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G = C42.252C23order 128 = 27

113rd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.252C23, C4⋊C4.74D4, C82C819C2, C86D427C2, (C2×D4).65D4, C4⋊D8.9C2, (C2×C8).190D4, C4.4D833C2, C4.D819C2, C4⋊Q8.73C22, D4.D436C2, C4.107(C4○D8), C4.10D817C2, C2.10(C82D4), C4⋊C8.191C22, C4.73(C8⋊C22), (C4×C8).285C22, (C4×D4).51C22, C41D4.38C22, C2.14(D4.2D4), C2.20(D4.3D4), C22.213(C4⋊D4), (C2×C4).37(C4○D4), (C2×C4).1287(C2×D4), SmallGroup(128,433)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.252C23
C1C2C22C2×C4C42C4×D4C86D4 — C42.252C23
C1C22C42 — C42.252C23
C1C22C42 — C42.252C23
C1C22C22C42 — C42.252C23

Generators and relations for C42.252C23
 G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=a-1b2, ab=ba, cac=a-1, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=a-1c, ece=bc, ede=a2d >

Subgroups: 240 in 88 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 [×2], C23 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×3], M4(2) [×2], D8 [×2], SD16 [×2], C22×C4, C2×D4, C2×D4 [×3], C2×Q8, C4×C8, C22⋊C8, D4⋊C4 [×3], Q8⋊C4, C4⋊C8 [×3], C4×D4, C41D4, C4⋊Q8, C2×M4(2), C2×D8, C2×SD16, C4.D8, C4.10D8, C82C8, C86D4, C4⋊D8, D4.D4, C4.4D8, C42.252C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4, C4⋊D4, C4○D8, C8⋊C22 [×3], D4.2D4, C82D4, D4.3D4, C42.252C23

Character table of C42.252C23

 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
ρ214-44-4000-4040000000000000    orthogonal lifted from C8⋊C22
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 C42.252C23
On 64 points
Generators in S64
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 43 13 47)(10 44 14 48)(11 45 15 41)(12 46 16 42)(17 54 21 50)(18 55 22 51)(19 56 23 52)(20 49 24 53)(33 64 37 60)(34 57 38 61)(35 58 39 62)(36 59 40 63)
(1 61 27 36)(2 62 28 37)(3 63 29 38)(4 64 30 39)(5 57 31 40)(6 58 32 33)(7 59 25 34)(8 60 26 35)(9 23 45 54)(10 24 46 55)(11 17 47 56)(12 18 48 49)(13 19 41 50)(14 20 42 51)(15 21 43 52)(16 22 44 53)
(2 26)(3 7)(4 32)(6 30)(8 28)(9 54)(10 22)(11 52)(12 20)(13 50)(14 18)(15 56)(16 24)(17 43)(19 41)(21 47)(23 45)(25 29)(33 39)(34 63)(35 37)(36 61)(38 59)(40 57)(42 49)(44 55)(46 53)(48 51)(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 23)(2 20)(3 17)(4 22)(5 19)(6 24)(7 21)(8 18)(9 61)(10 58)(11 63)(12 60)(13 57)(14 62)(15 59)(16 64)(25 52)(26 49)(27 54)(28 51)(29 56)(30 53)(31 50)(32 55)(33 46)(34 43)(35 48)(36 45)(37 42)(38 47)(39 44)(40 41)

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

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

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

Matrix representation of C42.252C23 in GL6(𝔽17)

100000
010000
0001150
0010015
0010016
0001160
,
010000
1600000
000100
0016000
0016001
0001160
,
100000
0160000
001000
0001600
0000160
000001
,
1300000
0130000
005507
001212100
0012055
00051212
,
330000
3140000
00106413
0010111313
0001166
00100107

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

C42.252C23 in GAP, Magma, Sage, TeX

C_4^2._{252}C_2^3
% in TeX

G:=Group("C4^2.252C2^3");
// GroupNames label

G:=SmallGroup(128,433);
// by ID

G=gap.SmallGroup(128,433);
# by ID

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

G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=a^-1*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 C42.252C23 in TeX

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