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G = C24.66D4order 128 = 27

21st non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.66D4, C223C4≀C2, C4○D4.41D4, D45(C22⋊C4), (C22×D4)⋊18C4, Q85(C22⋊C4), (C22×Q8)⋊15C4, C4.112C22≀C2, (C2×C42)⋊10C22, C22.9C22≀C2, C23.544(C2×D4), (C22×C4).676D4, C24.4C425C2, C2.14(C243C4), (C2×M4(2))⋊37C22, (C23×C4).234C22, C23.196(C22⋊C4), (C22×C4).1323C23, C2.38(C42⋊C22), (C2×C4≀C2)⋊9C2, C2.38(C2×C4≀C2), (C2×C4○D4)⋊12C4, C4.4(C2×C22⋊C4), (C4×C22⋊C4)⋊22C2, (C2×C4).974(C2×D4), (C2×D4).202(C2×C4), (C2×Q8).185(C2×C4), (C22×C4○D4).5C2, (C2×C4).361(C22×C4), (C22×C4).263(C2×C4), (C2×C4).121(C22⋊C4), (C2×C4○D4).254C22, C22.242(C2×C22⋊C4), SmallGroup(128,521)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.66D4
C1C2C4C2×C4C22×C4C23×C4C22×C4○D4 — C24.66D4
C1C2C2×C4 — C24.66D4
C1C2×C4C23×C4 — C24.66D4
C1C2C2C22×C4 — C24.66D4

Generators and relations for C24.66D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=db=ebe-1=bd, ab=ba, eae-1=faf-1=ac=ca, ad=da, bc=cb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be3 >

Subgroups: 620 in 309 conjugacy classes, 70 normal (26 characteristic)
C1, C2 [×3], C2 [×9], C4 [×4], C4 [×10], C22, C22 [×4], C22 [×27], C8 [×2], C2×C4 [×4], C2×C4 [×4], C2×C4 [×40], D4 [×4], D4 [×22], Q8 [×4], Q8 [×6], C23 [×3], C23 [×15], C42 [×4], C22⋊C4 [×4], C2×C8 [×2], M4(2) [×4], C22×C4 [×6], C22×C4 [×19], C2×D4 [×2], C2×D4 [×17], C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×8], C4○D4 [×28], C24, C24, C2.C42, C22⋊C8 [×2], C4≀C2 [×8], C2×C42 [×2], C2×C22⋊C4, C2×M4(2) [×2], C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4 [×4], C2×C4○D4 [×10], C4×C22⋊C4, C24.4C4, C2×C4≀C2 [×4], C22×C4○D4, C24.66D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, C22⋊C4 [×12], C22×C4, C2×D4 [×6], C4≀C2 [×2], C2×C22⋊C4 [×3], C22≀C2 [×4], C243C4, C2×C4≀C2, C42⋊C22, C24.66D4

Smallest permutation representation of C24.66D4
On 32 points
Generators in S32
(2 32)(4 26)(6 28)(8 30)(10 22)(12 24)(14 18)(16 20)
(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)(17 31)(18 28)(19 25)(20 30)(21 27)(22 32)(23 29)(24 26)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
(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 32 9 18)(2 21 14 31)(3 26 11 20)(4 23 16 25)(5 28 13 22)(6 17 10 27)(7 30 15 24)(8 19 12 29)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2L4A4B4C4D4E4F4G4H4I···4U8A8B8C8D
order122222222···2444444444···48888
size111122224···4111122224···48888

38 irreducible representations

dim1111111122224
type++++++++
imageC1C2C2C2C2C4C4C4D4D4D4C4≀C2C42⋊C22
kernelC24.66D4C4×C22⋊C4C24.4C4C2×C4≀C2C22×C4○D4C22×D4C22×Q8C2×C4○D4C22×C4C4○D4C24C22C2
# reps1114122438182

Matrix representation of C24.66D4 in GL4(𝔽17) generated by

1000
0100
0010
00216
,
0400
13000
0010
0001
,
1000
0100
00160
00016
,
16000
01600
0010
0001
,
111100
11600
00116
00216
,
11600
111100
00116
00016
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,2,0,0,0,16],[0,13,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[11,11,0,0,11,6,0,0,0,0,1,2,0,0,16,16],[11,11,0,0,6,11,0,0,0,0,1,0,0,0,16,16] >;

C24.66D4 in GAP, Magma, Sage, TeX

C_2^4._{66}D_4
% in TeX

G:=Group("C2^4.66D4");
// GroupNames label

G:=SmallGroup(128,521);
// by ID

G=gap.SmallGroup(128,521);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,1018,248,124]);
// Polycyclic

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

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