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G = C42.473C23order 128 = 27

334th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.473C23, C4.692+ 1+4, D42.6C2, C86D412C2, C8⋊D444C2, C4⋊C840C22, (C4×C8)⋊49C22, C4⋊C4.371D4, C4⋊Q826C22, C4⋊SD1623C2, D4⋊Q836C2, (C4×SD16)⋊56C2, (C2×D4).321D4, C4.4D834C2, C22⋊C4.54D4, (C4×Q8)⋊30C22, C2.D841C22, C4.Q871C22, D4.30(C4○D4), C22⋊SD1624C2, C4⋊C4.416C23, C4.75(C8⋊C22), C22⋊C836C22, (C2×C8).356C23, (C2×C4).516C24, C23.333(C2×D4), C22⋊Q821C22, D4⋊C449C22, C2.81(D4○SD16), Q8⋊C498C22, (C2×SD16)⋊35C22, (C2×D4).426C23, (C4×D4).166C22, C4⋊D4.90C22, C41D4.90C22, (C2×Q8).227C23, C2.152(D45D4), C42⋊C226C22, C23.37D418C2, C23.19D438C2, (C2×M4(2))⋊32C22, (C22×C4).329C23, C22.776(C22×D4), C22.50C246C2, (C22×D4).417C22, C4.241(C2×C4○D4), (C2×C4).611(C2×D4), C2.78(C2×C8⋊C22), SmallGroup(128,2056)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.473C23
C1C2C4C2×C4C22×C4C22×D4D42 — C42.473C23
C1C2C2×C4 — C42.473C23
C1C22C4×D4 — C42.473C23
C1C2C2C2×C4 — C42.473C23

Generators and relations for C42.473C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=a2b2, e2=b2, ab=ba, cac-1=a-1, dad=ab2, eae-1=a-1b2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2b2c, ede-1=b2d >

Subgroups: 536 in 224 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×24], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×11], D4 [×2], D4 [×18], Q8 [×4], C23 [×2], C23 [×13], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], SD16 [×4], C22×C4 [×2], C22×C4, C2×D4 [×2], C2×D4 [×2], C2×D4 [×16], C2×Q8, C2×Q8, C24 [×2], C4×C8, C22⋊C8 [×2], D4⋊C4, D4⋊C4 [×8], Q8⋊C4, C4⋊C8, C4.Q8, C2.D8 [×2], C42⋊C2 [×2], C4×D4 [×2], C4×Q8, C4×Q8, C22≀C2 [×2], C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C4.4D4, C422C2 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×SD16, C2×SD16 [×2], C22×D4 [×2], C22×D4, C23.37D4 [×2], C86D4, C4×SD16, C22⋊SD16 [×2], C4⋊SD16, C8⋊D4 [×2], D4⋊Q8, C23.19D4 [×2], C4.4D8, D42, C22.50C24, C42.473C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C8⋊C22 [×2], C22×D4, C2×C4○D4, 2+ 1+4, D45D4, C2×C8⋊C22, D4○SD16, C42.473C23

Character table of C42.473C23

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F
 size 11114444882222444448888444488
ρ111111111111111111111111111111    trivial
ρ211111-1111-1-111-1-1-11-1-1-1-1111-1-111-1    linear of order 2
ρ31111111-1-11-111-11-1-111-1-11-1-111-11-1    linear of order 2
ρ411111-11-1-1-11111-11-1-1-1111-1-1-1-1-111    linear of order 2
ρ51111-1-1-11-11-111-11-11-1111-1-11-1-111-1    linear of order 2
ρ61111-11-11-1-11111-1111-1-1-1-1-1111111    linear of order 2
ρ71111-1-1-1-111111111-1-11-1-1-11-1-1-1-111    linear of order 2
ρ81111-11-1-11-1-111-1-1-1-11-111-11-111-11-1    linear of order 2
ρ91111-11-1-11-1-111-11-1-111-111-11-1-11-11    linear of order 2
ρ101111-1-1-1-1111111-11-1-1-11-11-11111-1-1    linear of order 2
ρ111111-11-11-1-11111111111-111-1-1-1-1-1-1    linear of order 2
ρ121111-1-1-11-11-111-1-1-11-1-1-1111-111-1-11    linear of order 2
ρ1311111-11-1-1-1111111-1-11-11-111111-1-1    linear of order 2
ρ141111111-1-11-111-1-1-1-11-11-1-111-1-11-11    linear of order 2
ρ1511111-1111-1-111-11-11-111-1-1-1-111-1-11    linear of order 2
ρ1611111111111111-1111-1-11-1-1-1-1-1-1-1-1    linear of order 2
ρ172222020-200-2-2-2-2022-200000000000    orthogonal lifted from D4
ρ1822220-20-2002-2-220-22200000000000    orthogonal lifted from D4
ρ1922220-20200-2-2-2-202-2200000000000    orthogonal lifted from D4
ρ2022220202002-2-220-2-2-200000000000    orthogonal lifted from D4
ρ212-22-220-200002-202i000-2i000002i-2i000    complex lifted from C4○D4
ρ222-22-220-200002-20-2i0002i00000-2i2i000    complex lifted from C4○D4
ρ232-22-2-20200002-20-2i0002i000002i-2i000    complex lifted from C4○D4
ρ242-22-2-20200002-202i000-2i00000-2i2i000    complex lifted from C4○D4
ρ254-4-44000000400-4000000000000000    orthogonal lifted from C8⋊C22
ρ264-4-44000000-4004000000000000000    orthogonal lifted from C8⋊C22
ρ274-44-40000000-440000000000000000    orthogonal lifted from 2+ 1+4
ρ2844-4-40000000000000000000-2-2002-200    complex lifted from D4○SD16
ρ2944-4-400000000000000000002-200-2-200    complex lifted from D4○SD16

Smallest permutation representation of C42.473C23
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 28 21 18)(2 25 22 19)(3 26 23 20)(4 27 24 17)(5 12 15 30)(6 9 16 31)(7 10 13 32)(8 11 14 29)
(1 32 23 12)(2 31 24 11)(3 30 21 10)(4 29 22 9)(5 28 13 20)(6 27 14 19)(7 26 15 18)(8 25 16 17)
(1 23)(2 4)(3 21)(5 32)(6 11)(7 30)(8 9)(10 15)(12 13)(14 31)(16 29)(17 25)(18 20)(19 27)(22 24)(26 28)
(1 24 21 4)(2 3 22 23)(5 8 15 14)(6 13 16 7)(9 32 31 10)(11 30 29 12)(17 18 27 28)(19 20 25 26)

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

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

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

Matrix representation of C42.473C23 in GL8(ℤ)

01000000
-10000000
000-10000
00100000
00000100
0000-1000
0000000-1
00000010
,
-10000000
0-1000000
00-100000
000-10000
00000100
0000-1000
0000000-1
00000010
,
00100000
00010000
-10000000
0-1000000
00000010
00000001
00001000
00000100
,
-10000000
0-1000000
00100000
00010000
00001000
00000-100
0000000-1
000000-10
,
01000000
10000000
000-10000
00-100000
00000100
0000-1000
00000001
000000-10

G:=sub<GL(8,Integers())| [0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0],[0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0] >;

C42.473C23 in GAP, Magma, Sage, TeX

C_4^2._{473}C_2^3
% in TeX

G:=Group("C4^2.473C2^3");
// GroupNames label

G:=SmallGroup(128,2056);
// by ID

G=gap.SmallGroup(128,2056);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,723,2019,346,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.473C23 in TeX

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