Copied to
clipboard

G = C42.477C23order 128 = 27

338th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.477C23, C4.732+ 1+4, (D4×Q8).7C2, C4⋊C4.373D4, (C4×Q16)⋊43C2, C86D4.5C2, Q8⋊Q819C2, C42Q1640C2, (C2×D4).323D4, C8.D429C2, C2.52(Q8○D8), C22⋊C4.56D4, C4⋊C4.420C23, C4⋊C8.110C22, (C2×C4).520C24, (C4×C8).227C22, (C2×C8).192C23, Q8.30(C4○D4), C22⋊Q1634C2, C4.SD1620C2, C23.337(C2×D4), C4⋊Q8.155C22, C4.Q8.62C22, (C4×D4).169C22, C4.77(C8.C22), C22⋊C8.88C22, (C4×Q8).165C22, (C2×Q8).229C23, C2.156(D45D4), C2.D8.193C22, C22⋊Q8.91C22, C23.38D418C2, C23.20D441C2, (C22×C4).333C23, Q8⋊C4.15C22, (C2×Q16).136C22, C22.780(C22×D4), (C22×Q8).349C22, C42⋊C2.198C22, (C2×M4(2)).122C22, C22.50C24.4C2, C4.245(C2×C4○D4), (C2×C4).613(C2×D4), C2.78(C2×C8.C22), SmallGroup(128,2060)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.477C23
C1C2C4C2×C4C22×C4C22×Q8D4×Q8 — C42.477C23
C1C2C2×C4 — C42.477C23
C1C22C4×D4 — C42.477C23
C1C2C2C2×C4 — C42.477C23

Generators and relations for C42.477C23
 G = < a,b,c,d,e | a4=b4=1, c2=a2b2, d2=e2=b2, ab=ba, cac-1=a-1, dad-1=ab2, eae-1=a-1b2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, ede-1=b2d >

Subgroups: 336 in 187 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×2], Q8 [×2], Q8 [×13], C23 [×2], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], Q16 [×4], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×9], C4×C8, C22⋊C8 [×2], Q8⋊C4 [×2], Q8⋊C4 [×8], C4⋊C8, C4.Q8 [×2], C2.D8, C42⋊C2 [×2], C4×D4, C4×D4, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×4], C22⋊Q8 [×2], C4.4D4, C422C2 [×2], C4⋊Q8 [×2], C4⋊Q8, C2×M4(2) [×2], C2×Q16, C2×Q16 [×2], C22×Q8 [×2], C23.38D4 [×2], C86D4, C4×Q16, C22⋊Q16 [×2], C42Q16, C8.D4 [×2], Q8⋊Q8, C23.20D4 [×2], C4.SD16, D4×Q8, C22.50C24, C42.477C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C8.C22 [×2], C22×D4, C2×C4○D4, 2+ 1+4, D45D4, C2×C8.C22, Q8○D8, C42.477C23

