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G = C42.253C23order 128 = 27

114th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.253C23, C4⋊C4.75D4, C82C820C2, C84Q827C2, C42Q167C2, (C2×C8).191D4, (C2×Q8).63D4, C4⋊C8.36C22, C4.D8.6C2, C4⋊Q8.74C22, C4.108(C4○D8), C4.10D830C2, C2.11(C82D4), C4.74(C8⋊C22), (C4×C8).286C22, C4⋊SD16.11C2, C4.4D8.13C2, (C4×Q8).51C22, C41D4.39C22, C4.96(C8.C22), C2.21(D4.3D4), C2.14(Q8.D4), C22.214(C4⋊D4), (C2×C4).38(C4○D4), (C2×C4).1288(C2×D4), SmallGroup(128,434)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.253C23
C1C2C22C2×C4C42C4×Q8C84Q8 — C42.253C23
C1C22C42 — C42.253C23
C1C22C42 — C42.253C23
C1C22C22C42 — C42.253C23

Generators and relations for C42.253C23
 G = < a,b,c,d,e | a4=b4=c2=1, d2=a-1b2, e2=b2, ab=ba, cac=a-1, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=a-1c, ece-1=bc, ede-1=a2d >

Subgroups: 208 in 81 conjugacy classes, 32 normal (all characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×4], C22, C22 [×3], C8 [×5], C2×C4 [×3], C2×C4 [×3], D4 [×5], Q8 [×4], C23, C42, C42, C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×3], SD16 [×2], Q16 [×2], C2×D4 [×3], C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4 [×3], Q8⋊C4, C4⋊C8 [×3], C4⋊C8, C4×Q8, C41D4, C4⋊Q8, C2×SD16, C2×Q16, C4.D8, C4.10D8, C82C8, C84Q8, C4⋊SD16, C42Q16, C4.4D8, C42.253C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4, C4⋊D4, C4○D8, C8⋊C22 [×2], C8.C22, Q8.D4, C82D4, D4.3D4, C42.253C23

Character table of C42.253C23

 class 12A2B2C2D4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J
 size 1111162222488164444888888
ρ111111111111111111111111    trivial
ρ21111-111111-1-11-1-1-1-1-11-1111    linear of order 2
ρ31111111111-1-1-1-1-1-1-11-11-111    linear of order 2
ρ41111-11111111-11111-1-1-1-111    linear of order 2
ρ51111111111-1-111111-1-1-1-1-1-1    linear of order 2
ρ61111-111111111-1-1-1-11-11-1-1-1    linear of order 2
ρ7111111111111-1-1-1-1-1-11-11-1-1    linear of order 2
ρ81111-111111-1-1-111111111-1-1    linear of order 2
ρ9222202-22-2-20002-2-22000000    orthogonal lifted from D4
ρ10222202-22-2-2000-222-2000000    orthogonal lifted from D4
ρ1122220-22-22-22-200000000000    orthogonal lifted from D4
ρ1222220-22-22-2-2200000000000    orthogonal lifted from D4
ρ1322220-2-2-2-2200000000000-2i2i    complex lifted from C4○D4
ρ1422220-2-2-2-22000000000002i-2i    complex lifted from C4○D4
ρ152-2-220-2020000002i-2i0-2-22--200    complex lifted from C4○D8
ρ162-2-220-202000000-2i2i02-2-2--200    complex lifted from C4○D8
ρ172-2-220-2020000002i-2i02--2-2-200    complex lifted from C4○D8
ρ182-2-220-202000000-2i2i0-2--22-200    complex lifted from C4○D8
ρ194-44-40040-400000000000000    orthogonal lifted from C8⋊C22
ρ204-44-400-40400000000000000    orthogonal lifted from C8⋊C22
ρ214-4-44040-4000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2244-4-40000000002-200-2-2000000    complex lifted from D4.3D4
ρ2344-4-4000000000-2-2002-2000000    complex lifted from D4.3D4

Smallest permutation representation of C42.253C23
On 64 points
Generators in S64
(1 18 5 22)(2 19 6 23)(3 20 7 24)(4 21 8 17)(9 63 13 59)(10 64 14 60)(11 57 15 61)(12 58 16 62)(25 52 29 56)(26 53 30 49)(27 54 31 50)(28 55 32 51)(33 44 37 48)(34 45 38 41)(35 46 39 42)(36 47 40 43)
(1 61 20 13)(2 62 21 14)(3 63 22 15)(4 64 23 16)(5 57 24 9)(6 58 17 10)(7 59 18 11)(8 60 19 12)(25 41 54 36)(26 42 55 37)(27 43 56 38)(28 44 49 39)(29 45 50 40)(30 46 51 33)(31 47 52 34)(32 48 53 35)
(2 19)(3 7)(4 17)(6 23)(8 21)(9 57)(10 16)(11 63)(12 14)(13 61)(15 59)(18 22)(25 41)(26 35)(27 47)(28 33)(29 45)(30 39)(31 43)(32 37)(34 56)(36 54)(38 52)(40 50)(42 53)(44 51)(46 49)(48 55)(58 64)(60 62)
(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 50 20 29)(2 55 21 26)(3 52 22 31)(4 49 23 28)(5 54 24 25)(6 51 17 30)(7 56 18 27)(8 53 19 32)(9 36 57 41)(10 33 58 46)(11 38 59 43)(12 35 60 48)(13 40 61 45)(14 37 62 42)(15 34 63 47)(16 39 64 44)

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

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

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

Matrix representation of C42.253C23 in GL6(𝔽17)

100000
010000
000010
000001
0016000
0001600
,
010000
1600000
000100
0016000
000001
0000160
,
100000
0160000
001000
0001600
0000160
000001
,
400000
040000
000505
00120120
0001205
0050120
,
550000
5120000
0044125
0041355
0051244
001212413

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,12,0,5,0,0,5,0,12,0,0,0,0,12,0,12,0,0,5,0,5,0],[5,5,0,0,0,0,5,12,0,0,0,0,0,0,4,4,5,12,0,0,4,13,12,12,0,0,12,5,4,4,0,0,5,5,4,13] >;

C42.253C23 in GAP, Magma, Sage, TeX

C_4^2._{253}C_2^3
% in TeX

G:=Group("C4^2.253C2^3");
// GroupNames label

G:=SmallGroup(128,434);
// by ID

G=gap.SmallGroup(128,434);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,736,422,387,352,1123,136,2804,718,172]);
// Polycyclic

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

Export

Character table of C42.253C23 in TeX

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