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G = C24.56D4order 128 = 27

11st non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.56D4, C4⋊D45C4, (C22×D4)⋊4C4, C22⋊C842C22, (C22×C4).212D4, C23.500(C2×D4), C24.4C421C2, C22.SD1620C2, C23.34D44C2, C4⋊D4.136C22, C22.36(C8⋊C22), C22.27(C23⋊C4), (C23×C4).209C22, (C22×C4).632C23, C2.C421C22, C23.173(C22⋊C4), C2.9(C23.37D4), C2.15(C42⋊C22), (C2×C4⋊C4)⋊9C4, C4⋊C4.10(C2×C4), (C2×D4).9(C2×C4), (C2×C4⋊D4).4C2, C2.18(C2×C23⋊C4), (C2×C4).1156(C2×D4), (C2×C4).90(C22⋊C4), (C2×C4).122(C22×C4), (C22×C4).200(C2×C4), C22.186(C2×C22⋊C4), SmallGroup(128,242)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.56D4
C1C2C22C23C22×C4C23×C4C2×C4⋊D4 — C24.56D4
C1C22C2×C4 — C24.56D4
C1C22C23×C4 — C24.56D4
C1C2C22C22×C4 — C24.56D4

Generators and relations for C24.56D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, ebe-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bde3 >

Subgroups: 444 in 163 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2 [×7], C4 [×8], C22, C22 [×4], C22 [×21], C8 [×2], C2×C4 [×4], C2×C4 [×18], D4 [×12], C23 [×3], C23 [×13], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], M4(2) [×2], C22×C4 [×6], C22×C4 [×5], C2×D4 [×2], C2×D4 [×11], C24, C24, C2.C42 [×2], C2.C42, C22⋊C8 [×2], C22⋊C8, C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×4], C4⋊D4 [×2], C2×M4(2), C23×C4, C22×D4, C22×D4, C22.SD16 [×4], C23.34D4, C24.4C4, C2×C4⋊D4, C24.56D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C8⋊C22 [×2], C2×C23⋊C4, C23.37D4, C42⋊C22, C24.56D4

Character table of C24.56D4

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11112222488224448888888888
ρ111111111111111111111111111    trivial
ρ21111111111111111-1-1-11-11-1-1-1-1    linear of order 2
ρ311111-1-11-1-1111-11-1-11-111-111-1-1    linear of order 2
ρ411111-1-11-1-1111-11-11-111-1-1-1-111    linear of order 2
ρ511111-1-11-11-111-11-11-11-1-1111-1-1    linear of order 2
ρ611111-1-11-11-111-11-1-11-1-111-1-111    linear of order 2
ρ7111111111-1-111111-1-1-1-1-1-11111    linear of order 2
ρ8111111111-1-111111111-11-1-1-1-1-1    linear of order 2
ρ91111-1-1-1-11-11-1-111-1i-i-i-1i1-ii-ii    linear of order 4
ρ101111-111-1-1-1-1-1-1-111ii-i1-i1-iii-i    linear of order 4
ρ111111-1-1-1-111-1-1-111-1-iii1-i-1-ii-ii    linear of order 4
ρ121111-111-1-111-1-1-111-i-ii-1i-1-iii-i    linear of order 4
ρ131111-111-1-1-1-1-1-1-111-i-ii1i1i-i-ii    linear of order 4
ρ141111-1-1-1-11-11-1-111-1-iii-1-i1i-ii-i    linear of order 4
ρ151111-111-1-111-1-1-111ii-i-1-i-1i-i-ii    linear of order 4
ρ161111-1-1-1-111-1-1-111-1i-i-i1i-1i-ii-i    linear of order 4
ρ172222-2-2-2-220022-2-220000000000    orthogonal lifted from D4
ρ1822222222200-2-2-2-2-20000000000    orthogonal lifted from D4
ρ1922222-2-22-200-2-22-220000000000    orthogonal lifted from D4
ρ202222-222-2-200222-2-20000000000    orthogonal lifted from D4
ρ214-4-44400-4000000000000000000    orthogonal lifted from C8⋊C22
ρ2244-4-40-440000000000000000000    orthogonal lifted from C23⋊C4
ρ234-4-44-4004000000000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-404-40000000000000000000    orthogonal lifted from C23⋊C4
ρ254-44-40000000-4i4i0000000000000    complex lifted from C42⋊C22
ρ264-44-400000004i-4i0000000000000    complex lifted from C42⋊C22

Smallest permutation representation of C24.56D4
On 32 points
Generators in S32
(1 5)(3 7)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)
(1 14)(2 21)(3 16)(4 23)(5 10)(6 17)(7 12)(8 19)(9 25)(11 27)(13 29)(15 31)(18 28)(20 30)(22 32)(24 26)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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 14 8)(2 28 21 18)(3 11 16 27)(4 5 23 10)(6 32 17 22)(7 15 12 31)(9 24 25 26)(13 20 29 30)

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

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

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

Matrix representation of C24.56D4 in GL8(𝔽17)

160000000
016000000
00100000
00010000
000016000
000001600
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
000021600
00000010
000000216
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
00100000
000160000
016000000
160000000
000000116
000000016
000013400
00000400
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
000021600

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

C24.56D4 in GAP, Magma, Sage, TeX

C_2^4._{56}D_4
% in TeX

G:=Group("C2^4.56D4");
// GroupNames label

G:=SmallGroup(128,242);
// by ID

G=gap.SmallGroup(128,242);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1123,1018,248,1971]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=b,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*d*e^3>;
// generators/relations

Export

Character table of C24.56D4 in TeX

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