Character table of C42.477C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q8A8B8C8D8E8F
 size 11114422224444444888888444488
ρ111111111111111111111111111111    trivial
ρ211111-1-111-11-1-11-1-111-1-11-11-111-11-1    linear of order 2
ρ31111-1-1111111-1-1111-11-1-1-111111-1-1    linear of order 2
ρ41111-11-111-11-11-1-1-11-1-11-111-111-1-11    linear of order 2
ρ51111111111-1-111-11-1-1-1-1-1-1-1111111    linear of order 2
ρ611111-1-111-1-11-111-1-1-111-11-1-111-11-1    linear of order 2
ρ71111-1-11111-1-1-1-1-11-11-1111-11111-1-1    linear of order 2
ρ81111-11-111-1-111-11-1-111-11-1-1-111-1-11    linear of order 2
ρ91111-1-11111-11-1-111-11-11-1-11-1-1-1-111    linear of order 2
ρ101111-11-111-1-1-11-1-1-1-111-1-1111-1-111-1    linear of order 2
ρ111111111111-111111-1-1-1-1111-1-1-1-1-1-1    linear of order 2
ρ1211111-1-111-1-1-1-11-1-1-1-1111-111-1-11-11    linear of order 2
ρ131111-1-111111-1-1-1-111-11-111-1-1-1-1-111    linear of order 2
ρ141111-11-111-1111-11-11-1-111-1-11-1-111-1    linear of order 2
ρ1511111111111-111-111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ1611111-1-111-111-111-111-1-1-11-11-1-11-11    linear of order 2
ρ172222-22-2-2-2-200-22020000000000000    orthogonal lifted from D4
ρ182222222-2-2200-2-20-20000000000000    orthogonal lifted from D4
ρ1922222-2-2-2-2-2002-2020000000000000    orthogonal lifted from D4
ρ202222-2-22-2-2200220-20000000000000    orthogonal lifted from D4
ρ212-22-2000-220-2i-200202i0000002i00-2i00    complex lifted from C4○D4
ρ222-22-2000-2202i200-20-2i0000002i00-2i00    complex lifted from C4○D4
ρ232-22-2000-2202i-20020-2i000000-2i002i00    complex lifted from C4○D4
ρ242-22-2000-220-2i200-202i000000-2i002i00    complex lifted from C4○D4
ρ254-44-40004-400000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-4400400-40000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-4-4400-40040000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-400000000000000000000-2222000    symplectic lifted from Q8○D8, Schur index 2
ρ2944-4-40000000000000000000022-22000    symplectic lifted from Q8○D8, Schur index 2

Smallest permutation representation of C42.477C23
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 23 19 26)(2 24 20 27)(3 21 17 28)(4 22 18 25)(5 62 12 16)(6 63 9 13)(7 64 10 14)(8 61 11 15)(29 43 36 40)(30 44 33 37)(31 41 34 38)(32 42 35 39)(45 53 49 57)(46 54 50 58)(47 55 51 59)(48 56 52 60)
(1 58 17 56)(2 57 18 55)(3 60 19 54)(4 59 20 53)(5 44 10 39)(6 43 11 38)(7 42 12 37)(8 41 9 40)(13 36 61 31)(14 35 62 30)(15 34 63 29)(16 33 64 32)(21 52 26 46)(22 51 27 45)(23 50 28 48)(24 49 25 47)
(1 39 19 42)(2 43 20 40)(3 37 17 44)(4 41 18 38)(5 46 12 50)(6 51 9 47)(7 48 10 52)(8 49 11 45)(13 59 63 55)(14 56 64 60)(15 57 61 53)(16 54 62 58)(21 33 28 30)(22 31 25 34)(23 35 26 32)(24 29 27 36)
(1 26 19 23)(2 22 20 25)(3 28 17 21)(4 24 18 27)(5 14 12 64)(6 63 9 13)(7 16 10 62)(8 61 11 15)(29 41 36 38)(30 37 33 44)(31 43 34 40)(32 39 35 42)(45 53 49 57)(46 60 50 56)(47 55 51 59)(48 58 52 54)

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

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

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

Matrix representation of C42.477C23 in GL8(𝔽17)

01000000
160000000
00010000
001600000
000000016
00000010
00000100
000016000
,
160000000
016000000
001600000
000160000
000001600
00001000
000000016
00000010
,
012030000
120300000
014050000
140500000
00004000
000001300
000000130
00000004
,
00100000
00010000
10000000
01000000
000051200
0000121200
000000512
0000001212
,
160000000
01000000
001600000
00010000
00000100
000016000
000000016
00000010

G:=sub<GL(8,GF(17))| [0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[0,12,0,14,0,0,0,0,12,0,14,0,0,0,0,0,0,3,0,5,0,0,0,0,3,0,5,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0] >;

C42.477C23 in GAP, Magma, Sage, TeX

C_4^2._{477}C_2^3
% in TeX

G:=Group("C4^2.477C2^3");
// GroupNames label

G:=SmallGroup(128,2060);
// by ID

G=gap.SmallGroup(128,2060);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,723,352,2019,346,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.477C23 in TeX

׿
×
𝔽