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G = C42.58D4order 128 = 27

40th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.58D4, (C4×D4)⋊5C4, (C4×Q8)⋊5C4, C4.4D49C4, C42.C25C4, C428C41C2, C42.75(C2×C4), C23.502(C2×D4), (C22×C4).214D4, C42.6C431C2, C22.SD16.6C2, C23.31D421C2, C4⋊D4.137C22, C22⋊C8.133C22, C22.16(C8⋊C22), (C22×C4).634C23, (C2×C42).178C22, C22⋊Q8.142C22, C22.12(C8.C22), C2.C42.2C22, C2.17(C42⋊C22), C2.10(C23.36D4), C23.36C23.7C2, C2.18(C23.C23), C4⋊C4.12(C2×C4), (C2×D4).10(C2×C4), (C2×Q8).10(C2×C4), (C2×C4).1158(C2×D4), (C2×C4).92(C22⋊C4), (C2×C4).124(C22×C4), C22.188(C2×C22⋊C4), SmallGroup(128,244)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.58D4
C1C2C22C23C22×C4C2×C42C23.36C23 — C42.58D4
C1C22C2×C4 — C42.58D4
C1C22C2×C42 — C42.58D4
C1C2C22C22×C4 — C42.58D4

Generators and relations for C42.58D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=b-1c-1 >

Subgroups: 244 in 111 conjugacy classes, 44 normal (36 characteristic)
C1, C2 [×3], C2 [×3], C4 [×11], C22 [×3], C22 [×5], C8 [×2], C2×C4 [×6], C2×C4 [×15], D4 [×3], Q8, C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×2], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2.C42 [×2], C2.C42, C8⋊C4, C22⋊C8 [×2], C4⋊C8, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C22.SD16 [×2], C23.31D4 [×2], C428C4, C42.6C4, C23.36C23, C42.58D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C8⋊C22, C8.C22, C23.C23, C23.36D4, C42⋊C22, C42.58D4

Character table of C42.58D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D
 size 11112282222444488888888888
ρ111111111111111111111111111    trivial
ρ2111111-11-11-1-1-1-11-1-1111-11-1-111    linear of order 2
ρ3111111111111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ4111111-11-11-1-1-1-1111-1-11-1111-1-1    linear of order 2
ρ511111111-11-1-1-1-11-1-111-11-111-1-1    linear of order 2
ρ6111111-1111111111111-1-1-1-1-1-1-1    linear of order 2
ρ711111111-11-1-1-1-1111-1-1-11-1-1-111    linear of order 2
ρ8111111-111111111-1-1-1-1-1-1-11111    linear of order 2
ρ91111-1-11-1-1-1-11-111i-ii-i-1-11-ii-ii    linear of order 4
ρ101111-1-1-1-11-11-11-11-iii-i-111i-i-ii    linear of order 4
ρ111111-1-11-11-11-11-11i-i-ii1-1-1i-i-ii    linear of order 4
ρ121111-1-1-1-1-1-1-11-111-ii-ii11-1-ii-ii    linear of order 4
ρ131111-1-11-11-11-11-11-iii-i1-1-1-iii-i    linear of order 4
ρ141111-1-1-1-1-1-1-11-111i-ii-i11-1i-ii-i    linear of order 4
ρ151111-1-11-1-1-1-11-111-ii-ii-1-11i-ii-i    linear of order 4
ρ161111-1-1-1-11-11-11-11i-i-ii-111-iii-i    linear of order 4
ρ172222-2-202222-2-22-200000000000    orthogonal lifted from D4
ρ182222220-22-222-2-2-200000000000    orthogonal lifted from D4
ρ192222-2-202-22-222-2-200000000000    orthogonal lifted from D4
ρ202222220-2-2-2-2-222-200000000000    orthogonal lifted from D4
ρ214-4-444-400000000000000000000    orthogonal lifted from C8⋊C22
ρ224-4-44-4400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2344-4-400004i0-4i000000000000000    complex lifted from C23.C23
ρ244-44-4000-4i04i0000000000000000    complex lifted from C42⋊C22
ρ254-44-40004i0-4i0000000000000000    complex lifted from C42⋊C22
ρ2644-4-40000-4i04i000000000000000    complex lifted from C23.C23

Smallest permutation representation of C42.58D4
On 32 points
Generators in S32
(1 12 25 22)(2 9 26 19)(3 14 27 24)(4 11 28 21)(5 16 29 18)(6 13 30 23)(7 10 31 20)(8 15 32 17)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)
(2 28 30 8)(3 27)(4 6 32 26)(7 31)(9 17 23 11)(10 24)(12 16)(13 21 19 15)(14 20)(18 22)
(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)

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

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

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

Matrix representation of C42.58D4 in GL8(𝔽17)

115000000
016000000
001150000
000160000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
0000151600
000000160
00000021
,
10000000
116000000
001300000
001340000
00001100
000001600
00000011
0000001516
,
001300000
001340000
10000000
116000000
00000011
0000001516
00001100
000001600

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

C42.58D4 in GAP, Magma, Sage, TeX

C_4^2._{58}D_4
% in TeX

G:=Group("C4^2.58D4");
// GroupNames label

G:=SmallGroup(128,244);
// by ID

G=gap.SmallGroup(128,244);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,184,1123,1018,248,1971]);
// Polycyclic

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

Export

Character table of C42.58D4 in TeX

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