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G = C24.59D4order 128 = 27

14th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.59D4, C4⋊D47C4, C22⋊Q87C4, C42⋊C24C4, (C22×C4).127D4, C23.506(C2×D4), C22.SD161C2, C22.18(C4○D8), C23.34D45C2, C23.31D41C2, C22.7(C23⋊C4), C4⋊D4.140C22, C22⋊C8.165C22, C23.55(C22⋊C4), (C23×C4).212C22, C22.19C24.4C2, (C22×C4).638C23, C22⋊Q8.145C22, C2.C42.6C22, C2.21(C42⋊C22), C2.10(C23.24D4), (C2×C4○D4)⋊4C4, C4⋊C4.16(C2×C4), (C2×C22⋊C8)⋊7C2, (C2×D4).13(C2×C4), C2.21(C2×C23⋊C4), (C2×Q8).13(C2×C4), (C2×C4).1162(C2×D4), (C22×C4).203(C2×C4), (C2×C4).128(C22×C4), (C2×C4).174(C22⋊C4), C22.192(C2×C22⋊C4), SmallGroup(128,248)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.59D4
C1C2C22C23C22×C4C23×C4C22.19C24 — C24.59D4
C1C22C2×C4 — C24.59D4
C1C22C23×C4 — C24.59D4
C1C2C22C22×C4 — C24.59D4

Generators and relations for C24.59D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, faf-1=ad=da, eae-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: 340 in 142 conjugacy classes, 46 normal (34 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×4], C22 [×14], C8 [×2], C2×C4 [×4], C2×C4 [×19], D4 [×7], Q8, C23 [×3], C23 [×6], C42, C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C22×C4 [×6], C22×C4 [×5], C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C24, C2.C42 [×2], C2.C42, C22⋊C8 [×2], C22⋊C8, C2×C22⋊C4, C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C22×C8, C23×C4, C2×C4○D4, C22.SD16 [×2], C23.31D4 [×2], C23.34D4, C2×C22⋊C8, C22.19C24, C24.59D4
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, C4○D8 [×2], C2×C23⋊C4, C23.24D4, C42⋊C22, C24.59D4

Smallest permutation representation of C24.59D4
On 32 points
Generators in S32
(1 22)(2 9)(3 24)(4 11)(5 18)(6 13)(7 20)(8 15)(10 25)(12 27)(14 29)(16 31)(17 26)(19 28)(21 30)(23 32)
(2 32)(4 26)(6 28)(8 30)(9 23)(11 17)(13 19)(15 21)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(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)
(2 30 32 8)(3 29)(4 6 26 28)(7 25)(9 17 23 11)(10 24)(12 16)(13 21 19 15)(14 20)(18 22)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4F4G4H···4N8A···8H
order12222222224···444···48···8
size11112222482···248···84···4

32 irreducible representations

dim111111111122244
type+++++++++
imageC1C2C2C2C2C2C4C4C4C4D4D4C4○D8C23⋊C4C42⋊C22
kernelC24.59D4C22.SD16C23.31D4C23.34D4C2×C22⋊C8C22.19C24C42⋊C2C4⋊D4C22⋊Q8C2×C4○D4C22×C4C24C22C22C2
# reps122111222231822

Matrix representation of C24.59D4 in GL6(𝔽17)

180000
0160000
001000
000100
00150160
001516016
,
1600000
0160000
001000
000100
00150160
001516016
,
1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
1570000
090000
006224
00613149
003151513
0013980
,
1300000
140000
001000
00131600
002112
001601616

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,8,16,0,0,0,0,0,0,1,0,15,15,0,0,0,1,0,16,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,15,15,0,0,0,1,0,16,0,0,0,0,16,0,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,0,0,0,0,0,7,9,0,0,0,0,0,0,6,6,3,13,0,0,2,13,15,9,0,0,2,14,15,8,0,0,4,9,13,0],[13,1,0,0,0,0,0,4,0,0,0,0,0,0,1,13,2,16,0,0,0,16,1,0,0,0,0,0,1,16,0,0,0,0,2,16] >;

C24.59D4 in GAP, Magma, Sage, TeX

C_2^4._{59}D_4
% in TeX

G:=Group("C2^4.59D4");
// GroupNames label

G:=SmallGroup(128,248);
// by ID

G=gap.SmallGroup(128,248);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,387,1123,1018,248,1971]);
// 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,f*a*f^-1=a*d=d*a,e*a*e^-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

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