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G = C42.366D4order 128 = 27

62nd non-split extension by C42 of D4 acting via D4/C4=C2

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

Aliases: C42.366D4, C42.723C23, (C2×C4)⋊8D8, (C4×D8)⋊8C2, C4.94(C2×D8), C43(C87D4), C811(C4○D4), C42(C84D4), C84D425C2, C87D442C2, C42(C82Q8), C82Q836C2, C22.1(C2×D8), C43(C4.4D8), C4.26(C4○D8), C4.4D847C2, C2.13(C22×D8), C4⋊C4.108C23, (C2×C4).367C24, (C4×C8).411C22, (C2×C8).564C23, (C4×D4).89C22, (C22×C4).619D4, C23.395(C2×D4), C4⋊Q8.290C22, (C2×D8).133C22, (C2×D4).123C23, C2.D8.182C22, C41D4.155C22, C4⋊D4.172C22, (C22×C8).539C22, C22.627(C22×D4), D4⋊C4.147C22, (C22×C4).1572C23, (C2×C42).1136C22, C22.26C2412C2, C2.64(C22.26C24), (C2×C4×C8)⋊26C2, C2.36(C2×C4○D8), C4.52(C2×C4○D4), (C2×C4).700(C2×D4), SmallGroup(128,1901)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.366D4
C1C2C4C2×C4C42C4×C8C2×C4×C8 — C42.366D4
C1C2C2×C4 — C42.366D4
C1C2×C4C2×C42 — C42.366D4
C1C2C2C2×C4 — C42.366D4

Generators and relations for C42.366D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=a2c3 >

Subgroups: 484 in 228 conjugacy classes, 100 normal (26 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×6], C22, C22 [×2], C22 [×14], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×16], D4 [×24], Q8 [×4], C23, C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×4], D8 [×8], C22×C4 [×3], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×2], C4○D4 [×8], C4×C8 [×2], C4×C8 [×2], D4⋊C4 [×8], C2.D8 [×4], C2×C42, C4×D4 [×4], C4×D4 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C4.4D4 [×2], C41D4 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×D8 [×4], C2×C4○D4 [×2], C2×C4×C8, C4×D8 [×4], C87D4 [×4], C4.4D8 [×2], C84D4, C82Q8, C22.26C24 [×2], C42.366D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C4○D4 [×4], C24, C2×D8 [×6], C4○D8 [×2], C22×D4, C2×C4○D4 [×2], C22.26C24, C22×D8, C2×C4○D8, C42.366D4

Smallest permutation representation of C42.366D4
On 64 points
Generators in S64
(1 51 5 55)(2 52 6 56)(3 53 7 49)(4 54 8 50)(9 47 13 43)(10 48 14 44)(11 41 15 45)(12 42 16 46)(17 36 21 40)(18 37 22 33)(19 38 23 34)(20 39 24 35)(25 63 29 59)(26 64 30 60)(27 57 31 61)(28 58 32 62)
(1 33 31 12)(2 34 32 13)(3 35 25 14)(4 36 26 15)(5 37 27 16)(6 38 28 9)(7 39 29 10)(8 40 30 11)(17 60 41 50)(18 61 42 51)(19 62 43 52)(20 63 44 53)(21 64 45 54)(22 57 46 55)(23 58 47 56)(24 59 48 49)
(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 41)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 59)(10 58)(11 57)(12 64)(13 63)(14 62)(15 61)(16 60)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 32)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 56)(40 55)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4N4O4P4Q4R8A···8P
order122222222244444···444448···8
size111122888811112···288882···2

44 irreducible representations

dim1111111122222
type+++++++++++
imageC1C2C2C2C2C2C2C2D4D4C4○D4D8C4○D8
kernelC42.366D4C2×C4×C8C4×D8C87D4C4.4D8C84D4C82Q8C22.26C24C42C22×C4C8C2×C4C4
# reps1144211222888

Matrix representation of C42.366D4 in GL4(𝔽17) generated by

0400
13000
0040
0004
,
4000
0400
00013
0040
,
0100
16000
00314
0033
,
1000
01600
001414
00143
G:=sub<GL(4,GF(17))| [0,13,0,0,4,0,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,0,4,0,0,13,0],[0,16,0,0,1,0,0,0,0,0,3,3,0,0,14,3],[1,0,0,0,0,16,0,0,0,0,14,14,0,0,14,3] >;

C42.366D4 in GAP, Magma, Sage, TeX

C_4^2._{366}D_4
% in TeX

G:=Group("C4^2.366D4");
// GroupNames label

G:=SmallGroup(128,1901);
// by ID

G=gap.SmallGroup(128,1901);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,520,80,4037,124]);
// Polycyclic

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

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