,7],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,10,11,10,0,0,5,0,15,8,0,0,0,0,5,5,0,0,0,0,5,12],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,10,0,0,0,0,1,14,7,0,0,0,0,16,0,0,0,0,0,0,16] >;

C42.696C23 in GAP, Magma, Sage, TeX

C_4^2._{696}C_2^3
% in TeX

G:=Group("C4^2.696C2^3");
// GroupNames label

G:=SmallGroup(128,1717);
// by ID

G=gap.SmallGroup(128,1717);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,219,100,675,1018,124]);
// Polycyclic

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

׿
×
𝔽