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G = C24.21D4order 128 = 27

21st non-split extension by C24 of D4 acting faithfully

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

Aliases: C24.21D4, C4.50(C4×D4), (C2×D4).5Q8, C4⋊C4.300D4, C4.D42C4, C426C41C2, C23.4(C4⋊C4), M4(2)⋊1(C2×C4), C2.2(D44D4), C23.552(C2×D4), C4.23(C22⋊Q8), M4(2)⋊C41C2, C2.2(D4.9D4), C22.80C22≀C2, (C22×C4).672C23, C42⋊C2.8C22, C22.11C24.3C2, (C2×C42).263C22, (C22×D4).13C22, (C2×M4(2)).5C22, C24.3C22.4C2, C2.12(C23.8Q8), C22.45(C22.D4), (C2×C4).7(C2×Q8), (C2×D4).66(C2×C4), (C2×C4).982(C2×D4), C22.22(C2×C4⋊C4), (C2×C4).6(C22×C4), (C2×C4⋊C4).54C22, (C2×C4.D4).1C2, (C2×C4).315(C4○D4), SmallGroup(128,588)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.21D4
C1C2C22C23C22×C4C22×D4C22.11C24 — C24.21D4
C1C2C2×C4 — C24.21D4
C1C22C22×C4 — C24.21D4
C1C2C2C22×C4 — C24.21D4

Generators and relations for C24.21D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, 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=de3 >

Subgroups: 372 in 162 conjugacy classes, 56 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×7], C22 [×3], C22 [×16], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×15], D4 [×8], C23, C23 [×4], C23 [×8], C42 [×4], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C22×C4 [×6], C2×D4 [×4], C2×D4 [×4], C24 [×2], C4.D4 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4×D4 [×4], C2×M4(2) [×2], C22×D4, C426C4 [×2], C24.3C22, C2×C4.D4, M4(2)⋊C4 [×2], C22.11C24, C24.21D4
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, D44D4, D4.9D4, C24.21D4

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4P4Q4R8A8B8C8D
order122222222244444···4448888
size111122444422224···4888888

32 irreducible representations

dim1111111222244
type+++++++-++
imageC1C2C2C2C2C2C4D4Q8D4C4○D4D44D4D4.9D4
kernelC24.21D4C426C4C24.3C22C2×C4.D4M4(2)⋊C4C22.11C24C4.D4C4⋊C4C2×D4C24C2×C4C2C2
# reps1211218422422

Matrix representation of C24.21D4 in GL6(𝔽17)

1600000
0160000
001000
0001600
0000160
000001
,
100000
010000
001000
000100
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
730000
6100000
000010
0000016
000100
001000
,
0140000
600000
001000
0001600
000001
000010

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[7,6,0,0,0,0,3,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,16,0,0],[0,6,0,0,0,0,14,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C24.21D4 in GAP, Magma, Sage, TeX

C_2^4._{21}D_4
% in TeX

G:=Group("C2^4.21D4");
// GroupNames label

G:=SmallGroup(128,588);
// by ID

G=gap.SmallGroup(128,588);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,521,248,1411]);
// Polycyclic

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

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