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G = C42.476C23order 128 = 27

337th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.476C23, C4.722+ 1+4, C8⋊D446C2, C86D415C2, C4⋊C4.164D4, (C4×Q16)⋊42C2, Q8.Q839C2, (C2×D4).178D4, C8.D428C2, Q85D4.5C2, C2.51(Q8○D8), D4.7D449C2, C4⋊C4.419C23, C4⋊C8.109C22, (C2×C4).519C24, (C4×C8).226C22, (C2×C8).105C23, Q8.29(C4○D4), Q8.D445C2, C22⋊Q1633C2, C22⋊C4.174D4, C23.336(C2×D4), C4.Q8.61C22, (C2×D4).243C23, (C4×D4).168C22, C4⋊D4.92C22, C22⋊C8.87C22, (C4×Q8).164C22, (C2×Q8).228C23, C2.155(D45D4), C2.D8.124C22, C22⋊Q8.90C22, D4⋊C4.14C22, C23.36D424C2, C23.48D429C2, C23.19D441C2, C23.38D417C2, (C22×C4).332C23, (C2×Q16).135C22, Q8⋊C4.14C22, (C2×SD16).60C22, C4.4D4.73C22, C22.779(C22×D4), C42.C2.44C22, C2.93(D8⋊C22), C22.46C247C2, (C22×Q8).348C22, C42⋊C2.197C22, C42.78C2212C2, (C2×M4(2)).121C22, C4.244(C2×C4○D4), (C2×C4).930(C2×D4), (C2×C4⋊C4).673C22, (C2×C4○D4).219C22, SmallGroup(128,2059)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.476C23
C1C2C4C2×C4C22×C4C22×Q8Q85D4 — C42.476C23
C1C2C2×C4 — C42.476C23
C1C22C4×D4 — C42.476C23
C1C2C2C2×C4 — C42.476C23

Generators and relations for C42.476C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=a2b2, d2=b2, ab=ba, cac-1=a-1b2, dad-1=ab2, eae=a-1, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, ede=b2d >

Subgroups: 360 in 190 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×9], C8 [×4], C2×C4 [×5], C2×C4 [×17], D4 [×7], Q8 [×2], Q8 [×7], C23 [×2], C23, C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×5], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×2], SD16, Q16 [×3], C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×3], C2×Q8 [×4], C4○D4 [×3], C4×C8, C22⋊C8 [×2], D4⋊C4 [×3], Q8⋊C4 [×7], C4⋊C8, C4.Q8, C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8 [×2], C4⋊D4, C4⋊D4, C22⋊Q8 [×3], C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C422C2, C2×M4(2) [×2], C2×SD16, C2×Q16 [×2], C22×Q8, C2×C4○D4, C23.36D4, C23.38D4, C86D4, C4×Q16, C22⋊Q16, D4.7D4, Q8.D4, C8⋊D4, C8.D4, Q8.Q8, C23.19D4, C23.48D4, C42.78C22, Q85D4, C22.46C24, C42.476C23
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, D45D4, D8⋊C22, Q8○D8, C42.476C23

