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G = C42.254C23order 128 = 27

115th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.254C23, C4⋊C4.76D4, C82C821C2, (C2×D4).66D4, C86D4.9C2, (C2×C8).192D4, D4⋊Q8.9C2, C4⋊Q8.75C22, C4.109(C4○D8), C4.10D831C2, C4⋊C8.192C22, C4.94(C8⋊C22), (C4×C8).287C22, D42Q8.12C2, C4.SD1633C2, C4.6Q1610C2, (C4×D4).52C22, C2.10(C8.D4), C4.46(C8.C22), C2.22(D4.3D4), C2.15(D4.2D4), C22.215(C4⋊D4), (C2×C4).39(C4○D4), (C2×C4).1289(C2×D4), SmallGroup(128,435)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.254C23
C1C2C22C2×C4C42C4×D4C86D4 — C42.254C23
C1C22C42 — C42.254C23
C1C22C42 — C42.254C23
C1C22C22C42 — C42.254C23

Generators and relations for C42.254C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=a-1b2, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=a-1c, ece=bc, ede=a2d >

Subgroups: 176 in 77 conjugacy classes, 32 normal (all characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×4], C22, C22 [×3], C8 [×5], C2×C4 [×3], C2×C4 [×5], D4 [×2], Q8 [×3], C23, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×3], M4(2) [×2], C22×C4, C2×D4, C2×Q8 [×2], C4×C8, C22⋊C8, D4⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8 [×3], C4.Q8, C2.D8, C4×D4, C4⋊Q8 [×2], C2×M4(2), C4.10D8, C4.6Q16, C82C8, C86D4, D4⋊Q8, D42Q8, C4.SD16, C42.254C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4, C4⋊D4, C4○D8, C8⋊C22, C8.C22 [×2], D4.2D4, C8.D4, D4.3D4, C42.254C23

Character table of C42.254C23

 class 12A2B2C2D4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J
 size 1111822224816164444888888
ρ111111111111111111111111    trivial
ρ211111111111-11-1-1-1-11-11-1-1-1    linear of order 2
ρ31111-111111-1111111-1-1-1-1-1-1    linear of order 2
ρ41111-111111-1-11-1-1-1-1-11-1111    linear of order 2
ρ51111-111111-1-1-11111111-1-11    linear of order 2
ρ61111-111111-11-1-1-1-1-11-1111-1    linear of order 2
ρ711111111111-1-11111-1-1-111-1    linear of order 2
ρ8111111111111-1-1-1-1-1-11-1-1-11    linear of order 2
ρ9222202-22-2-20002-2-22000000    orthogonal lifted from D4
ρ102222-2-22-22-22000000000000    orthogonal lifted from D4
ρ11222202-22-2-2000-222-2000000    orthogonal lifted from D4
ρ1222222-22-22-2-2000000000000    orthogonal lifted from D4
ρ1322220-2-2-2-2200000000002i-2i0    complex lifted from C4○D4
ρ1422220-2-2-2-220000000000-2i2i0    complex lifted from C4○D4
ρ152-2-220-2020000002i-2i0-2-2--2002    complex lifted from C4○D8
ρ162-2-220-202000000-2i2i0--2-2-2002    complex lifted from C4○D8
ρ172-2-220-2020000002i-2i0--22-200-2    complex lifted from C4○D8
ρ182-2-220-202000000-2i2i0-22--200-2    complex lifted from C4○D8
ρ194-4-44040-4000000000000000    orthogonal lifted from C8⋊C22
ρ204-44-40040-400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ214-44-400-40400000000000000    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 C42.254C23
On 64 points
Generators in S64
(1 55 5 51)(2 56 6 52)(3 49 7 53)(4 50 8 54)(9 62 13 58)(10 63 14 59)(11 64 15 60)(12 57 16 61)(17 32 21 28)(18 25 22 29)(19 26 23 30)(20 27 24 31)(33 46 37 42)(34 47 38 43)(35 48 39 44)(36 41 40 45)
(1 42 49 35)(2 43 50 36)(3 44 51 37)(4 45 52 38)(5 46 53 39)(6 47 54 40)(7 48 55 33)(8 41 56 34)(9 19 64 28)(10 20 57 29)(11 21 58 30)(12 22 59 31)(13 23 60 32)(14 24 61 25)(15 17 62 26)(16 18 63 27)
(1 16 5 12)(2 62 6 58)(3 14 7 10)(4 60 8 64)(9 52 13 56)(11 50 15 54)(17 47 21 43)(18 39 22 35)(19 45 23 41)(20 37 24 33)(25 48 29 44)(26 40 30 36)(27 46 31 42)(28 38 32 34)(49 63 53 59)(51 61 55 57)
(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)
(2 6)(4 8)(9 32)(10 29)(11 26)(12 31)(13 28)(14 25)(15 30)(16 27)(17 58)(18 63)(19 60)(20 57)(21 62)(22 59)(23 64)(24 61)(33 48)(34 45)(35 42)(36 47)(37 44)(38 41)(39 46)(40 43)(50 54)(52 56)

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

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

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

Matrix representation of C42.254C23 in GL6(𝔽17)

100000
010000
000010
000001
0016000
0001600
,
010000
1600000
00161500
001100
00001615
000011
,
3140000
14140000
00101000
0012700
000077
0000510
,
400000
040000
00127127
005555
00510127
00121255
,
100000
0160000
001000
00161600
000010
00001616

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],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,10,12,0,0,0,0,10,7,0,0,0,0,0,0,7,5,0,0,0,0,7,10],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,12,5,5,12,0,0,7,5,10,12,0,0,12,5,12,5,0,0,7,5,7,5],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;

C42.254C23 in GAP, Magma, Sage, TeX

C_4^2._{254}C_2^3
% in TeX

G:=Group("C4^2.254C2^3");
// GroupNames label

G:=SmallGroup(128,435);
// by ID

G=gap.SmallGroup(128,435);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,736,422,387,1123,136,2804,718,172]);
// Polycyclic

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

Export

Character table of C42.254C23 in TeX

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