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G = C42.248C23order 128 = 27

109th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.248C23, C4⋊C4.70D4, C81C822C2, C86D426C2, (C2×D4).63D4, C4⋊D8.8C2, (C2×C8).186D4, D42Q834C2, C4.4D820C2, C4.D818C2, C4.6Q169C2, C4⋊Q8.69C22, C4.103(C4○D8), C2.12(C8⋊D4), C4⋊C8.189C22, C4.93(C8⋊C22), (C4×C8).214C22, (C4×D4).49C22, C41D4.36C22, C4.74(C8.C22), C2.16(D4.4D4), C2.12(D4.2D4), C22.209(C4⋊D4), (C2×C4).33(C4○D4), (C2×C4).1283(C2×D4), SmallGroup(128,429)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.248C23
C1C2C22C2×C4C42C4×D4C86D4 — C42.248C23
C1C22C42 — C42.248C23
C1C22C42 — C42.248C23
C1C22C22C42 — C42.248C23

Generators and relations for C42.248C23
 G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=ab2, ab=ba, cac=a-1, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=a-1c, ece=bc, ede=a2d >

Subgroups: 232 in 85 conjugacy classes, 32 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×3], C22, C22 [×6], C8 [×5], C2×C4 [×3], C2×C4 [×4], D4 [×7], Q8, C23 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×3], M4(2) [×2], D8 [×2], C22×C4, C2×D4, C2×D4 [×3], C2×Q8, C4×C8, C22⋊C8, D4⋊C4 [×4], C4⋊C8 [×3], C4.Q8, C4×D4, C41D4, C4⋊Q8, C2×M4(2), C2×D8, C4.D8, C4.6Q16, C81C8, C86D4, C4⋊D8, D42Q8, C4.4D8, C42.248C23
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, D4.4D4, C42.248C23

Character table of C42.248C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J
 size 1111816222248164444888888
ρ111111111111111111111111    trivial
ρ21111-1111111-111111-1-1-1-1-1-1    linear of order 2
ρ311111-11111111-1-1-1-11-11-1-1-1    linear of order 2
ρ41111-1-111111-11-1-1-1-1-11-1111    linear of order 2
ρ511111-1111111-11111-1-1-1-111    linear of order 2
ρ61111-1-111111-1-111111111-1-1    linear of order 2
ρ7111111111111-1-1-1-1-1-11-11-1-1    linear of order 2
ρ81111-1111111-1-1-1-1-1-11-11-111    linear of order 2
ρ92222002-22-2-2002-2-22000000    orthogonal lifted from D4
ρ102222002-22-2-200-222-2000000    orthogonal lifted from D4
ρ11222220-22-22-2-200000000000    orthogonal lifted from D4
ρ122222-20-22-22-2200000000000    orthogonal lifted from D4
ρ13222200-2-2-2-220000000000-2i2i    complex lifted from C4○D4
ρ14222200-2-2-2-2200000000002i-2i    complex lifted from C4○D4
ρ152-2-2200-202000002i-2i0-22--2-200    complex lifted from C4○D8
ρ162-2-2200-20200000-2i2i0--22-2-200    complex lifted from C4○D8
ρ172-2-2200-202000002i-2i0--2-2-2200    complex lifted from C4○D8
ρ182-2-2200-20200000-2i2i0-2-2--2200    complex lifted from C4○D8
ρ194-4-440040-400000000000000    orthogonal lifted from C8⋊C22
ρ204-44-400040-40000000000000    orthogonal lifted from C8⋊C22
ρ2144-4-40000000002200-22000000    orthogonal lifted from D4.4D4
ρ2244-4-4000000000-220022000000    orthogonal lifted from D4.4D4
ρ234-44-4000-4040000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C42.248C23
On 64 points
Generators in S64
(1 18 5 22)(2 19 6 23)(3 20 7 24)(4 21 8 17)(9 45 13 41)(10 46 14 42)(11 47 15 43)(12 48 16 44)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(33 62 37 58)(34 63 38 59)(35 64 39 60)(36 57 40 61)
(1 61 24 34)(2 62 17 35)(3 63 18 36)(4 64 19 37)(5 57 20 38)(6 58 21 39)(7 59 22 40)(8 60 23 33)(9 49 43 25)(10 50 44 26)(11 51 45 27)(12 52 46 28)(13 53 47 29)(14 54 48 30)(15 55 41 31)(16 56 42 32)
(2 19)(3 7)(4 17)(6 23)(8 21)(9 25)(10 52)(11 31)(12 50)(13 29)(14 56)(15 27)(16 54)(18 22)(26 46)(28 44)(30 42)(32 48)(33 39)(34 61)(35 37)(36 59)(38 57)(40 63)(41 51)(43 49)(45 55)(47 53)(58 60)(62 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 49)(2 54)(3 51)(4 56)(5 53)(6 50)(7 55)(8 52)(9 61)(10 58)(11 63)(12 60)(13 57)(14 62)(15 59)(16 64)(17 30)(18 27)(19 32)(20 29)(21 26)(22 31)(23 28)(24 25)(33 46)(34 43)(35 48)(36 45)(37 42)(38 47)(39 44)(40 41)

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

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

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

Matrix representation of C42.248C23 in GL6(𝔽17)

100000
010000
0001600
001000
00143016
003310
,
010000
1600000
000100
0016000
0000016
000010
,
100000
0160000
001000
0001600
000001
000010
,
400000
040000
00141400
0031400
009933
0098143
,
14140000
1430000
001414150
00143015
000033
0000314

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

C42.248C23 in GAP, Magma, Sage, TeX

C_4^2._{248}C_2^3
% in TeX

G:=Group("C4^2.248C2^3");
// GroupNames label

G:=SmallGroup(128,429);
// by ID

G=gap.SmallGroup(128,429);
# by ID

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

G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=a*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=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 C42.248C23 in TeX

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