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G = C42.278C23order 128 = 27

139th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.278C23, C22⋊C4SD16, (C4×C8)⋊47C22, C4⋊C4.401D4, C4.102(C4×D4), (C4×Q8)⋊2C22, (C2×SD16)⋊10C4, SD1611(C2×C4), (C4×SD16)⋊51C2, C8.21(C22×C4), C4.26(C23×C4), D4.9(C22×C4), C22.69(C4×D4), C8⋊C439C22, C2.D867C22, C4.Q874C22, SD16⋊C43C2, Q8.8(C22×C4), C4⋊C4.366C23, C82M4(2)⋊7C2, (C2×C8).417C23, (C2×C4).206C24, C22⋊C4.188D4, C2.6(D4○SD16), (C4×D4).58C22, C23.438(C2×D4), Q8⋊C493C22, (C2×D4).375C23, (C2×Q8).347C23, (C22×SD16).4C2, C23.25D425C2, C23.38D432C2, C22.11C24.7C2, (C22×C8).249C22, (C22×C4).927C23, C22.150(C22×D4), D4⋊C4.197C22, C23.32C236C2, C42⋊C2.83C22, (C2×SD16).111C22, C23.37D4.10C2, (C22×D4).323C22, (C22×Q8).259C22, (C2×M4(2)).353C22, C2.66(C2×C4×D4), (C2×C8)⋊15(C2×C4), (C2×Q8)⋊21(C2×C4), C4.14(C2×C4○D4), (C2×C4).913(C2×D4), (C2×D4).138(C2×C4), (C2×C4).265(C4○D4), (C2×C4).265(C22×C4), SmallGroup(128,1681)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C42.278C23
C1C2C22C2×C4C22×C4C42⋊C2C23.32C23 — C42.278C23
C1C2C4 — C42.278C23
C1C22C42⋊C2 — C42.278C23
C1C2C2C2×C4 — C42.278C23

Generators and relations for C42.278C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=b2, ab=ba, ac=ca, ad=da, eae=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=b2c, de=ed >

Subgroups: 436 in 246 conjugacy classes, 140 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×14], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×4], D4 [×6], Q8 [×4], Q8 [×6], C23, C23 [×8], C42 [×2], C42 [×6], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×6], M4(2) [×2], SD16 [×16], C22×C4, C22×C4 [×5], C2×D4 [×6], C2×D4 [×3], C2×Q8 [×6], C2×Q8 [×3], C24, C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C22⋊C4 [×2], C42⋊C2 [×3], C42⋊C2 [×2], C4×D4 [×4], C4×D4 [×2], C4×Q8 [×4], C4×Q8 [×2], C22×C8, C2×M4(2), C2×SD16 [×12], C22×D4, C22×Q8, C82M4(2), C23.37D4, C23.38D4, C23.25D4, C4×SD16 [×4], SD16⋊C4 [×4], C22.11C24, C23.32C23, C22×SD16, C42.278C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, D4○SD16 [×2], C42.278C23

Smallest permutation representation of C42.278C23
On 32 points
Generators in S32
(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 19 15 5)(2 20 16 6)(3 17 13 7)(4 18 14 8)(9 22 25 30)(10 23 26 31)(11 24 27 32)(12 21 28 29)
(1 4 3 2)(5 18 7 20)(6 19 8 17)(9 29 11 31)(10 30 12 32)(13 16 15 14)(21 27 23 25)(22 28 24 26)
(1 31 15 23)(2 32 16 24)(3 29 13 21)(4 30 14 22)(5 10 19 26)(6 11 20 27)(7 12 17 28)(8 9 18 25)
(1 3)(2 14)(4 16)(5 7)(6 18)(8 20)(9 27)(10 12)(11 25)(13 15)(17 19)(21 23)(22 32)(24 30)(26 28)(29 31)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4L4M···4X8A8B8C8D8E···8J
order12222222224···44···488888···8
size11112244442···24···422224···4

44 irreducible representations

dim111111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C4D4D4C4○D4D4○SD16
kernelC42.278C23C82M4(2)C23.37D4C23.38D4C23.25D4C4×SD16SD16⋊C4C22.11C24C23.32C23C22×SD16C2×SD16C22⋊C4C4⋊C4C2×C4C2
# reps1111144111162244

Matrix representation of C42.278C23 in GL6(𝔽17)

400000
040000
00160150
0000161
000010
000110
,
1600000
0160000
0011500
0011600
000101
00161160
,
1300000
240000
00160150
001601616
000010
0011610
,
16130000
010000
0010700
005700
000555
00125512
,
100000
010000
001000
000100
00160160
00160016

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,16,1,1,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,16,0,0,15,16,1,1,0,0,0,0,0,16,0,0,0,0,1,0],[13,2,0,0,0,0,0,4,0,0,0,0,0,0,16,16,0,1,0,0,0,0,0,16,0,0,15,16,1,1,0,0,0,16,0,0],[16,0,0,0,0,0,13,1,0,0,0,0,0,0,10,5,0,12,0,0,7,7,5,5,0,0,0,0,5,5,0,0,0,0,5,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,16,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;

C42.278C23 in GAP, Magma, Sage, TeX

C_4^2._{278}C_2^3
% in TeX

G:=Group("C4^2.278C2^3");
// GroupNames label

G:=SmallGroup(128,1681);
// by ID

G=gap.SmallGroup(128,1681);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,184,2019,521,2804,1411,172]);
// Polycyclic

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

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