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G = C42.408C23order 128 = 27

269th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.408C23, C4.1142+ 1+4, C85D425C2, C4⋊C821C22, (C4×C8)⋊43C22, C4⋊C4.131D4, C4⋊Q812C22, C4⋊SD1611C2, C8.2D413C2, C22⋊C4.23D4, (C4×Q8)⋊15C22, C8⋊C412C22, D4.7D429C2, C22⋊SD1612C2, C4⋊C4.161C23, (C2×C8).330C23, (C2×C4).420C24, Q8.D425C2, (C2×Q16)⋊26C22, C23.292(C2×D4), C42.C26C22, D4⋊C433C22, C2.46(D4○SD16), Q8⋊C448C22, (C2×SD16)⋊44C22, (C2×D4).169C23, C41D4.68C22, C22⋊C8.55C22, (C2×Q8).157C23, C22⋊Q8.43C22, (C22×C4).308C23, C4.4D4.40C22, C22.680(C22×D4), C42.7C2212C2, C22.35C245C2, C22.29C24.15C2, C42.29C224C2, (C22×D4).393C22, C42.78C2215C2, C42⋊C2.159C22, C2.91(C22.29C24), (C2×C4).549(C2×D4), (C2×C4○D4).179C22, SmallGroup(128,1954)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.408C23
C1C2C4C2×C4C22×C4C22×D4C22.29C24 — C42.408C23
C1C2C2×C4 — C42.408C23
C1C22C42⋊C2 — C42.408C23
C1C2C2C2×C4 — C42.408C23

Generators and relations for C42.408C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b2, ab=ba, cac-1=dad=a-1, eae=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece=a2c, de=ed >

Subgroups: 460 in 198 conjugacy classes, 84 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×16], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×9], D4 [×14], Q8 [×6], C23, C23 [×7], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], SD16 [×6], Q16 [×2], C22×C4, C22×C4, C2×D4 [×3], C2×D4 [×8], C2×Q8, C2×Q8 [×2], C4○D4 [×2], C24, C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×2], C42⋊C2, C4×Q8 [×2], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C42.C2, C42.C2 [×2], C422C2 [×2], C41D4 [×2], C4⋊Q8, C2×SD16 [×6], C2×Q16 [×2], C22×D4, C2×C4○D4, C42.7C22, C22⋊SD16 [×2], D4.7D4 [×2], C4⋊SD16 [×2], Q8.D4 [×2], C42.78C22, C42.29C22, C85D4, C8.2D4, C22.29C24, C22.35C24, C42.408C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×2], C22.29C24, D4○SD16 [×2], C42.408C23

Character table of C42.408C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F
 size 11114888224444488888444488
ρ111111111111111111111111111    trivial
ρ21111-1-1111111-1-1-1-1-11-111111-1-1    linear of order 2
ρ31111-111-111-1-111-1-1-1-111-111-1-11    linear of order 2
ρ411111-11-111-1-1-1-1111-1-11-111-11-1    linear of order 2
ρ51111-1-1-1111-1-111-1-11-1111-1-111-1    linear of order 2
ρ6111111-1111-1-1-1-111-1-1-111-1-11-11    linear of order 2
ρ711111-1-1-111111111-1111-1-1-1-1-1-1    linear of order 2
ρ81111-11-1-11111-1-1-1-111-11-1-1-1-111    linear of order 2
ρ911111-11-111-1-1-1-11-1111-11-1-11-11    linear of order 2
ρ101111-111-111-1-111-11-11-1-11-1-111-1    linear of order 2
ρ111111-1-1111111-1-1-11-1-11-1-1-1-1-111    linear of order 2
ρ12111111111111111-11-1-1-1-1-1-1-1-1-1    linear of order 2
ρ131111-11-1-11111-1-1-111-11-11111-1-1    linear of order 2
ρ1411111-1-1-11111111-1-1-1-1-1111111    linear of order 2
ρ15111111-1111-1-1-1-11-1-111-1-111-11-1    linear of order 2
ρ161111-1-1-1111-1-111-1111-1-1-111-1-11    linear of order 2
ρ1722222000-2-22-2-22-200000000000    orthogonal lifted from D4
ρ182222-2000-2-2-22-22200000000000    orthogonal lifted from D4
ρ1922222000-2-2-222-2-200000000000    orthogonal lifted from D4
ρ202222-2000-2-22-22-2200000000000    orthogonal lifted from D4
ρ214-44-40000-440000000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-400004-40000000000000000    orthogonal lifted from 2+ 1+4
ρ234-4-44000000000000000002-2-2-2000    complex lifted from D4○SD16
ρ2444-4-400000000000000002-200-2-200    complex lifted from D4○SD16
ρ254-4-4400000000000000000-2-22-2000    complex lifted from D4○SD16
ρ2644-4-40000000000000000-2-2002-200    complex lifted from D4○SD16

Smallest permutation representation of C42.408C23
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 19 23 25)(2 20 24 26)(3 17 21 27)(4 18 22 28)(5 15 9 32)(6 16 10 29)(7 13 11 30)(8 14 12 31)
(1 8 23 12)(2 7 24 11)(3 6 21 10)(4 5 22 9)(13 20 30 26)(14 19 31 25)(15 18 32 28)(16 17 29 27)
(1 25)(2 28)(3 27)(4 26)(5 7)(9 11)(13 32)(14 31)(15 30)(16 29)(17 21)(18 24)(19 23)(20 22)
(2 24)(4 22)(5 11)(6 8)(7 9)(10 12)(13 32)(14 16)(15 30)(18 28)(20 26)(29 31)

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

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

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

Matrix representation of C42.408C23 in GL8(𝔽17)

00100000
00010000
160000000
016000000
000016000
00000001
0000151611
00000100
,
01000000
160000000
00010000
001600000
0000161601
0000211616
00000001
000000160
,
005120000
0012120000
512000000
1212000000
0000105012
00007755
00000055
000000512
,
01000000
10000000
000160000
001600000
00001000
0000151611
00000001
00000010
,
10000000
01000000
001600000
000160000
0000101616
00000100
000000160
000000016

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

C42.408C23 in GAP, Magma, Sage, TeX

C_4^2._{408}C_2^3
% in TeX

G:=Group("C4^2.408C2^3");
// GroupNames label

G:=SmallGroup(128,1954);
// by ID

G=gap.SmallGroup(128,1954);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,675,1018,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.408C23 in TeX

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