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G = C24.23D4order 128 = 27

23rd non-split extension by C24 of D4 acting faithfully

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

Aliases: C24.23D4, C4.86(C4×D4), C4210(C2×C4), C4⋊C4.316D4, (C2×D4).73D4, C4.4D412C4, C4.5(C4⋊D4), C426C422C2, C23.560(C2×D4), C2.5(D4.9D4), C22.99C22≀C2, C23.9(C22⋊C4), C23.38D422C2, (C22×C4).683C23, (C2×C42).284C22, C22.11C24.5C2, (C22×D4).19C22, (C22×Q8).15C22, C42⋊C2.20C22, C2.24(C23.23D4), (C2×M4(2)).180C22, C22.48(C22.D4), (C2×Q8)⋊8(C2×C4), (C2×D4).76(C2×C4), (C2×C4).55(C4○D4), (C2×C4).1004(C2×D4), (C2×C4.D4).8C2, (C2×C4.4D4).5C2, (C2×C4).185(C22×C4), C22.40(C2×C22⋊C4), SmallGroup(128,617)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.23D4
C1C2C4C2×C4C22×C4C22×D4C22.11C24 — C24.23D4
C1C2C2×C4 — C24.23D4
C1C22C22×C4 — C24.23D4
C1C2C2C22×C4 — C24.23D4

Generators and relations for C24.23D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, ac=ca, ad=da, eae-1=faf-1=acd, bc=cb, bd=db, be=eb, bf=fb, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef-1=be3 >

Subgroups: 404 in 180 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×8], C22 [×3], C22 [×16], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×16], D4 [×8], Q8 [×6], C23, C23 [×4], C23 [×8], C42 [×2], C42 [×3], C22⋊C4 [×14], C4⋊C4 [×4], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×6], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×5], C24 [×2], C4.D4 [×2], Q8⋊C4 [×4], C2×C42, C2×C22⋊C4 [×4], C42⋊C2 [×2], C4×D4 [×4], C4.4D4 [×4], C4.4D4 [×2], C2×M4(2) [×2], C22×D4, C22×Q8, C426C4 [×2], C2×C4.D4, C23.38D4 [×2], C22.11C24, C2×C4.4D4, C24.23D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, D4.9D4 [×2], C24.23D4

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111122224
type+++++++++
imageC1C2C2C2C2C2C4D4D4D4C4○D4D4.9D4
kernelC24.23D4C426C4C2×C4.D4C23.38D4C22.11C24C2×C4.4D4C4.4D4C4⋊C4C2×D4C24C2×C4C2
# reps121211842244

Matrix representation of C24.23D4 in GL6(𝔽17)

100000
010000
0010013
0001600
000011
0000016
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0016004
00016013
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
1450000
1230000
00016013
000144
0041301
0008016
,
350000
12140000
0040160
0001311
0000130
000084

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,13,0,1,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,4,13,0,1],[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],[14,12,0,0,0,0,5,3,0,0,0,0,0,0,0,0,4,0,0,0,16,1,13,8,0,0,0,4,0,0,0,0,13,4,1,16],[3,12,0,0,0,0,5,14,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,16,1,13,8,0,0,0,1,0,4] >;

C24.23D4 in GAP, Magma, Sage, TeX

C_2^4._{23}D_4
% in TeX

G:=Group("C2^4.23D4");
// GroupNames label

G:=SmallGroup(128,617);
// by ID

G=gap.SmallGroup(128,617);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,521,248,2804,1411,2028,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=f^2=d,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=c*d=d*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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