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G = C42.257C23order 128 = 27

118th non-split extension by C42 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.257C23, C4⋊C848C22, C24.75(C2×C4), (C22×C4).63Q8, C4.60(C22×Q8), C23.31(C4⋊C4), (C2×C8).395C23, (C2×C4).634C24, C42.201(C2×C4), (C22×C4).414D4, C4.186(C22×D4), C4⋊M4(2)⋊30C2, C42⋊C2.27C4, C2.8(Q8○M4(2)), (C2×C42).752C22, (C22×C8).427C22, (C23×C4).517C22, C23.138(C22×C4), C22.163(C23×C4), (C22×C4).1502C23, C42.6C2226C2, (C22×M4(2)).29C2, C42⋊C2.283C22, (C2×M4(2)).337C22, C4.62(C2×C4⋊C4), (C2×C4⋊C4).66C4, C4⋊C4.213(C2×C4), (C2×C4).60(C4⋊C4), (C2×C4).840(C2×D4), C22.34(C2×C4⋊C4), C2.20(C22×C4⋊C4), (C2×C4).236(C2×Q8), (C2×C22⋊C4).43C4, C22⋊C4.64(C2×C4), (C2×C4).248(C22×C4), (C22×C4).328(C2×C4), (C2×C42⋊C2).52C2, SmallGroup(128,1637)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.257C23
C1C2C4C2×C4C22×C4C23×C4C2×C42⋊C2 — C42.257C23
C1C22 — C42.257C23
C1C2×C4 — C42.257C23
C1C2C2C2×C4 — C42.257C23

Generators and relations for C42.257C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1, dad=ab2, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=b2c, de=ed >

Subgroups: 332 in 242 conjugacy classes, 172 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×6], C22 [×10], C8 [×8], C2×C4 [×2], C2×C4 [×34], C2×C4 [×4], C23, C23 [×6], C23 [×2], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C2×C8 [×8], C2×C8 [×8], M4(2) [×16], C22×C4 [×2], C22×C4 [×16], C24, C4⋊C8 [×16], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C22×C8 [×4], C2×M4(2) [×8], C2×M4(2) [×8], C23×C4, C4⋊M4(2) [×4], C42.6C22 [×8], C2×C42⋊C2, C22×M4(2) [×2], C42.257C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, C22×C4⋊C4, Q8○M4(2) [×2], C42.257C23

Smallest permutation representation of C42.257C23
On 32 points
Generators in S32
(1 19 31 16)(2 9 32 20)(3 21 25 10)(4 11 26 22)(5 23 27 12)(6 13 28 24)(7 17 29 14)(8 15 30 18)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 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)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 24)(10 21)(11 18)(12 23)(13 20)(14 17)(15 22)(16 19)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J4K···4R8A···8P
order12222···244444···44···48···8
size11112···211112···24···44···4

44 irreducible representations

dim11111111224
type++++++-
imageC1C2C2C2C2C4C4C4D4Q8Q8○M4(2)
kernelC42.257C23C4⋊M4(2)C42.6C22C2×C42⋊C2C22×M4(2)C2×C22⋊C4C2×C4⋊C4C42⋊C2C22×C4C22×C4C2
# reps14812448444

Matrix representation of C42.257C23 in GL6(𝔽17)

400000
1130000
0051500
00121200
0040415
001131613
,
1600000
0160000
0013000
0001300
0000130
0000013
,
1680000
410000
000010
0015091
0013000
0021320
,
1600000
0160000
001000
0051600
000010
0040416
,
100000
010000
001000
000100
0000160
0040016

G:=sub<GL(6,GF(17))| [4,1,0,0,0,0,0,13,0,0,0,0,0,0,5,12,4,1,0,0,15,12,0,13,0,0,0,0,4,16,0,0,0,0,15,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,4,0,0,0,0,8,1,0,0,0,0,0,0,0,15,13,2,0,0,0,0,0,13,0,0,1,9,0,2,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,5,0,4,0,0,0,16,0,0,0,0,0,0,1,4,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;

C42.257C23 in GAP, Magma, Sage, TeX

C_4^2._{257}C_2^3
% in TeX

G:=Group("C4^2.257C2^3");
// GroupNames label

G:=SmallGroup(128,1637);
// by ID

G=gap.SmallGroup(128,1637);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,2019,124]);
// Polycyclic

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

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