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G = C42.410C23order 128 = 27

271st non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.410C23, C4.1162+ 1+4, C83D415C2, C4⋊D828C2, C4⋊C822C22, C4⋊C4.133D4, (C4×C8)⋊17C22, C4⋊Q813C22, D4⋊D430C2, C22⋊D824C2, C4⋊SD1612C2, C2.31(D4○D8), C4.4D818C2, C8.12D45C2, (C4×D4)⋊16C22, (C2×D8)⋊26C22, C22⋊C4.25D4, (C2×Q16)⋊6C22, (C4×Q8)⋊16C22, C8⋊C413C22, C22⋊SD1613C2, D4.7D430C2, D4.2D427C2, D4⋊C44C22, C4⋊C4.163C23, (C2×C8).163C23, (C2×C4).422C24, Q8.D427C2, C23.294(C2×D4), C2.47(D4○SD16), Q8⋊C435C22, (C2×SD16)⋊45C22, (C2×D4).171C23, C4⋊D4.45C22, C4.4D410C22, C41D4.69C22, C22⋊C8.57C22, (C2×Q8).159C23, C22.29C2417C2, C22⋊Q8.45C22, (C22×C4).310C23, C22.682(C22×D4), C42.28C225C2, C42.7C2214C2, C22.36C247C2, (C22×D4).394C22, C42⋊C2.161C22, C2.93(C22.29C24), (C2×C4).551(C2×D4), (C2×C4○D4).181C22, SmallGroup(128,1956)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.410C23
C1C2C4C2×C4C22×C4C22×D4C22.29C24 — C42.410C23
C1C2C2×C4 — C42.410C23
C1C22C42⋊C2 — C42.410C23
C1C2C2C2×C4 — C42.410C23

Generators and relations for C42.410C23
 G = < a,b,c,d,e | a4=b4=c2=d2=e2=1, ab=ba, cac=dad=a-1b2, eae=ab2, cbc=dbd=b-1, be=eb, dcd=bc, ece=a2b2c, de=ed >

Subgroups: 500 in 204 conjugacy classes, 84 normal (all characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×19], C8 [×4], C2×C4 [×6], C2×C4 [×9], D4 [×17], Q8 [×5], C23, C23 [×8], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×4], C4⋊C4 [×3], C2×C8 [×4], D8 [×3], SD16 [×4], Q16, C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×9], C2×Q8 [×2], C2×Q8, C4○D4 [×2], C24, C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×2], C42⋊C2, C4×D4, C4×Q8, C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4 [×3], C4.4D4, C422C2, C41D4 [×2], C4⋊Q8, C2×D8 [×3], C2×SD16 [×4], C2×Q16, C22×D4, C2×C4○D4, C42.7C22, C22⋊D8, D4⋊D4, C22⋊SD16, D4.7D4, C4⋊D8, C4⋊SD16, D4.2D4, Q8.D4, C4.4D8, C42.28C22, C8.12D4, C83D4, C22.29C24, C22.36C24, C42.410C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×2], C22.29C24, D4○D8, D4○SD16, C42.410C23

Character table of C42.410C23

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F
 size 11114888822444448888444488
ρ111111111111111111111111111    trivial
ρ2111111-1-1111-1-1-1-11-1-111-11-11-11    linear of order 2
ρ31111-1-1-1-111111-1-1-111-11-1-1-1-111    linear of order 2
ρ41111-1-111111-1-111-1-1-1-111-11-1-11    linear of order 2
ρ51111-1111-11111-1-1-1-1-11-1-1-1-1-111    linear of order 2
ρ61111-11-1-1-111-1-111-1111-11-11-1-11    linear of order 2
ρ711111-1-1-1-11111111-1-1-1-1111111    linear of order 2
ρ811111-111-111-1-1-1-1111-1-1-11-11-11    linear of order 2
ρ9111111-11111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ101111111-1111-1-1-1-11-11-1-11-11-11-1    linear of order 2
ρ111111-1-11-111111-1-1-11-11-11111-1-1    linear of order 2
ρ121111-1-1-11111-1-111-1-111-1-11-111-1    linear of order 2
ρ131111-11-11-11111-1-1-1-11-111111-1-1    linear of order 2
ρ141111-111-1-111-1-111-11-1-11-11-111-1    linear of order 2
ρ1511111-11-1-11111111-1111-1-1-1-1-1-1    linear of order 2
ρ1611111-1-11-111-1-1-1-111-1111-11-11-1    linear of order 2
ρ17222220000-2-2-222-2-20000000000    orthogonal lifted from D4
ρ18222220000-2-22-2-22-20000000000    orthogonal lifted from D4
ρ192222-20000-2-22-22-220000000000    orthogonal lifted from D4
ρ202222-20000-2-2-22-2220000000000    orthogonal lifted from D4
ρ214-44-4000004-4000000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-400000-44000000000000000    orthogonal lifted from 2+ 1+4
ρ234-4-440000000000000000-22022000    orthogonal lifted from D4○D8
ρ244-4-440000000000000000220-22000    orthogonal lifted from D4○D8
ρ2544-4-400000000000000000-2-202-200    complex lifted from D4○SD16
ρ2644-4-4000000000000000002-20-2-200    complex lifted from D4○SD16

