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G = C42.509C23order 128 = 27

370th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.509C23, C4.302- 1+4, C84Q85C2, C4⋊C4.174D4, Q89(C4○D4), Q83Q85C2, D4.Q847C2, C4⋊SD1625C2, D4⋊Q843C2, (C4×SD16)⋊61C2, C4.4D836C2, (C2×Q8).242D4, Q86D4.9C2, C4⋊C4.436C23, C4⋊C8.133C22, C4.78(C8⋊C22), (C2×C4).560C24, (C2×C8).116C23, (C4×C8).296C22, C4⋊Q8.189C22, SD16⋊C444C2, C8⋊C4.59C22, C2.68(Q85D4), (C2×D4).271C23, (C4×D4).199C22, C41D4.99C22, (C2×Q8).404C23, (C4×Q8).191C22, C4.Q8.178C22, C2.D8.134C22, C2.100(D4○SD16), D4⋊C4.88C22, (C2×SD16).68C22, C22.820(C22×D4), C42.C2.64C22, Q8⋊C4.209C22, C42.29C2213C2, C4.261(C2×C4○D4), (C2×C4).636(C2×D4), C2.88(C2×C8⋊C22), SmallGroup(128,2100)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.509C23
C1C2C4C2×C4C42C4×Q8Q83Q8 — C42.509C23
C1C2C2×C4 — C42.509C23
C1C22C4×Q8 — C42.509C23
C1C2C2C2×C4 — C42.509C23

Generators and relations for C42.509C23
 G = < a,b,c,d,e | a4=b4=c2=1, d2=a2, e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=a2c, ece-1=bc, ede-1=b2d >

Subgroups: 392 in 191 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C2×C4 [×3], C2×C4 [×4], C2×C4 [×11], D4 [×15], Q8 [×2], Q8 [×5], C23 [×3], C42, C42 [×2], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×2], SD16 [×6], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8 [×2], C2×Q8, C4○D4 [×4], C4×C8, C8⋊C4 [×2], D4⋊C4, D4⋊C4 [×8], Q8⋊C4, C4⋊C8, C4⋊C8 [×2], C4.Q8, C2.D8 [×2], C4×D4, C4×D4 [×2], C4×Q8 [×2], C4×Q8 [×2], C4×Q8, C4⋊D4 [×3], C42.C2 [×2], C42.C2 [×2], C41D4, C41D4 [×2], C4⋊Q8, C4⋊Q8, C2×SD16, C2×SD16 [×2], C2×C4○D4, C4×SD16, SD16⋊C4 [×2], C84Q8, C4⋊SD16, C4⋊SD16 [×2], D4⋊Q8, D4.Q8 [×2], C4.4D8, C42.29C22 [×2], Q86D4, Q83Q8, C42.509C23
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, Q85D4, C2×C8⋊C22, D4○SD16, C42.509C23

