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G = C42.267D4order 128 = 27

249th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.267D4, C42.398C23, C4.1012+ 1+4, (C2×C4)⋊4Q16, C4⋊Q168C2, C42Q168C2, C4.47(C2×Q16), (C4×C8).74C22, C4⋊C8.286C22, C4⋊C4.148C23, (C2×C8).160C23, (C2×C4).407C24, C4.SD1612C2, (C22×C4).497D4, C23.691(C2×D4), C4⋊Q8.301C22, C22⋊Q16.2C2, C2.15(C22×Q16), C22.18(C2×Q16), C4.26(C8.C22), (C2×Q8).144C23, Q8⋊C4.3C22, (C2×Q16).25C22, (C4×Q8).100C22, C22⋊C8.178C22, (C2×C42).874C22, C22.667(C22×D4), C22⋊Q8.192C22, C42.12C4.35C2, (C22×C4).1078C23, (C22×Q8).320C22, C2.78(C22.29C24), C23.37C23.37C2, (C2×C4⋊Q8).53C2, (C2×C4).867(C2×D4), C2.54(C2×C8.C22), SmallGroup(128,1941)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.267D4
C1C2C4C2×C4C22×C4C22×Q8C2×C4⋊Q8 — C42.267D4
C1C2C2×C4 — C42.267D4
C1C22C2×C42 — C42.267D4
C1C2C2C2×C4 — C42.267D4

Generators and relations for C42.267D4
 G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=b2c3 >

Subgroups: 348 in 192 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×16], Q8 [×18], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×15], C2×C8 [×4], Q16 [×8], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×7], C4×C8 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C2×C42, C2×C4⋊C4 [×2], C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C2×Q16 [×8], C22×Q8 [×2], C42.12C4, C22⋊Q16 [×4], C42Q16 [×4], C4.SD16 [×2], C4⋊Q16 [×2], C2×C4⋊Q8, C23.37C23, C42.267D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C24, C2×Q16 [×6], C8.C22 [×2], C22×D4, 2+ 1+4 [×2], C22.29C24, C22×Q16, C2×C8.C22, C42.267D4

Smallest permutation representation of C42.267D4
On 64 points
Generators in S64
(1 40 19 48)(2 33 20 41)(3 34 21 42)(4 35 22 43)(5 36 23 44)(6 37 24 45)(7 38 17 46)(8 39 18 47)(9 25 49 63)(10 26 50 64)(11 27 51 57)(12 28 52 58)(13 29 53 59)(14 30 54 60)(15 31 55 61)(16 32 56 62)
(1 7 5 3)(2 22 6 18)(4 24 8 20)(9 15 13 11)(10 52 14 56)(12 54 16 50)(17 23 21 19)(25 31 29 27)(26 58 30 62)(28 60 32 64)(33 43 37 47)(34 40 38 36)(35 45 39 41)(42 48 46 44)(49 55 53 51)(57 63 61 59)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 14 5 10)(2 13 6 9)(3 12 7 16)(4 11 8 15)(17 56 21 52)(18 55 22 51)(19 54 23 50)(20 53 24 49)(25 41 29 45)(26 48 30 44)(27 47 31 43)(28 46 32 42)(33 59 37 63)(34 58 38 62)(35 57 39 61)(36 64 40 60)

G:=sub<Sym(64)| (1,40,19,48)(2,33,20,41)(3,34,21,42)(4,35,22,43)(5,36,23,44)(6,37,24,45)(7,38,17,46)(8,39,18,47)(9,25,49,63)(10,26,50,64)(11,27,51,57)(12,28,52,58)(13,29,53,59)(14,30,54,60)(15,31,55,61)(16,32,56,62), (1,7,5,3)(2,22,6,18)(4,24,8,20)(9,15,13,11)(10,52,14,56)(12,54,16,50)(17,23,21,19)(25,31,29,27)(26,58,30,62)(28,60,32,64)(33,43,37,47)(34,40,38,36)(35,45,39,41)(42,48,46,44)(49,55,53,51)(57,63,61,59), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,14,5,10)(2,13,6,9)(3,12,7,16)(4,11,8,15)(17,56,21,52)(18,55,22,51)(19,54,23,50)(20,53,24,49)(25,41,29,45)(26,48,30,44)(27,47,31,43)(28,46,32,42)(33,59,37,63)(34,58,38,62)(35,57,39,61)(36,64,40,60)>;

