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G = C24.103D4order 128 = 27

58th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.103D4, C4○D413D4, C4(D4⋊D4), C4(C22⋊D8), C4(Q8⋊D4), D4.39(C2×D4), C4⋊C4.5C23, Q8.39(C2×D4), D4⋊D451C2, C22⋊D835C2, Q8⋊D436C2, C4(D4.7D4), C4(C22⋊SD16), (C2×D8)⋊37C22, C225(C4○D8), C4(C22⋊Q16), C4.39(C22×D4), C4.129C22≀C2, D4.7D452C2, C22⋊SD1636C2, C4⋊D449C22, C22⋊C858C22, (C2×C4).221C24, (C2×C8).128C23, C22⋊Q1635C2, (C22×C8)⋊14C22, (C2×Q16)⋊38C22, C22.2C22≀C2, (C22×C4).714D4, C23.645(C2×D4), C22⋊Q861C22, (C2×Q8).18C23, D4⋊C471C22, C42⋊C25C22, C22.19C242C2, Q8⋊C465C22, (C2×SD16)⋊69C22, (C2×D4).380C23, C23.24D414C2, C2.6(D8⋊C22), (C23×C4).541C22, (C22×C4).959C23, C22.481(C22×D4), (C22×D4).562C22, (C22×Q8).466C22, (C2×C4○D8)⋊2C2, C2.9(C2×C4○D8), (C2×C4)(C22⋊D8), (C2×C4)(D4⋊D4), (C2×C22⋊C8)⋊25C2, (C2×C4)(Q8⋊D4), (C2×C4).449(C2×D4), (C22×C4○D4)⋊8C2, (C2×C4○D4)⋊1C22, C2.39(C2×C22≀C2), (C2×C4)(D4.7D4), (C2×C4)(C22⋊SD16), (C2×C4)(C22⋊Q16), SmallGroup(128,1734)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.103D4
C1C2C22C2×C4C22×C4C23×C4C22×C4○D4 — C24.103D4
C1C2C2×C4 — C24.103D4
C1C2×C4C23×C4 — C24.103D4
C1C2C2C2×C4 — C24.103D4

Generators and relations for C24.103D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, fbf=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de3 >

Subgroups: 740 in 382 conjugacy classes, 110 normal (42 characteristic)
C1, C2 [×3], C2 [×10], C4 [×4], C4 [×9], C22, C22 [×4], C22 [×30], C8 [×4], C2×C4 [×4], C2×C4 [×4], C2×C4 [×37], D4 [×4], D4 [×30], Q8 [×4], Q8 [×8], C23 [×3], C23 [×16], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4 [×6], C22×C4 [×18], C2×D4, C2×D4 [×2], C2×D4 [×20], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×8], C4○D4 [×32], C24, C24, C22⋊C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C22×C8 [×2], C2×D8 [×2], C2×SD16 [×4], C2×Q16 [×2], C4○D8 [×8], C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4 [×4], C2×C4○D4 [×10], C2×C22⋊C8, C23.24D4 [×2], C22⋊D8, Q8⋊D4, D4⋊D4 [×2], C22⋊SD16, C22⋊Q16, D4.7D4 [×2], C22.19C24, C2×C4○D8 [×2], C22×C4○D4, C24.103D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C4○D8 [×2], C22×D4 [×3], C2×C22≀C2, C2×C4○D8, D8⋊C22, C24.103D4

Smallest permutation representation of C24.103D4
On 32 points
Generators in S32
(2 25)(4 27)(6 29)(8 31)(9 24)(11 18)(13 20)(15 22)
(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 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(8 16)(17 29)(18 28)(19 27)(20 26)(21 25)(22 32)(23 31)(24 30)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2L2M4A4B4C4D4E4F4G4H4I···4M4N4O4P8A···8H
order122222222···22444444444···44448···8
size111122224···48111122224···48884···4

38 irreducible representations

dim11111111111122224
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4C4○D8D8⋊C22
kernelC24.103D4C2×C22⋊C8C23.24D4C22⋊D8Q8⋊D4D4⋊D4C22⋊SD16C22⋊Q16D4.7D4C22.19C24C2×C4○D8C22×C4○D4C22×C4C4○D4C24C22C2
# reps11211211212138182

Matrix representation of C24.103D4 in GL4(𝔽17) generated by

1000
0100
0010
00016
,
1000
01600
00160
00016
,
1000
0100
00160
00016
,
16000
01600
0010
0001
,
15000
0800
0001
0010
,
0800
15000
0001
0010
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[15,0,0,0,0,8,0,0,0,0,0,1,0,0,1,0],[0,15,0,0,8,0,0,0,0,0,0,1,0,0,1,0] >;

C24.103D4 in GAP, Magma, Sage, TeX

C_2^4._{103}D_4
% in TeX

G:=Group("C2^4.103D4");
// GroupNames label

G:=SmallGroup(128,1734);
// by ID

G=gap.SmallGroup(128,1734);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,248,2804,1411,172]);
// Polycyclic

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

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