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G = C42.46C23  order 128 = 27

46th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.46C23, C4.562+ (1+4), C8⋊8D4⋊50C2, C8⋊9D4⋊14C2, C4⋊C8⋊34C22, C4⋊C4.365D4, C4⋊Q8⋊23C22, D4⋊2Q8⋊18C2, (C2×D4).169D4, D4⋊5D4.3C2, C8.D4⋊23C2, (C4×Q8)⋊27C22, C4.Q8⋊25C22, C2.D8⋊36C22, C8⋊C4⋊23C22, D4.23(C4○D4), C22⋊SD16⋊21C2, D4.7D4⋊42C2, C4⋊C4.408C23, C22⋊C8⋊30C22, (C2×C8).353C23, (C2×C4).503C24, Q8.D4⋊40C2, (C22×C8)⋊43C22, C22⋊C4.165D4, (C2×Q16)⋊32C22, C23.322(C2×D4), C22⋊Q8⋊18C22, SD16⋊C4⋊34C2, C2.75(D4○SD16), Q8⋊C4⋊43C22, (C4×D4).156C22, (C2×D4).232C23, C4⋊D4.82C22, (C2×Q8).216C23, C2.139(D4⋊5D4), C42⋊C2⋊23C22, C23.24D4⋊31C2, C23.20D4⋊32C2, C23.19D4⋊32C2, C23.37D4⋊13C2, (C2×SD16).54C22, C4.4D4.64C22, C22.763(C22×D4), D4⋊C4.187C22, C2.84(D8⋊C22), C22.50C24⋊4C2, (C22×C4).1147C23, (C22×D4).411C22, C42.28C22⋊14C2, (C2×M4(2)).111C22, C4.228(C2×C4○D4), (C2×C4).600(C2×D4), (C2×C4○D4).208C22, SmallGroup(128,2043)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C42.46C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — D4⋊5D4 — C42.46C23
Lower central
C1 — C2 — C2×C4 — C42.46C23
Upper central
C1 — C22 — C4×D4 — C42.46C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.46C23

Subgroups: 432 in 201 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×17], C8 [×4], C2×C4 [×5], C2×C4 [×13], D4 [×2], D4 [×9], Q8 [×6], C23 [×2], C23 [×7], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×5], C4⋊C4 [×5], C2×C8 [×4], C2×C8, M4(2), SD16 [×3], Q16, C22×C4 [×2], C22×C4 [×2], C2×D4 [×3], C2×D4 [×6], C2×Q8 [×2], C2×Q8, C4○D4 [×3], C24, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×4], C4⋊C8, C4.Q8 [×2], C2.D8, C2×C22⋊C4, C42⋊C2 [×2], C4×D4 [×2], C4×Q8, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8 [×3], C22.D4, C4.4D4, C4.4D4, C42⋊2C2 [×2], C4⋊Q8, C22×C8, C2×M4(2), C2×SD16 [×2], C2×Q16, C22×D4, C2×C4○D4, C23.24D4, C23.37D4, C8⋊9D4, SD16⋊C4, C22⋊SD16, D4.7D4, Q8.D4, C8⋊8D4, C8.D4, D4⋊2Q8, C23.19D4, C23.20D4, C42.28C22, D4⋊5D4, C22.50C24, C42.46C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C22×D4, C2×C4○D4, 2+ (1+4), D4⋊5D4, D8⋊C22, D4○SD16, C42.46C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=d2=1, c2=a2, e2=b2, ab=ba, cac-1=eae-1=a-1, dad=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, ede-1=b2d >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

1300000
240000
000400
0013000
000004
0000130
,
100000
010000
000100
0016000
0000016
000010
,
140000
8160000
000010
000001
001000
000100
,
100000
010000
001000
0001600
0000016
0000160
,
140000
0160000
0001600
001000
0000016
000010

G:=sub<GL(6,GF(17))| [13,2,0,0,0,0,0,4,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,8,0,0,0,0,4,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[1,0,0,0,0,0,4,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0] >;
 

Character table of C42.46C23

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F
 size 11114444822224444488888444488
ρ111111111111111111111111111111    trivial
ρ211111-11111-1-111-1-1-1-1-11-11-11-11-1-11    linear of order 2
ρ3111111111111111-11-11-1-1-11-1-1-1-1-1-1    linear of order 2
ρ411111-11111-1-111-11-11-1-11-1-1-11-111-1    linear of order 2
ρ51111-1-1-1-111-1-11-1111111-1-1-11-11-11-1    linear of order 2
ρ61111-11-1-111111-1-1-1-1-1-111-111111-1-1    linear of order 2
ρ71111-1-1-1-111-1-11-11-11-11-111-1-11-11-11    linear of order 2
ρ81111-11-1-111111-1-11-11-1-1-111-1-1-1-111    linear of order 2
ρ91111-11-11-1111111-11-1-1-1-1-1-1111111    linear of order 2
ρ101111-1-1-11-11-1-111-11-111-11-111-11-1-11    linear of order 2
ρ111111-11-11-1111111111-1111-1-1-1-1-1-1-1    linear of order 2
ρ121111-1-1-11-11-1-111-1-1-1-111-111-11-111-1    linear of order 2
ρ1311111-11-1-11-1-11-11-11-1-1-11111-11-11-1    linear of order 2
ρ141111111-1-11111-1-11-111-1-11-11111-1-1    linear of order 2
ρ1511111-11-1-11-1-11-11111-11-1-11-11-11-11    linear of order 2
ρ161111111-1-11111-1-1-1-1-1111-1-1-1-1-1-111    linear of order 2
ρ172222020-20-2-2-2-2220-2000000000000    orthogonal lifted from D4
ρ18222202020-2-2-2-2-2-202000000000000    orthogonal lifted from D4
ρ1922220-20-20-222-22-202000000000000    orthogonal lifted from D4
ρ2022220-2020-222-2-220-2000000000000    orthogonal lifted from D4
ρ212-22-220-200-2002002i02i0000002i02i00    complex lifted from C4○D4
ρ222-22-2-20200-2002002i02i0000002i02i00    complex lifted from C4○D4
ρ232-22-2-20200-2002002i02i0000002i02i00    complex lifted from C4○D4
ρ242-22-220-200-2002002i02i0000002i02i00    complex lifted from C4○D4
ρ254-44-400000400-40000000000000000    orthogonal lifted from 2+ (1+4)
ρ264-4-440000004i4i00000000000000000    complex lifted from D8⋊C22
ρ274-4-440000004i4i00000000000000000    complex lifted from D8⋊C22
ρ2844-4-400000000000000000002√-202√-2000    complex lifted from D4○SD16
ρ2944-4-400000000000000000002√-202√-2000    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

C_4^2._{46}C_2^3
 
% in TeX
 
G:=Group("C4^2.46C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2043);
 
// by ID
 
G=gap.SmallGroup(128,2043);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,723,346,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2,e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1,d*a*d=a*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*c,e*d*e^-1=b^2*d>;
 
// generators/relations
 

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