G:=Group( (1,40,19,48)(2,33,20,41)(3,34,21,42)(4,35,22,43)(5,36,23,44)(6,37,24,45)(7,38,17,46)(8,39,18,47)(9,25,49,63)(10,26,50,64)(11,27,51,57)(12,28,52,58)(13,29,53,59)(14,30,54,60)(15,31,55,61)(16,32,56,62), (1,7,5,3)(2,22,6,18)(4,24,8,20)(9,15,13,11)(10,52,14,56)(12,54,16,50)(17,23,21,19)(25,31,29,27)(26,58,30,62)(28,60,32,64)(33,43,37,47)(34,40,38,36)(35,45,39,41)(42,48,46,44)(49,55,53,51)(57,63,61,59), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,14,5,10)(2,13,6,9)(3,12,7,16)(4,11,8,15)(17,56,21,52)(18,55,22,51)(19,54,23,50)(20,53,24,49)(25,41,29,45)(26,48,30,44)(27,47,31,43)(28,46,32,42)(33,59,37,63)(34,58,38,62)(35,57,39,61)(36,64,40,60) );

G=PermutationGroup([(1,40,19,48),(2,33,20,41),(3,34,21,42),(4,35,22,43),(5,36,23,44),(6,37,24,45),(7,38,17,46),(8,39,18,47),(9,25,49,63),(10,26,50,64),(11,27,51,57),(12,28,52,58),(13,29,53,59),(14,30,54,60),(15,31,55,61),(16,32,56,62)], [(1,7,5,3),(2,22,6,18),(4,24,8,20),(9,15,13,11),(10,52,14,56),(12,54,16,50),(17,23,21,19),(25,31,29,27),(26,58,30,62),(28,60,32,64),(33,43,37,47),(34,40,38,36),(35,45,39,41),(42,48,46,44),(49,55,53,51),(57,63,61,59)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,14,5,10),(2,13,6,9),(3,12,7,16),(4,11,8,15),(17,56,21,52),(18,55,22,51),(19,54,23,50),(20,53,24,49),(25,41,29,45),(26,48,30,44),(27,47,31,43),(28,46,32,42),(33,59,37,63),(34,58,38,62),(35,57,39,61),(36,64,40,60)])

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J4K···4R8A···8H
order1222224···4444···48···8
size1111222···2448···84···4

32 irreducible representations

dim1111111122244
type++++++++++--+
imageC1C2C2C2C2C2C2C2D4D4Q16C8.C222+ 1+4
kernelC42.267D4C42.12C4C22⋊Q16C42Q16C4.SD16C4⋊Q16C2×C4⋊Q8C23.37C23C42C22×C4C2×C4C4C4
# reps1144221122822

Matrix representation of C42.267D4 in GL6(𝔽17)

010000
1600000
0001600
001000
0014001
00014160
,
0160000
100000
0016000
0001600
000010
0011001
,
1430000
14140000
0011002
000020
000800
007006
,
550000
5120000
000020
0011002
009000
000960

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,14,0,0,0,16,0,0,14,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,11,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,11,0,0,7,0,0,0,0,8,0,0,0,0,2,0,0,0,0,2,0,0,6],[5,5,0,0,0,0,5,12,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,9,0,0,2,0,0,6,0,0,0,2,0,0] >;

C42.267D4 in GAP, Magma, Sage, TeX

C_4^2._{267}D_4
% in TeX

G:=Group("C4^2.267D4");
// GroupNames label

G:=SmallGroup(128,1941);
// by ID

G=gap.SmallGroup(128,1941);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,352,675,4037,1027,124]);
// Polycyclic

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

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