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G = C24.48D4order 128 = 27

3rd non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.48D4, (C2×C4).9C42, C22.11C4≀C2, (C22×C4).5Q8, (C2×M4(2))⋊1C4, C4.33(C23⋊C4), C23.14(C4⋊C4), (C22×C4).640D4, C2.C4210C4, C24.4C4.6C2, C2.10(C426C4), (C23×C4).193C22, C2.7(C23.9D4), C23.141(C22⋊C4), C22.47(C2.C42), (C2×C4).71(C4⋊C4), (C4×C22⋊C4).9C2, (C22×C4).152(C2×C4), (C2×C4).373(C22⋊C4), SmallGroup(128,29)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.48D4
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C24.48D4
C1C22C2×C4 — C24.48D4
C1C2×C4C23×C4 — C24.48D4
C1C2C22C23×C4 — C24.48D4

Generators and relations for C24.48D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=c, f2=abd, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, bc=cb, bd=db, ebe-1=fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=acde3 >

Subgroups: 280 in 123 conjugacy classes, 36 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×11], C22, C22 [×4], C22 [×11], C8 [×2], C2×C4 [×2], C2×C4 [×2], C2×C4 [×25], C23, C23 [×2], C23 [×5], C42 [×4], C22⋊C4 [×8], C2×C8 [×2], M4(2) [×2], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C24, C2.C42 [×4], C22⋊C8 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×M4(2) [×2], C23×C4, C4×C22⋊C4 [×2], C24.4C4, C24.48D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C23⋊C4 [×2], C4≀C2 [×4], C426C4 [×2], C23.9D4, C24.48D4

Smallest permutation representation of C24.48D4
On 32 points
Generators in S32
(1 26)(2 31)(3 28)(4 25)(5 30)(6 27)(7 32)(8 29)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)
(1 30)(2 14)(3 32)(4 16)(5 26)(6 10)(7 28)(8 12)(9 18)(11 20)(13 22)(15 24)(17 25)(19 27)(21 29)(23 31)
(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 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)
(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 30 22 13)(2 4 6 8)(3 32 24 15)(5 26 18 9)(7 28 20 11)(10 29 14 25)(12 31 16 27)(17 19 21 23)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I···4Y8A8B8C8D
order122222222444444444···48888
size111122224111122224···48888

38 irreducible representations

dim1111122224
type++++-++
imageC1C2C2C4C4D4Q8D4C4≀C2C23⋊C4
kernelC24.48D4C4×C22⋊C4C24.4C4C2.C42C2×M4(2)C22×C4C22×C4C24C22C4
# reps12184211162

Matrix representation of C24.48D4 in GL4(𝔽17) generated by

16000
0100
0012
00016
,
1000
0100
001615
0001
,
16000
01600
00160
00016
,
16000
01600
0010
0001
,
01300
1000
001612
0011
,
1000
01300
0012
001616
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,2,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,15,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,13,0,0,0,0,0,16,1,0,0,12,1],[1,0,0,0,0,13,0,0,0,0,1,16,0,0,2,16] >;

C24.48D4 in GAP, Magma, Sage, TeX

C_2^4._{48}D_4
% in TeX

G:=Group("C2^4.48D4");
// GroupNames label

G:=SmallGroup(128,29);
// by ID

G=gap.SmallGroup(128,29);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,136,3924]);
// Polycyclic

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

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