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G = C42.498C23order 128 = 27

359th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.498C23, C4.922- 1+4, C86D425C2, C8⋊D456C2, C4⋊C4.386D4, D43Q89C2, C82Q832C2, D42Q824C2, C4.Q1641C2, (C4×SD16)⋊24C2, (C2×D4).336D4, C8.19(C4○D4), C2.60(D4○D8), C22⋊C4.69D4, C4⋊C4.262C23, C4⋊C8.130C22, (C2×C8).113C23, (C4×C8).199C22, (C2×C4).549C24, C23.354(C2×D4), C4⋊Q8.178C22, C2.D8.69C22, (C4×D4).189C22, (C2×D4).264C23, C22.D836C2, C4.50(C8.C22), (C2×Q8).249C23, (C4×Q8).186C22, C2.102(D46D4), M4(2)⋊C444C2, C4.Q8.140C22, C23.20D451C2, C4⋊D4.113C22, C22⋊C8.108C22, (C22×C4).349C23, Q8⋊C4.85C22, C22.809(C22×D4), C22⋊Q8.112C22, D4⋊C4.171C22, (C2×SD16).122C22, C42⋊C2.220C22, (C2×M4(2)).142C22, C22.49C24.6C2, C4.131(C2×C4○D4), (C2×C4).633(C2×D4), C2.85(C2×C8.C22), (C2×C4⋊C4).698C22, SmallGroup(128,2089)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.498C23
C1C2C4C2×C4C22×C4C2×C4⋊C4D43Q8 — C42.498C23
C1C2C2×C4 — C42.498C23
C1C22C4×D4 — C42.498C23
C1C2C2C2×C4 — C42.498C23

Generators and relations for C42.498C23
 G = < a,b,c,d,e | a4=b4=1, c2=d2=a2, 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 177 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×9], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×15], D4 [×6], Q8 [×4], C23 [×2], C23, C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8, C2×Q8 [×2], C4×C8, C22⋊C8 [×2], D4⋊C4, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4 [×2], C4⋊C8, C4.Q8, C4.Q8 [×2], C2.D8 [×6], C2×C4⋊C4 [×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], C42.C2, C4⋊Q8 [×2], C2×M4(2) [×2], C2×SD16, M4(2)⋊C4 [×2], C86D4, C4×SD16, C8⋊D4 [×2], D42Q8, C4.Q16, C22.D8 [×2], C23.20D4 [×2], C82Q8, D43Q8, C22.49C24, C42.498C23
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, D46D4, C2×C8.C22, D4○D8, C42.498C23

Character table of C42.498C23

 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
ρ212-22-20000-220-2i2i000-2i2i00000200-200    complex lifted from C4○D4
ρ222-22-20000-2202i2i000-2i-2i00000-200200    complex lifted from C4○D4
ρ232-22-20000-2202i-2i0002i-2i00000200-200    complex lifted from C4○D4
ρ242-22-20000-220-2i-2i0002i2i00000-200200    complex lifted from C4○D4
ρ2544-4-40000000000000000000022-22000    orthogonal lifted from D4○D8
ρ2644-4-400000000000000000000-2222000    orthogonal lifted from D4○D8
ρ274-44-400004-40000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ284-4-44000-4004000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ294-4-44000400-4000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C42.498C23
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 20 39 23)(2 17 40 24)(3 18 37 21)(4 19 38 22)(5 10 61 29)(6 11 62 30)(7 12 63 31)(8 9 64 32)(13 53 26 58)(14 54 27 59)(15 55 28 60)(16 56 25 57)(33 43 49 46)(34 44 50 47)(35 41 51 48)(36 42 52 45)
(1 16 3 14)(2 15 4 13)(5 33 7 35)(6 36 8 34)(9 47 11 45)(10 46 12 48)(17 60 19 58)(18 59 20 57)(21 54 23 56)(22 53 24 55)(25 37 27 39)(26 40 28 38)(29 43 31 41)(30 42 32 44)(49 63 51 61)(50 62 52 64)
(1 50 3 52)(2 35 4 33)(5 58 7 60)(6 54 8 56)(9 16 11 14)(10 26 12 28)(13 31 15 29)(17 48 19 46)(18 42 20 44)(21 45 23 47)(22 43 24 41)(25 30 27 32)(34 37 36 39)(38 49 40 51)(53 63 55 61)(57 62 59 64)
(1 25 39 16)(2 15 40 28)(3 27 37 14)(4 13 38 26)(5 41 61 48)(6 47 62 44)(7 43 63 46)(8 45 64 42)(9 36 32 52)(10 51 29 35)(11 34 30 50)(12 49 31 33)(17 55 24 60)(18 59 21 54)(19 53 22 58)(20 57 23 56)

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

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

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

Matrix representation of C42.498C23 in GL6(𝔽17)

400000
4130000
0000016
000010
000100
0016000
,
100000
010000
000100
0016000
000001
0000160
,
490000
0130000
000100
001000
0000016
0000160
,
1300000
0130000
0031400
00141400
0000314
00001414
,
490000
4130000
000100
0016000
0000016
000010

G:=sub<GL(6,GF(17))| [4,4,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[4,0,0,0,0,0,9,13,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,0,0,0,3,14,0,0,0,0,14,14],[4,4,0,0,0,0,9,13,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0] >;

C42.498C23 in GAP, Magma, Sage, TeX

C_4^2._{498}C_2^3
% in TeX

G:=Group("C4^2.498C2^3");
// GroupNames label

G:=SmallGroup(128,2089);
// by ID

G=gap.SmallGroup(128,2089);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,436,2019,346,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=d^2=a^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

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Character table of C42.498C23 in TeX

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