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G = C24.58D4order 128 = 27

13rd non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.58D4, C4⋊D46C4, C22⋊Q86C4, C42⋊C23C4, C4.28(C23⋊C4), C23.503(C2×D4), (C22×C4).215D4, C23.7Q81C2, C24.4C422C2, C22.SD1621C2, C4⋊D4.138C22, C23.31D422C2, C22⋊C8.134C22, C22.17(C8⋊C22), C23.54(C22⋊C4), (C22×C4).635C23, C22.19C24.3C2, (C23×C4).211C22, C22⋊Q8.143C22, C22.13(C8.C22), C2.C42.3C22, C2.18(C42⋊C22), C2.11(C23.36D4), (C2×C4○D4)⋊3C4, C4⋊C4.13(C2×C4), (C2×D4).11(C2×C4), C2.20(C2×C23⋊C4), (C2×Q8).11(C2×C4), (C2×C4).1159(C2×D4), (C2×C4).125(C22×C4), (C22×C4).202(C2×C4), (C2×C4).352(C22⋊C4), C22.189(C2×C22⋊C4), SmallGroup(128,245)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.58D4
C1C2C22C23C22×C4C23×C4C22.19C24 — C24.58D4
C1C22C2×C4 — C24.58D4
C1C22C23×C4 — C24.58D4
C1C2C22C22×C4 — C24.58D4

Generators and relations for C24.58D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, bc=cb, bd=db, ebe-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bde3 >

Subgroups: 348 in 143 conjugacy classes, 46 normal (36 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×8], C22 [×3], C22 [×15], C8 [×2], C2×C4 [×4], C2×C4 [×20], D4 [×7], Q8, C23 [×3], C23 [×5], C42, C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], M4(2) [×2], C22×C4 [×6], C22×C4 [×5], C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C24, C2.C42 [×2], C22⋊C8 [×2], C22⋊C8, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C2×M4(2), C23×C4, C2×C4○D4, C22.SD16 [×2], C23.31D4 [×2], C23.7Q8, C24.4C4, C22.19C24, C24.58D4
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, C8.C22, C2×C23⋊C4, C23.36D4, C42⋊C22, C24.58D4

Character table of C24.58D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 11112244822224488888888888
ρ111111111111111111111111111    trivial
ρ211111111-1111111-11111-1-1-1-1-1-1    linear of order 2
ρ3111111-1-11-11-111-1-1-11-111-1-1-111    linear of order 2
ρ4111111-1-1-1-11-111-11-11-11-1111-1-1    linear of order 2
ρ51111111111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ611111111-1111111-1-1-1-1-1-1-11111    linear of order 2
ρ7111111-1-1-1-11-111-111-11-1-11-1-111    linear of order 2
ρ8111111-1-11-11-111-1-11-11-11-111-1-1    linear of order 2
ρ91111-1-1-11-11-11-11-1-1ii-i-i11i-ii-i    linear of order 4
ρ101111-1-1-1111-11-11-11ii-i-i-1-1-ii-ii    linear of order 4
ρ111111-1-11-11-1-1-1-111-1-iii-i-11i-i-ii    linear of order 4
ρ121111-1-11-1-1-1-1-1-1111-iii-i1-1-iii-i    linear of order 4
ρ131111-1-1-1111-11-11-11-i-iii-1-1i-ii-i    linear of order 4
ρ141111-1-1-11-11-11-11-1-1-i-iii11-ii-ii    linear of order 4
ρ151111-1-11-1-1-1-1-1-1111i-i-ii1-1i-i-ii    linear of order 4
ρ161111-1-11-11-1-1-1-111-1i-i-ii-11-iii-i    linear of order 4
ρ172222-2-2-220-22-22-2200000000000    orthogonal lifted from D4
ρ18222222-2-202-22-2-2200000000000    orthogonal lifted from D4
ρ192222-2-22-202222-2-200000000000    orthogonal lifted from D4
ρ20222222220-2-2-2-2-2-200000000000    orthogonal lifted from D4
ρ214-44-40000040-400000000000000    orthogonal lifted from C23⋊C4
ρ224-44-400000-40400000000000000    orthogonal lifted from C23⋊C4
ρ2344-4-44-400000000000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-4-4400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-4-44000000-4i04i0000000000000    complex lifted from C42⋊C22
ρ264-4-440000004i0-4i0000000000000    complex lifted from C42⋊C22

Smallest permutation representation of C24.58D4
On 32 points
Generators in S32
(1 19)(2 14)(3 21)(4 16)(5 23)(6 10)(7 17)(8 12)(9 26)(11 28)(13 30)(15 32)(18 29)(20 31)(22 25)(24 27)
(2 27)(4 29)(6 31)(8 25)(10 20)(12 22)(14 24)(16 18)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(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 17)(2 14 27 24)(3 13)(4 22 29 12)(5 21)(6 10 31 20)(7 9)(8 18 25 16)(11 30)(15 26)(19 32)(23 28)

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

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

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

Matrix representation of C24.58D4 in GL8(𝔽17)

04000000
130000000
000130000
00400000
00000100
00001000
000016161616
00000001
,
10000000
01000000
001600000
000160000
00001000
00000100
000000160
00001515016
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
000130000
001300000
01000000
10000000
00001111
00000010
00001000
00001501516
,
01000000
10000000
000130000
001300000
00001000
000001600
00001111
00001501516

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

C24.58D4 in GAP, Magma, Sage, TeX

C_2^4._{58}D_4
% in TeX

G:=Group("C2^4.58D4");
// GroupNames label

G:=SmallGroup(128,245);
// by ID

G=gap.SmallGroup(128,245);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,352,1123,1018,248,1971]);
// 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,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,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 C24.58D4 in TeX

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