Smallest permutation representation of C42.410C23
On 32 points
Generators in S32
(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)
(1 24 26 19)(2 21 27 20)(3 22 28 17)(4 23 25 18)(5 12 31 15)(6 9 32 16)(7 10 29 13)(8 11 30 14)
(1 5)(2 30)(3 7)(4 32)(6 25)(8 27)(9 23)(10 17)(11 21)(12 19)(13 22)(14 20)(15 24)(16 18)(26 31)(28 29)
(1 19)(2 23)(3 17)(4 21)(6 30)(8 32)(9 11)(10 13)(12 15)(14 16)(18 27)(20 25)(22 28)(24 26)
(2 27)(4 25)(5 29)(6 8)(7 31)(9 11)(10 15)(12 13)(14 16)(18 23)(20 21)(30 32)

G:=sub<Sym(32)| (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), (1,24,26,19)(2,21,27,20)(3,22,28,17)(4,23,25,18)(5,12,31,15)(6,9,32,16)(7,10,29,13)(8,11,30,14), (1,5)(2,30)(3,7)(4,32)(6,25)(8,27)(9,23)(10,17)(11,21)(12,19)(13,22)(14,20)(15,24)(16,18)(26,31)(28,29), (1,19)(2,23)(3,17)(4,21)(6,30)(8,32)(9,11)(10,13)(12,15)(14,16)(18,27)(20,25)(22,28)(24,26), (2,27)(4,25)(5,29)(6,8)(7,31)(9,11)(10,15)(12,13)(14,16)(18,23)(20,21)(30,32)>;

G:=Group( (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), (1,24,26,19)(2,21,27,20)(3,22,28,17)(4,23,25,18)(5,12,31,15)(6,9,32,16)(7,10,29,13)(8,11,30,14), (1,5)(2,30)(3,7)(4,32)(6,25)(8,27)(9,23)(10,17)(11,21)(12,19)(13,22)(14,20)(15,24)(16,18)(26,31)(28,29), (1,19)(2,23)(3,17)(4,21)(6,30)(8,32)(9,11)(10,13)(12,15)(14,16)(18,27)(20,25)(22,28)(24,26), (2,27)(4,25)(5,29)(6,8)(7,31)(9,11)(10,15)(12,13)(14,16)(18,23)(20,21)(30,32) );

G=PermutationGroup([(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)], [(1,24,26,19),(2,21,27,20),(3,22,28,17),(4,23,25,18),(5,12,31,15),(6,9,32,16),(7,10,29,13),(8,11,30,14)], [(1,5),(2,30),(3,7),(4,32),(6,25),(8,27),(9,23),(10,17),(11,21),(12,19),(13,22),(14,20),(15,24),(16,18),(26,31),(28,29)], [(1,19),(2,23),(3,17),(4,21),(6,30),(8,32),(9,11),(10,13),(12,15),(14,16),(18,27),(20,25),(22,28),(24,26)], [(2,27),(4,25),(5,29),(6,8),(7,31),(9,11),(10,15),(12,13),(14,16),(18,23),(20,21),(30,32)])

Matrix representation of C42.410C23 in GL8(𝔽17)

101500000
001610000
001600000
011600000
0000160150
000000161
00001010
000011610
,
115000000
116000000
016010000
1161600000
000016200
000016100
0000016016
000011610
,
007100000
50700000
051250000
1251250000
000061100
000031100
00000333
0000143314
,
115000000
016000000
1160160000
1161600000
00001000
000011600
00000010
0000160016
,
10000000
01000000
101600000
100160000
00001000
00000100
0000160160
0000160016

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,15,16,1,1,0,0,0,0,0,1,0,0],[1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,16,0,1,0,0,0,0,2,1,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[0,5,0,12,0,0,0,0,0,0,5,5,0,0,0,0,7,7,12,12,0,0,0,0,10,0,5,5,0,0,0,0,0,0,0,0,6,3,0,14,0,0,0,0,11,11,3,3,0,0,0,0,0,0,3,3,0,0,0,0,0,0,3,14],[1,0,1,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,1,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16],[1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16] >;

C42.410C23 in GAP, Magma, Sage, TeX

C_4^2._{410}C_2^3
% in TeX

G:=Group("C4^2.410C2^3");
// GroupNames label

G:=SmallGroup(128,1956);
// by ID

G=gap.SmallGroup(128,1956);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,675,1018,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.410C23 in TeX

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