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G = C24.22D4order 128 = 27

22nd non-split extension by C24 of D4 acting faithfully

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

Aliases: C24.22D4, C24.170C23, C23⋊(C4⋊C4), C23⋊C44C4, (C2×D4).8Q8, C2.2C2≀C22, C23.4(C2×Q8), C22.33(C4×D4), (C22×C4).22D4, C23.6(C4○D4), C23.9D45C2, C23.556(C2×D4), C23.5(C22×C4), C23.8Q81C2, C22.88C22≀C2, (C23×C4).19C22, C22.11C24.4C2, C23.23D4.2C2, C2.2(C23.7D4), (C22×D4).14C22, C22.51(C22⋊Q8), C2.23(C23.8Q8), C22.47(C22.D4), (C2×C4)⋊(C4⋊C4), C22⋊C44(C2×C4), (C2×D4).68(C2×C4), (C2×C23⋊C4).5C2, C22.25(C2×C4⋊C4), (C2×C22⋊C4).9C22, SmallGroup(128,599)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.22D4
C1C2C22C23C24C22×D4C22.11C24 — C24.22D4
C1C2C23 — C24.22D4
C1C22C24 — C24.22D4
C1C2C24 — C24.22D4

Generators and relations for C24.22D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=c, eae-1=faf-1=ab=ba, ac=ca, ad=da, bc=cb, ebe-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 436 in 187 conjugacy classes, 56 normal (20 characteristic)
C1, C2 [×3], C2 [×8], C4 [×13], C22 [×3], C22 [×4], C22 [×17], C2×C4 [×2], C2×C4 [×33], D4 [×8], C23 [×3], C23 [×6], C23 [×6], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×12], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C22×C4 [×10], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2.C42 [×3], C23⋊C4 [×4], C2×C22⋊C4, C2×C22⋊C4 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2, C4×D4 [×4], C23×C4, C22×D4, C23.9D4 [×2], C23.8Q8 [×2], C23.23D4, C2×C23⋊C4, C22.11C24, C24.22D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C23.8Q8, C2≀C22, C23.7D4, C24.22D4

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A···4N4O···4T
order12222···2224···44···4
size11112···2444···48···8

32 irreducible representations

dim1111111222244
type+++++++-++
imageC1C2C2C2C2C2C4D4Q8D4C4○D4C2≀C22C23.7D4
kernelC24.22D4C23.9D4C23.8Q8C23.23D4C2×C23⋊C4C22.11C24C23⋊C4C22×C4C2×D4C24C23C2C2
# reps1221118521422

Matrix representation of C24.22D4 in GL6(𝔽5)

400000
040000
001000
000400
000040
000411
,
100000
010000
001000
000100
000040
001104
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000040
000004
,
010000
400000
000040
004412
000400
000001
,
030000
300000
004000
000400
004412
000004

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

C24.22D4 in GAP, Magma, Sage, TeX

C_2^4._{22}D_4
% in TeX

G:=Group("C2^4.22D4");
// GroupNames label

G:=SmallGroup(128,599);
// by ID

G=gap.SmallGroup(128,599);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,521,1411]);
// Polycyclic

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

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