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G = C24.14D4order 128 = 27

14th non-split extension by C24 of D4 acting faithfully

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

Aliases: C24.14D4, C23.7SD16, C4⋊C4.11D4, C23⋊C8.8C2, (C2×Q8).15D4, (C22×C4).49D4, C2.8(Q8⋊D4), C23.525(C2×D4), C232Q8.2C2, C22⋊Q8.9C22, (C22×C4).14C23, C22.14(C2×SD16), C22.135C22≀C2, C2.10(D4.9D4), C23.47D425C2, C23.31D414C2, C22⋊C8.114C22, C23.11D4.2C2, C2.5(C23.7D4), C22.31(C8.C22), C2.C42.21C22, (C2×C4).203(C2×D4), (C2×C4⋊C4).19C22, (C2×C22⋊C4).97C22, SmallGroup(128,340)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C24.14D4
C1C2C22C23C22×C4C2×C22⋊C4C232Q8 — C24.14D4
C1C22C22×C4 — C24.14D4
C1C22C22×C4 — C24.14D4
C1C2C22C22×C4 — C24.14D4

Generators and relations for C24.14D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=c, f2=d, eae-1=faf-1=ab=ba, ac=ca, ad=da, bc=cb, ebe-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de3 >

Subgroups: 284 in 114 conjugacy classes, 34 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×9], C22 [×3], C22 [×9], C8 [×2], C2×C4 [×2], C2×C4 [×15], Q8 [×3], C23, C23 [×2], C23 [×4], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C22×C4 [×2], C22×C4 [×4], C2×Q8 [×2], C2×Q8, C24, C2.C42, C2.C42, C22⋊C8 [×2], Q8⋊C4 [×2], C4.Q8 [×2], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, C22⋊Q8 [×2], C22⋊Q8 [×5], C23⋊C8, C23.31D4 [×2], C23.11D4, C23.47D4 [×2], C232Q8, C24.14D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, SD16 [×2], C2×D4 [×3], C22≀C2, C2×SD16, C8.C22, Q8⋊D4, D4.9D4, C23.7D4, C24.14D4

Character table of C24.14D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11112244448888888888888
ρ111111111111111111111111    trivial
ρ2111111-1-111-11-11-11-11-1-11-11    linear of order 2
ρ311111111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ4111111-1-111-1-11-111-11-11-11-1    linear of order 2
ρ51111111111-11111-11-1-1-1-1-1-1    linear of order 2
ρ6111111-1-11111-11-1-1-1-111-11-1    linear of order 2
ρ71111111111-1-1-1-1-1-11-1-11111    linear of order 2
ρ8111111-1-1111-11-11-1-1-11-11-11    linear of order 2
ρ92222-2-200-2200-20200000000    orthogonal lifted from D4
ρ1022222222-2-2000000-2000000    orthogonal lifted from D4
ρ112222-2-2002-2020-2000000000    orthogonal lifted from D4
ρ122222-2-2002-20-202000000000    orthogonal lifted from D4
ρ13222222-2-2-2-20000002000000    orthogonal lifted from D4
ρ142222-2-200-220020-200000000    orthogonal lifted from D4
ρ1522-2-2-222-200000000000-2-2--2--2    complex lifted from SD16
ρ1622-2-2-22-2200000000000--2-2-2--2    complex lifted from SD16
ρ1722-2-2-222-200000000000--2--2-2-2    complex lifted from SD16
ρ1822-2-2-22-2200000000000-2--2--2-2    complex lifted from SD16
ρ1944-4-44-400000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ204-4-440000002i0000000-2i0000    complex lifted from D4.9D4
ρ214-44-4000000000002i0-2i00000    complex lifted from C23.7D4
ρ224-44-400000000000-2i02i00000    complex lifted from C23.7D4
ρ234-4-44000000-2i00000002i0000    complex lifted from D4.9D4

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

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

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

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

Matrix representation of C24.14D4 in GL6(𝔽17)

1600000
0160000
00161500
000100
00001615
000001
,
100000
010000
0016000
0001600
000010
000001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
070000
570000
000048
0000013
00161500
001100
,
1600000
1610000
001200
00161600
000048
0000013

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,15,1,0,0,0,0,0,0,16,0,0,0,0,0,15,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,5,0,0,0,0,7,7,0,0,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,4,0,0,0,0,0,8,13,0,0],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,2,16,0,0,0,0,0,0,4,0,0,0,0,0,8,13] >;

C24.14D4 in GAP, Magma, Sage, TeX

C_2^4._{14}D_4
% in TeX

G:=Group("C2^4.14D4");
// GroupNames label

G:=SmallGroup(128,340);
// by ID

G=gap.SmallGroup(128,340);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,232,422,1123,570,521,136,1411]);
// Polycyclic

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

Export

Character table of C24.14D4 in TeX

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