Character table of C42.476C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D8E8F
 size 11114482222444444488888444488
ρ111111111111111111111111111111    trivial
ρ211111-11-111-1-1-1-11-1-1-11-111-11-1-11-11    linear of order 2
ρ3111111-111111-111-1111-1-111-1-1-1-1-1-1    linear of order 2
ρ411111-1-1-111-1-11-111-1-111-11-1-111-11-1    linear of order 2
ρ51111-1-1-11111111-11-11-1-111-11111-1-1    linear of order 2
ρ61111-11-1-111-1-1-1-1-1-11-1-111111-1-111-1    linear of order 2
ρ71111-1-1111111-11-1-1-11-11-11-1-1-1-1-111    linear of order 2
ρ81111-111-111-1-11-1-111-1-1-1-111-111-1-11    linear of order 2
ρ9111111-11111-1-111-11-1-1-1-1-1-1111111    linear of order 2
ρ1011111-1-1-111-111-111-11-11-1-111-1-11-11    linear of order 2
ρ1111111111111-111111-1-111-1-1-1-1-1-1-1-1    linear of order 2
ρ1211111-11-111-11-1-11-1-11-1-11-11-111-11-1    linear of order 2
ρ131111-1-111111-1-11-1-1-1-111-1-111111-1-1    linear of order 2
ρ141111-111-111-111-1-11111-1-1-1-11-1-111-1    linear of order 2
ρ151111-1-1-11111-111-11-1-11-11-11-1-1-1-111    linear of order 2
ρ161111-11-1-111-11-1-1-1-111111-1-1-111-1-11    linear of order 2
ρ172222-2-20-2-2-2-2002202000000000000    orthogonal lifted from D4
ρ182222220-2-2-2-2002-20-2000000000000    orthogonal lifted from D4
ρ192222-2202-2-2200-220-2000000000000    orthogonal lifted from D4
ρ2022222-202-2-2200-2-202000000000000    orthogonal lifted from D4
ρ212-22-200002-20-2i-200202i000002i00-2i00    complex lifted from C4○D4
ρ222-22-200002-20-2i200-202i00000-2i002i00    complex lifted from C4○D4
ρ232-22-200002-202i200-20-2i000002i00-2i00    complex lifted from C4○D4
ρ242-22-200002-202i-20020-2i00000-2i002i00    complex lifted from C4○D4
ρ254-44-40000-440000000000000000000    orthogonal lifted from 2+ 1+4
ρ2644-4-400000000000000000000-2222000    symplectic lifted from Q8○D8, Schur index 2
ρ2744-4-40000000000000000000022-22000    symplectic lifted from Q8○D8, Schur index 2
ρ284-4-440004i00-4i000000000000000000    complex lifted from D8⋊C22
ρ294-4-44000-4i004i000000000000000000    complex lifted from D8⋊C22

Smallest permutation representation of C42.476C23
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 9 51 41)(2 10 52 42)(3 11 49 43)(4 12 50 44)(5 36 56 27)(6 33 53 28)(7 34 54 25)(8 35 55 26)(13 21 47 17)(14 22 48 18)(15 23 45 19)(16 24 46 20)(29 62 58 38)(30 63 59 39)(31 64 60 40)(32 61 57 37)
(1 55 49 6)(2 7 50 56)(3 53 51 8)(4 5 52 54)(9 35 43 28)(10 25 44 36)(11 33 41 26)(12 27 42 34)(13 57 45 30)(14 31 46 58)(15 59 47 32)(16 29 48 60)(17 37 23 63)(18 64 24 38)(19 39 21 61)(20 62 22 40)
(1 24 51 20)(2 17 52 21)(3 22 49 18)(4 19 50 23)(5 59 56 30)(6 31 53 60)(7 57 54 32)(8 29 55 58)(9 16 41 46)(10 47 42 13)(11 14 43 48)(12 45 44 15)(25 37 34 61)(26 62 35 38)(27 39 36 63)(28 64 33 40)
(1 20)(2 19)(3 18)(4 17)(5 39)(6 38)(7 37)(8 40)(9 16)(10 15)(11 14)(12 13)(21 50)(22 49)(23 52)(24 51)(25 57)(26 60)(27 59)(28 58)(29 33)(30 36)(31 35)(32 34)(41 46)(42 45)(43 48)(44 47)(53 62)(54 61)(55 64)(56 63)

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

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

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

Matrix representation of C42.476C23 in GL6(𝔽17)

1150000
1160000
0016007
00016100
0001010
007001
,
100000
010000
000100
0016000
000001
0000160
,
1380000
040000
00141400
0014300
000033
0000314
,
1600000
0160000
0001010
00100016
0010010
00016100
,
1600000
1610000
0001010
007001
0016007
00016100

G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,7,0,0,0,16,10,0,0,0,0,10,1,0,0,0,7,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,14,14,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,3,14],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,10,1,0,0,0,10,0,0,16,0,0,1,0,0,10,0,0,0,16,10,0],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,7,16,0,0,0,10,0,0,16,0,0,1,0,0,10,0,0,0,1,7,0] >;

C42.476C23 in GAP, Magma, Sage, TeX

C_4^2._{476}C_2^3
% in TeX

G:=Group("C4^2.476C2^3");
// GroupNames label

G:=SmallGroup(128,2059);
// by ID

G=gap.SmallGroup(128,2059);
# by ID

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

G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2*b^2,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,e*a*e=a^-1,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 C42.476C23 in TeX

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