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G = C42.250C23order 128 = 27

111st non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.250C23, C4⋊C4.72D4, C81C824C2, (C2×D4).64D4, C86D4.8C2, (C2×C8).188D4, D4⋊Q8.8C2, C4⋊Q8.71C22, C4.105(C4○D8), C4.10D816C2, C2.14(C8⋊D4), C4⋊C8.190C22, C4.45(C8⋊C22), (C4×C8).216C22, C4.SD1619C2, (C4×D4).50C22, D4.D4.11C2, C4.76(C8.C22), C2.13(D4.2D4), C2.16(D4.5D4), C22.211(C4⋊D4), (C2×C4).35(C4○D4), (C2×C4).1285(C2×D4), SmallGroup(128,431)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.250C23
C1C2C22C2×C4C42C4×D4C86D4 — C42.250C23
C1C22C42 — C42.250C23
C1C22C42 — C42.250C23
C1C22C22C42 — C42.250C23

Generators and relations for C42.250C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=ab2, 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: 184 in 80 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 [×4], C23, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×3], M4(2) [×2], SD16 [×2], C22×C4, C2×D4, C2×Q8 [×2], C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4 [×3], C4⋊C8 [×3], C2.D8, C4×D4, C4⋊Q8 [×2], C2×M4(2), C2×SD16, C4.10D8 [×2], C81C8, C86D4, D4.D4, D4⋊Q8, C4.SD16, C42.250C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4, C4⋊D4, C4○D8, C8⋊C22 [×2], C8.C22, D4.2D4, C8⋊D4, D4.5D4, C42.250C23

Character table of C42.250C23

 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
ρ194-44-400-40400000000000000    orthogonal lifted from C8⋊C22
ρ204-4-44040-4000000000000000    orthogonal lifted from C8⋊C22
ρ214-44-40040-400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2244-4-40000000002200-22000000    symplectic lifted from D4.5D4, Schur index 2
ρ2344-4-4000000000-220022000000    symplectic lifted from D4.5D4, Schur index 2

Smallest permutation representation of C42.250C23
On 64 points
Generators in S64
(1 48 5 44)(2 41 6 45)(3 42 7 46)(4 43 8 47)(9 25 13 29)(10 26 14 30)(11 27 15 31)(12 28 16 32)(17 57 21 61)(18 58 22 62)(19 59 23 63)(20 60 24 64)(33 54 37 50)(34 55 38 51)(35 56 39 52)(36 49 40 53)
(1 50 46 39)(2 51 47 40)(3 52 48 33)(4 53 41 34)(5 54 42 35)(6 55 43 36)(7 56 44 37)(8 49 45 38)(9 19 31 57)(10 20 32 58)(11 21 25 59)(12 22 26 60)(13 23 27 61)(14 24 28 62)(15 17 29 63)(16 18 30 64)
(1 63 5 59)(2 20 6 24)(3 61 7 57)(4 18 8 22)(9 33 13 37)(10 55 14 51)(11 39 15 35)(12 53 16 49)(17 42 21 46)(19 48 23 44)(25 50 29 54)(26 34 30 38)(27 56 31 52)(28 40 32 36)(41 64 45 60)(43 62 47 58)
(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 19)(10 24)(11 21)(12 18)(13 23)(14 20)(15 17)(16 22)(25 59)(26 64)(27 61)(28 58)(29 63)(30 60)(31 57)(32 62)(33 52)(34 49)(35 54)(36 51)(37 56)(38 53)(39 50)(40 55)(41 45)(43 47)

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

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

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

Matrix representation of C42.250C23 in GL8(𝔽17)

160000000
016000000
001600000
000160000
0000131500
00000400
0000001315
00000004
,
01000000
160000000
60010000
0111600000
000016000
000001600
000000160
000000016
,
314000000
1414000000
5131430000
1314330000
000000145
000000123
000031200
000051400
,
7111500000
1170150000
891060000
986100000
00000010
00000001
0000131500
00000400
,
10000000
016000000
06100000
1100160000
00001000
00000100
000000160
000000016

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,15,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,15,4],[0,16,6,0,0,0,0,0,1,0,0,11,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[3,14,5,13,0,0,0,0,14,14,13,14,0,0,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,0,3,5,0,0,0,0,0,0,12,14,0,0,0,0,14,12,0,0,0,0,0,0,5,3,0,0],[7,11,8,9,0,0,0,0,11,7,9,8,0,0,0,0,15,0,10,6,0,0,0,0,0,15,6,10,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,15,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,11,0,0,0,0,0,16,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16] >;

C42.250C23 in GAP, Magma, Sage, TeX

C_4^2._{250}C_2^3
% in TeX

G:=Group("C4^2.250C2^3");
// GroupNames label

G:=SmallGroup(128,431);
// by ID

G=gap.SmallGroup(128,431);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,512,422,387,1123,136,2804,718,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2,d^2=a*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 C42.250C23 in TeX

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