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G = C42.478C23order 128 = 27

339th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.478C23, C4.742+ 1+4, (D4×Q8)⋊9C2, C8⋊D447C2, C86D416C2, C4⋊C4.374D4, C4.Q1636C2, Q8⋊D424C2, (C4×SD16)⋊58C2, (C2×D4).324D4, C22⋊C4.57D4, D4.D423C2, C4⋊C4.421C23, C4⋊C8.111C22, (C2×C4).521C24, (C4×C8).293C22, (C2×C8).357C23, Q8.31(C4○D4), C4.SD1634C2, C23.338(C2×D4), C4⋊Q8.156C22, C2.83(D4○SD16), (C2×D4).244C23, (C4×D4).170C22, C4⋊D4.93C22, C4.48(C8.C22), C22⋊C8.89C22, (C4×Q8).166C22, (C2×Q8).400C23, C2.157(D45D4), C2.D8.125C22, C4.Q8.168C22, C22⋊Q8.92C22, C23.38D419C2, C23.20D442C2, (C22×C4).334C23, C22.781(C22×D4), D4⋊C4.208C22, Q8⋊C4.116C22, (C2×SD16).161C22, (C22×Q8).350C22, C42⋊C2.199C22, (C2×M4(2)).123C22, C22.49C24.3C2, C4.246(C2×C4○D4), (C2×C4).614(C2×D4), C2.79(C2×C8.C22), SmallGroup(128,2061)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.478C23
C1C2C4C2×C4C22×C4C22×Q8D4×Q8 — C42.478C23
C1C2C2×C4 — C42.478C23
C1C22C4×D4 — C42.478C23
C1C2C2C2×C4 — C42.478C23

Generators and relations for C42.478C23
 G = < a,b,c,d,e | a4=b4=1, c2=a2, 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: 376 in 193 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×17], D4 [×6], Q8 [×2], Q8 [×10], C23 [×2], C23, C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], SD16 [×4], C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8, C2×Q8 [×2], C2×Q8 [×9], C4×C8, C22⋊C8 [×2], D4⋊C4, Q8⋊C4, Q8⋊C4 [×8], C4⋊C8, C4.Q8, C2.D8 [×2], C42⋊C2 [×2], C42⋊C2, C4×D4 [×2], C4×D4, C4×Q8, C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C4⋊Q8, C2×M4(2) [×2], C2×SD16, C2×SD16 [×2], C22×Q8 [×2], C23.38D4 [×2], C86D4, C4×SD16, Q8⋊D4 [×2], D4.D4, C8⋊D4 [×2], C4.Q16, C23.20D4 [×2], C4.SD16, D4×Q8, C22.49C24, C42.478C23
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, D4○SD16, C42.478C23

Character table of C42.478C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D8E8F
 size 11114482222444444488888444488
ρ111111111111111111111111111111    trivial
ρ21111-111-111-1-1-1-1-11-1-1-1111-11-1-11-11    linear of order 2
ρ31111-1-11111111-11-111-1-11-1-11111-1-1    linear of order 2
ρ411111-11-111-1-1-11-1-1-1-11-11-111-1-111-1    linear of order 2
ρ511111-1-1-111-1111-1-111-11-11-11-1-111-1    linear of order 2
ρ61111-1-1-11111-1-1-11-1-1-111-1111111-1-1    linear of order 2
ρ71111-11-1-111-111-1-11111-1-1-111-1-11-11    linear of order 2
ρ8111111-11111-1-1111-1-1-1-1-1-1-1111111    linear of order 2
ρ9111111-11111-111111-1111-1-1-1-1-1-1-1-1    linear of order 2
ρ101111-11-1-111-11-1-1-11-11-111-11-111-11-1    linear of order 2
ρ111111-1-1-11111-11-11-11-1-1-1111-1-1-1-111    linear of order 2
ρ1211111-1-1-111-11-11-1-1-111-111-1-111-1-11    linear of order 2
ρ1311111-11-111-1-111-1-11-1-11-1-11-111-1-11    linear of order 2
ρ141111-1-1111111-1-11-1-1111-1-1-1-1-1-1-111    linear of order 2
ρ151111-111-111-1-11-1-111-11-1-11-1-111-11-1    linear of order 2
ρ16111111111111-1111-11-1-1-111-1-1-1-1-1-1    linear of order 2
ρ172222-220-2-2-2-20022-20000000000000    orthogonal lifted from D4
ρ182222-2-202-2-22002-220000000000000    orthogonal lifted from D4
ρ1922222-20-2-2-2-200-2220000000000000    orthogonal lifted from D4
ρ2022222202-2-2200-2-2-20000000000000    orthogonal lifted from D4
ρ212-22-20000-220-2i2000-22i00000-2i002i00    complex lifted from C4○D4
ρ222-22-20000-2202i2000-2-2i000002i00-2i00    complex lifted from C4○D4
ρ232-22-20000-2202i-20002-2i00000-2i002i00    complex lifted from C4○D4
ρ242-22-20000-220-2i-200022i000002i00-2i00    complex lifted from C4○D4
ρ254-44-400004-40000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-44000400-4000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-4-44000-4004000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-400000000000000000000-2-22-2000    complex lifted from D4○SD16
ρ2944-4-4000000000000000000002-2-2-2000    complex lifted from D4○SD16

