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G = C42.63C23order 128 = 27

63rd non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.63C23, C4.842- 1+4, C8⋊Q829C2, C4⋊C4.381D4, C89D4.3C2, Q8.Q842C2, C4.Q1639C2, (C2×D4).181D4, C8.36(C4○D4), C8.D432C2, C2.58(Q8○D8), Q16⋊C426C2, C8.18D432C2, C4⋊C4.254C23, C4⋊C8.122C22, (C2×C8).108C23, (C2×C4).541C24, C22⋊C4.178D4, C23.346(C2×D4), C4⋊Q8.173C22, C2.94(D46D4), C8⋊C4.55C22, C4.Q8.69C22, (C4×D4).181C22, (C2×Q16).89C22, (C4×Q8).180C22, (C2×Q8).243C23, M4(2)⋊C436C2, C2.D8.197C22, C23.20D446C2, C23.48D434C2, C23.25D432C2, C22⋊C8.100C22, (C22×C8).292C22, Q8⋊C4.80C22, C22.801(C22×D4), C22⋊Q8.106C22, C42.C2.54C22, C2.96(D8⋊C22), (C22×C4).1169C23, C42⋊C2.212C22, (C2×M4(2)).134C22, C22.50C24.6C2, C22.46C24.4C2, C4.123(C2×C4○D4), (C2×C4).625(C2×D4), (C2×C4⋊C4).690C22, SmallGroup(128,2081)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.63C23
C1C2C4C2×C4C22×C4C42⋊C2C22.46C24 — C42.63C23
C1C2C2×C4 — C42.63C23
C1C22C4×D4 — C42.63C23
C1C2C2C2×C4 — C42.63C23

Generators and relations for C42.63C23
 G = < a,b,c,d,e | a4=b4=1, c2=d2=a2, e2=b2, ab=ba, cac-1=eae-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2c, ede-1=b2d >

Subgroups: 280 in 168 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×14], D4, Q8 [×6], C23 [×2], C42, C42 [×5], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×7], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], Q16 [×2], C22×C4 [×2], C22×C4, C2×D4, C2×Q8 [×2], C2×Q8, C8⋊C4, C22⋊C8 [×2], Q8⋊C4 [×6], C4⋊C8, C4.Q8 [×4], C2.D8 [×5], C2×C4⋊C4, C42⋊C2 [×3], C42⋊C2, C4×D4, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×4], C22.D4, C4.4D4, C42.C2, C42.C2, C422C2 [×3], C4⋊Q8, C22×C8, C2×M4(2), C2×Q16, C23.25D4, M4(2)⋊C4, C89D4, Q16⋊C4, C8.18D4, C8.D4, C4.Q16, Q8.Q8, C23.48D4, C23.20D4 [×3], C8⋊Q8, C22.46C24, C22.50C24, C42.63C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C22×D4, C2×C4○D4, 2- 1+4, D46D4, D8⋊C22, Q8○D8, C42.63C23

Character table of C42.63C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q8A8B8C8D8E8F
 size 11114422224444444888888444488
ρ111111111111111111111111111111    trivial
ρ211111111111-111-1-1-1-1-1-1-1-1-1111111    linear of order 2
ρ311111-11-11-111-1-1-11-1111-1-1-1-1-111-11    linear of order 2
ρ411111-11-11-11-1-1-11-11-1-1-1111-1-111-11    linear of order 2
ρ511111111111111-11-1-11-1-111-1-1-1-1-1-1    linear of order 2
ρ611111111111-1111-111-111-1-1-1-1-1-1-1-1    linear of order 2
ρ711111-11-11-111-1-1111-11-11-1-111-1-11-1    linear of order 2
ρ811111-11-11-11-1-1-1-1-1-11-11-11111-1-11-1    linear of order 2
ρ91111-111111-1-1-1-11-1111-1-11-1-1-1-1-111    linear of order 2
ρ101111-111111-11-1-1-11-1-1-111-11-1-1-1-111    linear of order 2
ρ111111-1-11-11-1-1-111-1-1-111-11-1111-1-1-11    linear of order 2
ρ121111-1-11-11-1-1111111-1-11-11-111-1-1-11    linear of order 2
ρ131111-111111-1-1-1-1-1-1-1-11111-11111-1-1    linear of order 2
ρ141111-111111-11-1-11111-1-1-1-111111-1-1    linear of order 2
ρ151111-1-11-11-1-1-1111-11-111-1-11-1-1111-1    linear of order 2
ρ161111-1-11-11-1-1111-11-11-1-111-1-1-1111-1    linear of order 2
ρ172222-22-2-2-2-220-22000000000000000    orthogonal lifted from D4
ρ182222-2-2-22-22202-2000000000000000    orthogonal lifted from D4
ρ1922222-2-22-22-20-22000000000000000    orthogonal lifted from D4
ρ20222222-2-2-2-2-202-2000000000000000    orthogonal lifted from D4
ρ212-22-200-202002i00-2i-2i2i0000002-20000    complex lifted from C4○D4
ρ222-22-200-20200-2i002i2i-2i0000002-20000    complex lifted from C4○D4
ρ232-22-200-202002i002i-2i-2i000000-220000    complex lifted from C4○D4
ρ242-22-200-20200-2i00-2i2i2i000000-220000    complex lifted from C4○D4
ρ254-44-40040-400000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2644-4-400000000000000000000022-2200    symplectic lifted from Q8○D8, Schur index 2
ρ2744-4-4000000000000000000000-222200    symplectic lifted from Q8○D8, Schur index 2
ρ284-4-44000-4i04i0000000000000000000    complex lifted from D8⋊C22
ρ294-4-440004i0-4i0000000000000000000    complex lifted from D8⋊C22

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

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

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

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

Matrix representation of C42.63C23 in GL6(𝔽17)

010000
1600000
0001300
004000
0000013
000040
,
100000
010000
000100
0016000
0000016
000010
,
0130000
1300000
000010
000001
001000
000100
,
400000
040000
004600
0061300
0000613
00001311
,
1600000
010000
0001600
001000
0000016
000010

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,6,0,0,0,0,6,13,0,0,0,0,0,0,6,13,0,0,0,0,13,11],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0] >;

C42.63C23 in GAP, Magma, Sage, TeX

C_4^2._{63}C_2^3
% in TeX

G:=Group("C4^2.63C2^3");
// GroupNames label

G:=SmallGroup(128,2081);
// by ID

G=gap.SmallGroup(128,2081);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,723,100,346,248,4037,1027,124]);
// 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=e*a*e^-1=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^-1=a^2*c,e*d*e^-1=b^2*d>;
// generators/relations

Export

Character table of C42.63C23 in TeX

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