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G = C42.58C23order 128 = 27

58th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.58C23, C4.792- 1+4, C8⋊Q824C2, C4⋊C4.166D4, C89D4.2C2, Q8.Q840C2, Q8⋊Q821C2, (C2×D4).330D4, C8.32(C4○D4), C8.D431C2, D43Q8.6C2, Q16⋊C425C2, C8.18D431C2, C4⋊C8.117C22, C4⋊C4.249C23, (C2×C8).364C23, (C2×C4).536C24, C22⋊C4.176D4, C23.481(C2×D4), C4⋊Q8.168C22, C2.89(D46D4), C8⋊C4.50C22, C4.Q8.66C22, C2.89(D4○SD16), (C4×D4).176C22, C22⋊C8.95C22, (C4×Q8).177C22, (C2×Q16).88C22, (C2×Q8).240C23, M4(2)⋊C431C2, C2.D8.129C22, C23.48D432C2, C23.20D443C2, C23.47D421C2, (C22×C4).340C23, (C22×C8).287C22, Q8⋊C4.77C22, C22.796(C22×D4), C22.6(C8.C22), C22⋊Q8.103C22, C42.C2.49C22, C42⋊C2.207C22, (C2×M4(2)).129C22, C22.46C24.3C2, (C2×C4.Q8)⋊12C2, C4.118(C2×C4○D4), (C2×C4).620(C2×D4), C2.82(C2×C8.C22), (C2×C4⋊C4).685C22, SmallGroup(128,2076)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.58C23
C1C2C4C2×C4C22×C4C2×C4⋊C4C22.46C24 — C42.58C23
C1C2C2×C4 — C42.58C23
C1C22C4×D4 — C42.58C23
C1C2C2C2×C4 — C42.58C23

Generators and relations for C42.58C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=a2b2, d2=a2, ab=ba, cac-1=eae=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, ede=b2d >

Subgroups: 296 in 172 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×5], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×16], D4 [×2], Q8 [×5], C23 [×2], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×7], C4⋊C4 [×12], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], Q16 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×Q8 [×2], C2×Q8, C8⋊C4, C22⋊C8 [×2], Q8⋊C4 [×6], C4⋊C8, C4.Q8 [×6], C2.D8 [×3], C2×C4⋊C4 [×3], C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C22.D4, C42.C2, C42.C2 [×2], C422C2, C4⋊Q8, C22×C8, C2×M4(2), C2×Q16, C2×C4.Q8, M4(2)⋊C4, C89D4, Q16⋊C4, C8.18D4, C8.D4, Q8⋊Q8, Q8.Q8, C23.47D4 [×2], C23.48D4, C23.20D4, C8⋊Q8, C22.46C24, D43Q8, C42.58C23
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○SD16, C42.58C23

Character table of C42.58C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D8E8F
 size 11112242244444444888888444488
ρ111111111111111111111111111111    trivial
ρ2111111111111-1111-1-1-1111-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-11111-1-1-11-11-1-11-11-111-11-1-11    linear of order 2
ρ41111-1-11111-1-111-1111-1-11-1-1-11-111-1    linear of order 2
ρ5111111-1111-11-1-1-11-111-1-11-1-1-1-1-111    linear of order 2
ρ6111111-1111-111-1-111-1-1-1-1111111-1-1    linear of order 2
ρ71111-1-1-11111-11-1111-111-1-1-1-11-11-11    linear of order 2
ρ81111-1-1-11111-1-1-111-11-11-1-111-11-11-1    linear of order 2
ρ9111111-111-1-11-1-1-1-1-11111-1-11111-1-1    linear of order 2
ρ10111111-111-1-111-1-1-11-1-111-11-1-1-1-111    linear of order 2
ρ111111-1-1-111-11-11-11-11-11-111-11-11-11-1    linear of order 2
ρ121111-1-1-111-11-1-1-11-1-11-1-1111-11-11-11    linear of order 2
ρ13111111111-111111-1111-1-1-11-1-1-1-1-1-1    linear of order 2
ρ14111111111-111-111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ151111-1-1111-1-1-1-11-1-1-1-111-111-11-111-1    linear of order 2
ρ161111-1-1111-1-1-111-1-111-11-11-11-11-1-11    linear of order 2
ρ17222222-2-2-20-2-202200000000000000    orthogonal lifted from D4
ρ182222222-2-202-20-2-200000000000000    orthogonal lifted from D4
ρ192222-2-22-2-20-220-2200000000000000    orthogonal lifted from D4
ρ202222-2-2-2-2-202202-200000000000000    orthogonal lifted from D4
ρ212-22-20002-2-2i00-2i002i2i0000000-20200    complex lifted from C4○D4
ρ222-22-20002-22i00-2i00-2i2i000000020-200    complex lifted from C4○D4
ρ232-22-20002-2-2i002i002i-2i000000020-200    complex lifted from C4○D4
ρ242-22-20002-22i002i00-2i-2i0000000-20200    complex lifted from C4○D4
ρ254-44-4000-4400000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ264-4-444-400000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-4-44-4400000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-40000000000000000000-2-202-2000    complex lifted from D4○SD16
ρ2944-4-400000000000000000002-20-2-2000    complex lifted from D4○SD16

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

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

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

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

Matrix representation of C42.58C23 in GL6(𝔽17)

1150000
1160000
0016000
0001600
000010
000001
,
100000
010000
000100
0016000
0000016
000010
,
1380000
040000
00101600
0016700
0000110
00001016
,
400000
040000
000010
000001
001000
000100
,
100000
1160000
001000
000100
0000160
0000016

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

C42.58C23 in GAP, Magma, Sage, TeX

C_4^2._{58}C_2^3
% in TeX

G:=Group("C4^2.58C2^3");
// GroupNames label

G:=SmallGroup(128,2076);
// by ID

G=gap.SmallGroup(128,2076);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,100,346,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.58C23 in TeX

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