Smallest permutation representation of C42.478C23
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 60 34 23)(2 57 35 24)(3 58 36 21)(4 59 33 22)(5 12 47 26)(6 9 48 27)(7 10 45 28)(8 11 46 25)(13 18 50 53)(14 19 51 54)(15 20 52 55)(16 17 49 56)(29 61 39 44)(30 62 40 41)(31 63 37 42)(32 64 38 43)
(1 49 3 51)(2 52 4 50)(5 40 7 38)(6 39 8 37)(9 61 11 63)(10 64 12 62)(13 35 15 33)(14 34 16 36)(17 58 19 60)(18 57 20 59)(21 54 23 56)(22 53 24 55)(25 42 27 44)(26 41 28 43)(29 46 31 48)(30 45 32 47)
(1 28 34 10)(2 11 35 25)(3 26 36 12)(4 9 33 27)(5 58 47 21)(6 22 48 59)(7 60 45 23)(8 24 46 57)(13 39 50 29)(14 30 51 40)(15 37 52 31)(16 32 49 38)(17 43 56 64)(18 61 53 44)(19 41 54 62)(20 63 55 42)
(1 16 34 49)(2 52 35 15)(3 14 36 51)(4 50 33 13)(5 41 47 62)(6 61 48 44)(7 43 45 64)(8 63 46 42)(9 39 27 29)(10 32 28 38)(11 37 25 31)(12 30 26 40)(17 23 56 60)(18 59 53 22)(19 21 54 58)(20 57 55 24)

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

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

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

Matrix representation of C42.478C23 in GL8(𝔽17)

503110000
5123140000
311500000
3145120000
00000503
0000120140
0000014012
00003050
,
160000000
016000000
001600000
000160000
00000100
000016000
00000001
000000160
,
57300000
012030000
14012100000
014050000
0000014012
0000140120
00000503
00005030
,
00100000
00010000
10000000
01000000
000012500
00005500
000000125
00000055
,
12101400000
1251430000
14012100000
1431250000
0000014012
00003050
00000503
0000120140

G:=sub<GL(8,GF(17))| [5,5,3,3,0,0,0,0,0,12,11,14,0,0,0,0,3,3,5,5,0,0,0,0,11,14,0,12,0,0,0,0,0,0,0,0,0,12,0,3,0,0,0,0,5,0,14,0,0,0,0,0,0,14,0,5,0,0,0,0,3,0,12,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[5,0,14,0,0,0,0,0,7,12,0,14,0,0,0,0,3,0,12,0,0,0,0,0,0,3,10,5,0,0,0,0,0,0,0,0,0,14,0,5,0,0,0,0,14,0,5,0,0,0,0,0,0,12,0,3,0,0,0,0,12,0,3,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,5,5],[12,12,14,14,0,0,0,0,10,5,0,3,0,0,0,0,14,14,12,12,0,0,0,0,0,3,10,5,0,0,0,0,0,0,0,0,0,3,0,12,0,0,0,0,14,0,5,0,0,0,0,0,0,5,0,14,0,0,0,0,12,0,3,0] >;

C42.478C23 in GAP, Magma, Sage, TeX

C_4^2._{478}C_2^3
% in TeX

G:=Group("C4^2.478C2^3");
// GroupNames label

G:=SmallGroup(128,2061);
// by ID

G=gap.SmallGroup(128,2061);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,352,2019,346,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2,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 C42.478C23 in TeX

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