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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, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C422C2, C2×M4(2), C2×SD16, C2×Q16, 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, C22, D4, C23, C2×D4, C4○D4, 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 45 17 41)(2 46 18 42)(3 47 19 43)(4 48 20 44)(5 38 34 62)(6 39 35 63)(7 40 36 64)(8 37 33 61)(9 52 22 14)(10 49 23 15)(11 50 24 16)(12 51 21 13)(25 54 58 31)(26 55 59 32)(27 56 60 29)(28 53 57 30)
(1 32 19 53)(2 54 20 29)(3 30 17 55)(4 56 18 31)(5 52 36 16)(6 13 33 49)(7 50 34 14)(8 15 35 51)(9 64 24 38)(10 39 21 61)(11 62 22 40)(12 37 23 63)(25 44 60 46)(26 47 57 41)(27 42 58 48)(28 45 59 43)
(1 11 17 24)(2 21 18 12)(3 9 19 22)(4 23 20 10)(5 32 34 55)(6 56 35 29)(7 30 36 53)(8 54 33 31)(13 46 51 42)(14 43 52 47)(15 48 49 44)(16 41 50 45)(25 61 58 37)(26 38 59 62)(27 63 60 39)(28 40 57 64)
(1 24)(2 23)(3 22)(4 21)(5 57)(6 60)(7 59)(8 58)(9 19)(10 18)(11 17)(12 20)(13 48)(14 47)(15 46)(16 45)(25 33)(26 36)(27 35)(28 34)(29 39)(30 38)(31 37)(32 40)(41 50)(42 49)(43 52)(44 51)(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), (1,45,17,41)(2,46,18,42)(3,47,19,43)(4,48,20,44)(5,38,34,62)(6,39,35,63)(7,40,36,64)(8,37,33,61)(9,52,22,14)(10,49,23,15)(11,50,24,16)(12,51,21,13)(25,54,58,31)(26,55,59,32)(27,56,60,29)(28,53,57,30), (1,32,19,53)(2,54,20,29)(3,30,17,55)(4,56,18,31)(5,52,36,16)(6,13,33,49)(7,50,34,14)(8,15,35,51)(9,64,24,38)(10,39,21,61)(11,62,22,40)(12,37,23,63)(25,44,60,46)(26,47,57,41)(27,42,58,48)(28,45,59,43), (1,11,17,24)(2,21,18,12)(3,9,19,22)(4,23,20,10)(5,32,34,55)(6,56,35,29)(7,30,36,53)(8,54,33,31)(13,46,51,42)(14,43,52,47)(15,48,49,44)(16,41,50,45)(25,61,58,37)(26,38,59,62)(27,63,60,39)(28,40,57,64), (1,24)(2,23)(3,22)(4,21)(5,57)(6,60)(7,59)(8,58)(9,19)(10,18)(11,17)(12,20)(13,48)(14,47)(15,46)(16,45)(25,33)(26,36)(27,35)(28,34)(29,39)(30,38)(31,37)(32,40)(41,50)(42,49)(43,52)(44,51)(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), (1,45,17,41)(2,46,18,42)(3,47,19,43)(4,48,20,44)(5,38,34,62)(6,39,35,63)(7,40,36,64)(8,37,33,61)(9,52,22,14)(10,49,23,15)(11,50,24,16)(12,51,21,13)(25,54,58,31)(26,55,59,32)(27,56,60,29)(28,53,57,30), (1,32,19,53)(2,54,20,29)(3,30,17,55)(4,56,18,31)(5,52,36,16)(6,13,33,49)(7,50,34,14)(8,15,35,51)(9,64,24,38)(10,39,21,61)(11,62,22,40)(12,37,23,63)(25,44,60,46)(26,47,57,41)(27,42,58,48)(28,45,59,43), (1,11,17,24)(2,21,18,12)(3,9,19,22)(4,23,20,10)(5,32,34,55)(6,56,35,29)(7,30,36,53)(8,54,33,31)(13,46,51,42)(14,43,52,47)(15,48,49,44)(16,41,50,45)(25,61,58,37)(26,38,59,62)(27,63,60,39)(28,40,57,64), (1,24)(2,23)(3,22)(4,21)(5,57)(6,60)(7,59)(8,58)(9,19)(10,18)(11,17)(12,20)(13,48)(14,47)(15,46)(16,45)(25,33)(26,36)(27,35)(28,34)(29,39)(30,38)(31,37)(32,40)(41,50)(42,49)(43,52)(44,51)(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)], [(1,45,17,41),(2,46,18,42),(3,47,19,43),(4,48,20,44),(5,38,34,62),(6,39,35,63),(7,40,36,64),(8,37,33,61),(9,52,22,14),(10,49,23,15),(11,50,24,16),(12,51,21,13),(25,54,58,31),(26,55,59,32),(27,56,60,29),(28,53,57,30)], [(1,32,19,53),(2,54,20,29),(3,30,17,55),(4,56,18,31),(5,52,36,16),(6,13,33,49),(7,50,34,14),(8,15,35,51),(9,64,24,38),(10,39,21,61),(11,62,22,40),(12,37,23,63),(25,44,60,46),(26,47,57,41),(27,42,58,48),(28,45,59,43)], [(1,11,17,24),(2,21,18,12),(3,9,19,22),(4,23,20,10),(5,32,34,55),(6,56,35,29),(7,30,36,53),(8,54,33,31),(13,46,51,42),(14,43,52,47),(15,48,49,44),(16,41,50,45),(25,61,58,37),(26,38,59,62),(27,63,60,39),(28,40,57,64)], [(1,24),(2,23),(3,22),(4,21),(5,57),(6,60),(7,59),(8,58),(9,19),(10,18),(11,17),(12,20),(13,48),(14,47),(15,46),(16,45),(25,33),(26,36),(27,35),(28,34),(29,39),(30,38),(31,37),(32,40),(41,50),(42,49),(43,52),(44,51),(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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