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G = C24.57D4order 128 = 27

12nd non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.57D4, C22⋊Q85C4, (C22×Q8)⋊4C4, (C22×C4).213D4, C23.501(C2×D4), C23.31D420C2, C22⋊C8.132C22, C22.28(C23⋊C4), C24.4C4.14C2, (C23×C4).210C22, (C22×C4).633C23, C23.34D4.4C2, C22⋊Q8.141C22, C23.174(C22⋊C4), C2.9(C23.38D4), C22.25(C8.C22), C2.C42.1C22, C2.16(C42⋊C22), (C2×C4⋊C4)⋊10C4, C4⋊C4.11(C2×C4), (C2×Q8).9(C2×C4), C2.19(C2×C23⋊C4), (C2×C22⋊Q8).4C2, (C2×C4).1157(C2×D4), (C2×C4).91(C22⋊C4), (C2×C4).123(C22×C4), (C22×C4).201(C2×C4), C22.187(C2×C22⋊C4), SmallGroup(128,243)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.57D4
C1C2C22C23C22×C4C23×C4C2×C22⋊Q8 — C24.57D4
C1C22C2×C4 — C24.57D4
C1C22C23×C4 — C24.57D4
C1C2C22C22×C4 — C24.57D4

Generators and relations for C24.57D4
 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=bcde3 >

Subgroups: 316 in 137 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2 [×5], C4 [×10], C22, C22 [×4], C22 [×11], C8 [×2], C2×C4 [×4], C2×C4 [×22], Q8 [×4], C23 [×3], C23 [×5], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×2], M4(2) [×2], C22×C4 [×6], C22×C4 [×6], C2×Q8 [×2], C2×Q8 [×3], C24, C2.C42 [×2], C2.C42, C22⋊C8 [×2], C22⋊C8, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8 [×4], C22⋊Q8 [×2], C2×M4(2), C23×C4, C22×Q8, C23.31D4 [×4], C23.34D4, C24.4C4, C2×C22⋊Q8, C24.57D4
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, C23.38D4, C42⋊C22, C24.57D4

Character table of C24.57D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 11112222422444888888888888
ρ111111111111111111111111111    trivial
ρ211111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ31111-11-11-1111-1-1-11-111-1-11-1-111    linear of order 2
ρ41111-11-11-1111-1-11-11-1-111-1-1-111    linear of order 2
ρ5111111111111111-1111-1-1-1-1-1-1-1    linear of order 2
ρ611111111111111-11-1-1-1111-1-1-1-1    linear of order 2
ρ71111-11-11-1111-1-1-1-1-11111-111-1-1    linear of order 2
ρ81111-11-11-1111-1-1111-1-1-1-1111-1-1    linear of order 2
ρ911111-11-1-1-1-111-1-i-1ii-i1-11-iii-i    linear of order 4
ρ1011111-11-1-1-1-111-1i-1-i-ii1-11i-i-ii    linear of order 4
ρ111111-1-1-1-11-1-11-11i-1-ii-i-111i-ii-i    linear of order 4
ρ121111-1-1-1-11-1-11-11-i-1i-ii-111-ii-ii    linear of order 4
ρ1311111-11-1-1-1-111-1-i1ii-i-11-1i-i-ii    linear of order 4
ρ1411111-11-1-1-1-111-1i1-i-ii-11-1-iii-i    linear of order 4
ρ151111-1-1-1-11-1-11-11i1-ii-i1-1-1-ii-ii    linear of order 4
ρ161111-1-1-1-11-1-11-11-i1i-ii1-1-1i-ii-i    linear of order 4
ρ17222222222-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ182222-22-22-2-2-2-222000000000000    orthogonal lifted from D4
ρ192222-2-2-2-2222-22-2000000000000    orthogonal lifted from D4
ρ2022222-22-2-222-2-22000000000000    orthogonal lifted from D4
ρ2144-4-4-4040000000000000000000    orthogonal lifted from C23⋊C4
ρ2244-4-440-40000000000000000000    orthogonal lifted from C23⋊C4
ρ234-4-44040-4000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ244-4-440-404000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-44-4000004i-4i000000000000000    complex lifted from C42⋊C22
ρ264-44-400000-4i4i000000000000000    complex lifted from C42⋊C22

Smallest permutation representation of C24.57D4
On 32 points
Generators in S32
(1 24)(2 21)(3 18)(4 23)(5 20)(6 17)(7 22)(8 19)(9 29)(10 26)(11 31)(12 28)(13 25)(14 30)(15 27)(16 32)
(2 12)(4 14)(6 16)(8 10)(17 32)(19 26)(21 28)(23 30)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)
(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)
(2 4 12 14)(3 9)(6 8 16 10)(7 13)(17 23 32 30)(18 25)(19 28 26 21)(20 24)(22 29)(27 31)

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

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

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

Matrix representation of C24.57D4 in GL8(𝔽17)

01000000
10000000
00010000
00100000
000016000
000001600
000000160
000000016
,
10000000
01000000
001600000
000160000
00001000
00000100
0000160160
0000160016
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
00400000
000130000
10000000
016000000
00001002
00001011
000000016
0000161016
,
10000000
016000000
00400000
000130000
00001000
000011600
0000160016
00000010

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

C24.57D4 in GAP, Magma, Sage, TeX

C_2^4._{57}D_4
% in TeX

G:=Group("C2^4.57D4");
// GroupNames label

G:=SmallGroup(128,243);
// by ID

G=gap.SmallGroup(128,243);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,1123,1018,248,1971]);
// 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*c*d*e^3>;
// generators/relations

Export

Character table of C24.57D4 in TeX

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