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G = C24.166D4order 128 = 27

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

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C24.166D4, C25.91C22, C23.749C24, C24.596C23, (C24×C4)⋊3C2, (C22×C4)⋊53D4, C243C429C2, C23.631(C2×D4), C22.56C22≀C2, C23.249(C4○D4), C23.34D463C2, (C22×C4).259C23, (C23×C4).649C22, C23.8Q8146C2, C22.459(C22×D4), C23.23D4112C2, C2.C4247C22, (C22×D4).308C22, C224(C22.D4), C2.92(C22.19C24), (C2×C4⋊C4)⋊42C22, C2.32(C2×C22≀C2), (C2×C4).1204(C2×D4), (C2×C22≀C2).16C2, (C2×C22⋊C4)⋊35C22, C22.590(C2×C4○D4), (C2×C22.D4)⋊45C2, C2.44(C2×C22.D4), SmallGroup(128,1581)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.166D4
C1C2C22C23C24C23×C4C24×C4 — C24.166D4
C1C23 — C24.166D4
C1C23 — C24.166D4
C1C23 — C24.166D4

Generators and relations for C24.166D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, ebe-1=fbf=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=ce-1 >

Subgroups: 1028 in 551 conjugacy classes, 132 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×13], C4 [×15], C22, C22 [×18], C22 [×75], C2×C4 [×8], C2×C4 [×77], D4 [×12], C23, C23 [×18], C23 [×75], C22⋊C4 [×26], C4⋊C4 [×8], C22×C4, C22×C4 [×18], C22×C4 [×64], C2×D4 [×18], C24 [×2], C24 [×6], C24 [×12], C2.C42 [×8], C2×C22⋊C4, C2×C22⋊C4 [×12], C2×C4⋊C4 [×4], C22≀C2 [×4], C22.D4 [×8], C23×C4 [×6], C23×C4 [×12], C22×D4, C22×D4 [×2], C25, C243C4, C23.34D4 [×2], C23.8Q8 [×4], C23.23D4 [×4], C2×C22≀C2, C2×C22.D4 [×2], C24×C4, C24.166D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×8], C24, C22≀C2 [×4], C22.D4 [×8], C22×D4 [×3], C2×C4○D4 [×4], C2×C22≀C2, C2×C22.D4 [×2], C22.19C24 [×4], C24.166D4

Smallest permutation representation of C24.166D4
On 32 points
Generators in S32
(1 8)(2 29)(3 6)(4 31)(5 14)(7 16)(9 24)(10 26)(11 22)(12 28)(13 32)(15 30)(17 25)(18 21)(19 27)(20 23)
(1 3)(2 29)(4 31)(5 14)(6 8)(7 16)(9 24)(10 12)(11 22)(13 15)(17 25)(18 20)(19 27)(21 23)(26 28)(30 32)
(1 13)(2 14)(3 15)(4 16)(5 29)(6 30)(7 31)(8 32)(9 17)(10 18)(11 19)(12 20)(21 26)(22 27)(23 28)(24 25)
(1 30)(2 31)(3 32)(4 29)(5 16)(6 13)(7 14)(8 15)(9 22)(10 23)(11 24)(12 21)(17 27)(18 28)(19 25)(20 26)
(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 12)(3 19)(4 10)(5 28)(6 22)(7 26)(8 24)(9 13)(11 15)(14 20)(16 18)(21 31)(23 29)(25 32)(27 30)

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2S2T4A···4P4Q···4W
order12···22···224···44···4
size11···12···282···28···8

44 irreducible representations

dim11111111222
type++++++++++
imageC1C2C2C2C2C2C2C2D4D4C4○D4
kernelC24.166D4C243C4C23.34D4C23.8Q8C23.23D4C2×C22≀C2C2×C22.D4C24×C4C22×C4C24C23
# reps112441218416

Matrix representation of C24.166D4 in GL6(𝔽5)

400000
040000
001000
000100
000010
000004
,
100000
040000
004000
000100
000010
000004
,
100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000040
000004
,
010000
100000
000400
001000
000003
000020
,
010000
100000
000100
001000
000001
000010

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,3,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C24.166D4 in GAP, Magma, Sage, TeX

C_2^4._{166}D_4
% in TeX

G:=Group("C2^4.166D4");
// GroupNames label

G:=SmallGroup(128,1581);
// by ID

G=gap.SmallGroup(128,1581);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,100,2019]);
// Polycyclic

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

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