Character table of C42.509C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D8E8F
 size 11118882222444444444888444488
ρ111111111111111111111111111111    trivial
ρ21111-111-11-111-1-111-1-11-1-11-1-1-111-11    linear of order 2
ρ311111-111111-1-11-1-1-1-1-1-1-1111111-1-1    linear of order 2
ρ41111-1-11-11-11-11-1-1-111-1111-1-1-1111-1    linear of order 2
ρ511111111111111-1111-11-1-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-111-11-111-1-1-11-1-1-1-11-1111-1-11-1    linear of order 2
ρ711111-111111-1-111-1-1-11-11-1-1-1-1-1-111    linear of order 2
ρ81111-1-11-11-11-11-11-11111-1-1111-1-1-11    linear of order 2
ρ91111-1-1-11111111111-11-1111-1-1-1-1-1-1    linear of order 2
ρ1011111-1-1-11-111-1-111-1111-11-111-1-11-1    linear of order 2
ρ111111-11-11111-1-11-1-1-11-11-111-1-1-1-111    linear of order 2
ρ12111111-1-11-11-11-1-1-11-1-1-111-111-1-1-11    linear of order 2
ρ131111-1-1-11111111-111-1-1-1-1-1-1111111    linear of order 2
ρ1411111-1-1-11-111-1-1-11-11-111-11-1-111-11    linear of order 2
ρ151111-11-11111-1-111-1-11111-1-11111-1-1    linear of order 2
ρ16111111-1-11-11-11-11-11-11-1-1-11-1-1111-1    linear of order 2
ρ172222000-2-2-2-22-220-22000000000000    orthogonal lifted from D4
ρ182222000-2-2-2-2-22202-2000000000000    orthogonal lifted from D4
ρ1922220002-22-222-20-2-2000000000000    orthogonal lifted from D4
ρ2022220002-22-2-2-2-2022000000000000    orthogonal lifted from D4
ρ212-22-2000020-20002002i-2-2i000-2i2i0000    complex lifted from C4○D4
ρ222-22-2000020-2000200-2i-22i0002i-2i0000    complex lifted from C4○D4
ρ232-22-2000020-2000-2002i2-2i0002i-2i0000    complex lifted from C4○D4
ρ242-22-2000020-2000-200-2i22i000-2i2i0000    complex lifted from C4○D4
ρ254-4-4400040-40000000000000000000    orthogonal lifted from C8⋊C22
ρ264-4-44000-4040000000000000000000    orthogonal lifted from C8⋊C22
ρ274-44-40000-404000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2844-4-40000000000000000000002-2-2-200    complex lifted from D4○SD16
ρ2944-4-4000000000000000000000-2-22-200    complex lifted from D4○SD16

Smallest permutation representation of C42.509C23
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 23 27 19)(2 24 28 20)(3 21 25 17)(4 22 26 18)(5 15 9 64)(6 16 10 61)(7 13 11 62)(8 14 12 63)(29 35 37 41)(30 36 38 42)(31 33 39 43)(32 34 40 44)(45 53 51 57)(46 54 52 58)(47 55 49 59)(48 56 50 60)
(1 55)(2 56)(3 53)(4 54)(5 42)(6 43)(7 44)(8 41)(9 36)(10 33)(11 34)(12 35)(13 40)(14 37)(15 38)(16 39)(17 51)(18 52)(19 49)(20 50)(21 45)(22 46)(23 47)(24 48)(25 57)(26 58)(27 59)(28 60)(29 63)(30 64)(31 61)(32 62)
(1 53 3 55)(2 56 4 54)(5 32 7 30)(6 31 8 29)(9 40 11 38)(10 39 12 37)(13 36 15 34)(14 35 16 33)(17 47 19 45)(18 46 20 48)(21 49 23 51)(22 52 24 50)(25 59 27 57)(26 58 28 60)(41 61 43 63)(42 64 44 62)
(1 29 27 37)(2 30 28 38)(3 31 25 39)(4 32 26 40)(5 56 9 60)(6 53 10 57)(7 54 11 58)(8 55 12 59)(13 46 62 52)(14 47 63 49)(15 48 64 50)(16 45 61 51)(17 33 21 43)(18 34 22 44)(19 35 23 41)(20 36 24 42)

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

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

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

Matrix representation of C42.509C23 in GL6(𝔽17)

1620000
1610000
000010
000001
0016000
0001600
,
100000
010000
000100
0016000
000001
0000160
,
1380000
1340000
0001370
00130010
00100013
0007130
,
1380000
040000
0001370
004007
007004
0007130
,
100000
010000
0031411
001414116
001616314
001611414

G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,2,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[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],[13,13,0,0,0,0,8,4,0,0,0,0,0,0,0,13,10,0,0,0,13,0,0,7,0,0,7,0,0,13,0,0,0,10,13,0],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,0,4,7,0,0,0,13,0,0,7,0,0,7,0,0,13,0,0,0,7,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,14,16,16,0,0,14,14,16,1,0,0,1,1,3,14,0,0,1,16,14,14] >;

C42.509C23 in GAP, Magma, Sage, TeX

C_4^2._{509}C_2^3
% in TeX

G:=Group("C4^2.509C2^3");
// GroupNames label

G:=SmallGroup(128,2100);
// by ID

G=gap.SmallGroup(128,2100);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,352,346,80,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.509C23 in TeX

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