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G = C42.54C23order 128 = 27

54th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.54C23, C4.642+ 1+4, C87D440C2, C89D421C2, C8⋊D440C2, C4⋊C837C22, C4⋊C4.369D4, C4⋊Q825C22, D4⋊D445C2, D8⋊C424C2, C22⋊D832C2, D45D410C2, D4⋊Q835C2, (C2×D4).173D4, C2.50(D4○D8), (C4×D4)⋊27C22, C2.D838C22, C4.Q827C22, C8⋊C426C22, D4.27(C4○D4), D4.2D442C2, C4⋊D418C22, C4⋊C4.412C23, C22⋊C833C22, (C2×C8).190C23, (C2×C4).511C24, C22⋊C4.169D4, (C22×C8)⋊33C22, (C2×D8).85C22, C23.328(C2×D4), D4⋊C455C22, Q8⋊C445C22, (C2×SD16)⋊33C22, (C2×D4).237C23, (C2×Q8).223C23, C2.147(D45D4), C42⋊C225C22, C22⋊Q8.85C22, C23.24D420C2, C23.37D415C2, C23.20D436C2, C23.19D436C2, C4.4D4.68C22, C22.771(C22×D4), C2.88(D8⋊C22), C22.49C245C2, (C22×C4).1155C23, (C22×D4).414C22, C42.28C2218C2, (C2×M4(2)).117C22, C4.236(C2×C4○D4), (C2×C4).608(C2×D4), (C2×C4○D4).214C22, SmallGroup(128,2051)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.54C23
C1C2C4C2×C4C22×C4C2×C4○D4D45D4 — C42.54C23
C1C2C2×C4 — C42.54C23
C1C22C4×D4 — C42.54C23
C1C2C2C2×C4 — C42.54C23

Generators and relations for C42.54C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=a2b2, 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 >

Subgroups: 472 in 207 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×9], C22, C22 [×20], C8 [×4], C2×C4 [×5], C2×C4 [×14], D4 [×2], D4 [×13], Q8 [×3], C23 [×2], C23 [×8], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×5], C4⋊C4 [×2], C2×C8 [×4], C2×C8, M4(2), D8 [×3], SD16, C22×C4 [×2], C22×C4 [×3], C2×D4 [×4], C2×D4 [×8], C2×Q8, C2×Q8, C4○D4 [×3], C24, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×7], Q8⋊C4 [×3], C4⋊C8, C4.Q8, C2.D8 [×2], C2×C22⋊C4, C42⋊C2 [×2], C42⋊C2, C4×D4 [×3], C22≀C2, C4⋊D4 [×3], C4⋊D4 [×2], C22⋊Q8, C22.D4, C4.4D4, C4.4D4 [×2], C4⋊Q8, C22×C8, C2×M4(2), C2×D8 [×2], C2×SD16, C22×D4, C2×C4○D4, C23.24D4, C23.37D4, C89D4, D8⋊C4, C22⋊D8, D4⋊D4, D4.2D4, C87D4, C8⋊D4, D4⋊Q8, C23.19D4, C23.20D4, C42.28C22, D45D4, C22.49C24, C42.54C23
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, D45D4, D8⋊C22, D4○D8, C42.54C23

Character table of C42.54C23

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F
 size 11114444882222444448888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-11-1-11-1-111-111-1-11111-11-11-1    linear of order 2
ρ311111-11-1-1-11-1-11-11-1-11111-11-11-1-11    linear of order 2
ρ41111-11-1-1111111-1-1-1-1-1-111-11111-1-1    linear of order 2
ρ51111-1-1-1-1-111-1-11-11-1-1111-11-11-111-1    linear of order 2
ρ61111111-11-11111-1-1-1-1-1-11-11-1-1-1-111    linear of order 2
ρ71111-11-111-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ811111-111-111-1-111-111-1-11-1-1-11-11-11    linear of order 2
ρ91111-11-1-1-111111-1-111-11-11-1-1-1-1-111    linear of order 2
ρ1011111-11-11-11-1-11-11111-1-11-1-11-111-1    linear of order 2
ρ111111-1-1-111-11-1-111-1-1-1-11-111-11-11-11    linear of order 2
ρ1211111111-11111111-1-11-1-111-1-1-1-1-1-1    linear of order 2
ρ1311111-111111-1-111-1-1-1-11-1-1-11-11-11-1    linear of order 2
ρ141111-11-11-1-1111111-1-11-1-1-1-1111111    linear of order 2
ρ151111111-1-1-11111-1-111-11-1-111111-1-1    linear of order 2
ρ161111-1-1-1-1111-1-11-11111-1-1-111-11-1-11    linear of order 2
ρ1722220-20200-222-2-2200-20000000000    orthogonal lifted from D4
ρ182222020200-2-2-2-2-2-20020000000000    orthogonal lifted from D4
ρ1922220-20-200-222-22-20020000000000    orthogonal lifted from D4
ρ202222020-200-2-2-2-22200-20000000000    orthogonal lifted from D4
ρ212-22-220-2000-2002002i-2i0000002i0-2i00    complex lifted from C4○D4
ρ222-22-220-2000-200200-2i2i000000-2i02i00    complex lifted from C4○D4
ρ232-22-2-202000-200200-2i2i0000002i0-2i00    complex lifted from C4○D4
ρ242-22-2-202000-2002002i-2i000000-2i02i00    complex lifted from C4○D4
ρ254-44-4000000400-4000000000000000    orthogonal lifted from 2+ 1+4
ρ2644-4-40000000000000000000220-22000    orthogonal lifted from D4○D8
ρ2744-4-40000000000000000000-22022000    orthogonal lifted from D4○D8
ρ284-4-440000000-4i4i0000000000000000    complex lifted from D8⋊C22
ρ294-4-4400000004i-4i0000000000000000    complex lifted from D8⋊C22

Smallest permutation representation of C42.54C23
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 25 18 22)(2 26 19 23)(3 27 20 24)(4 28 17 21)(5 10 29 16)(6 11 30 13)(7 12 31 14)(8 9 32 15)
(1 15 20 11)(2 14 17 10)(3 13 18 9)(4 16 19 12)(5 26 31 21)(6 25 32 24)(7 28 29 23)(8 27 30 22)
(1 20)(2 4)(3 18)(5 12)(6 15)(7 10)(8 13)(9 30)(11 32)(14 29)(16 31)(17 19)(21 26)(22 24)(23 28)(25 27)
(1 22 18 25)(2 21 19 28)(3 24 20 27)(4 23 17 26)(5 10 29 16)(6 9 30 15)(7 12 31 14)(8 11 32 13)

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

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

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

Matrix representation of C42.54C23 in GL6(𝔽17)

16160000
210000
0000115
0000116
0011500
0011600
,
100000
010000
0011500
0011600
0000115
0000116
,
1300000
840000
000600
003000
000006
000030
,
100000
010000
001000
0011600
000010
0000116
,
110000
0160000
0016200
0016100
0000115
0000116

G:=sub<GL(6,GF(17))| [16,2,0,0,0,0,16,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,1,1,0,0,0,0,15,16,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[13,8,0,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,6,0,0,0,0,0,0,0,0,3,0,0,0,0,6,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16],[1,0,0,0,0,0,1,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16] >;

C42.54C23 in GAP, Magma, Sage, TeX

C_4^2._{54}C_2^3
% in TeX

G:=Group("C4^2.54C2^3");
// GroupNames label

G:=SmallGroup(128,2051);
// by ID

G=gap.SmallGroup(128,2051);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,346,248,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2*b^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

Export

Character table of C42.54C23 in TeX

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