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G = C24.53D4order 128 = 27

8th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.53D4, C4⋊D42C4, C22⋊Q82C4, C22.7C4≀C2, C42⋊C21C4, C4.45(C23⋊C4), C23.491(C2×D4), (C22×C4).661D4, C42(C22.SD16), C22.17(C4○D8), C22.SD1623C2, C42(C23.31D4), C4⋊D4.130C22, C23.31D424C2, C22⋊C8.162C22, C23.51(C22⋊C4), (C23×C4).205C22, (C22×C4).623C23, C22.19C24.1C2, C22⋊Q8.135C22, C2.9(C23.24D4), C2.C42.500C22, (C2×C4○D4)⋊1C4, C4⋊C4.2(C2×C4), C2.18(C2×C4≀C2), (C2×C22⋊C8)⋊6C2, (C2×D4).4(C2×C4), (C2×Q8).4(C2×C4), (C4×C22⋊C4)⋊20C2, C2.14(C2×C23⋊C4), (C2×C4).1147(C2×D4), (C22×C4).196(C2×C4), (C2×C4).113(C22×C4), (C2×C4).390(C22⋊C4), C22.177(C2×C22⋊C4), SmallGroup(128,233)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.53D4
C1C2C22C23C22×C4C23×C4C22.19C24 — C24.53D4
C1C22C2×C4 — C24.53D4
C1C2×C4C23×C4 — C24.53D4
C1C2C22C22×C4 — C24.53D4

Generators and relations for C24.53D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, faf-1=ac=ca, ad=da, eae-1=acd, bc=cb, bd=db, ebe-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bde3 >

Subgroups: 348 in 152 conjugacy classes, 48 normal (34 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×10], C22, C22 [×4], C22 [×14], C8 [×2], C2×C4 [×4], C2×C4 [×22], D4 [×7], Q8, C23 [×3], C23 [×6], C42 [×3], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C22×C4 [×6], C22×C4 [×5], C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C24, C2.C42 [×2], C22⋊C8 [×2], C22⋊C8, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C22×C8, C23×C4, C2×C4○D4, C22.SD16 [×2], C23.31D4 [×2], C4×C22⋊C4, C2×C22⋊C8, C22.19C24, C24.53D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C4≀C2 [×2], C2×C22⋊C4, C4○D8 [×2], C2×C23⋊C4, C23.24D4, C2×C4≀C2, C24.53D4

Smallest permutation representation of C24.53D4
On 32 points
Generators in S32
(1 19)(2 11)(3 21)(4 13)(5 23)(6 15)(7 17)(8 9)(10 25)(12 27)(14 29)(16 31)(18 32)(20 26)(22 28)(24 30)
(1 5)(2 30)(3 7)(4 32)(6 26)(8 28)(9 22)(10 14)(11 24)(12 16)(13 18)(15 20)(17 21)(19 23)(25 29)(27 31)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(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 17 5 21)(2 11 30 24)(3 14 7 10)(4 18 32 13)(6 15 26 20)(8 22 28 9)(12 25 16 29)(19 27 23 31)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I···4Q4R4S4T8A···8H
order1222222222444444444···44448···8
size1111222248111122224···48884···4

38 irreducible representations

dim111111111122224
type+++++++++
imageC1C2C2C2C2C2C4C4C4C4D4D4C4≀C2C4○D8C23⋊C4
kernelC24.53D4C22.SD16C23.31D4C4×C22⋊C4C2×C22⋊C8C22.19C24C42⋊C2C4⋊D4C22⋊Q8C2×C4○D4C22×C4C24C22C22C4
# reps122111222231882

Matrix representation of C24.53D4 in GL4(𝔽17) generated by

1000
01600
00013
0040
,
16000
0100
00160
00016
,
1000
0100
00160
00016
,
16000
01600
00160
00016
,
01600
4000
00512
0055
,
4000
01600
00130
0004
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,0,4,0,0,13,0],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,4,0,0,16,0,0,0,0,0,5,5,0,0,12,5],[4,0,0,0,0,16,0,0,0,0,13,0,0,0,0,4] >;

C24.53D4 in GAP, Magma, Sage, TeX

C_2^4._{53}D_4
% in TeX

G:=Group("C2^4.53D4");
// GroupNames label

G:=SmallGroup(128,233);
// by ID

G=gap.SmallGroup(128,233);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,352,1123,1018,248,1971]);
// Polycyclic

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

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