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G = C42.450D4order 128 = 27

83rd non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.450D4, C42.341C23, (C4×D8)⋊21C2, D42(C4○D4), C43(C4⋊D8), C4⋊D847C2, Q82(C4○D4), (C4×SD16)⋊2C2, C43(D4⋊D4), D4⋊D452C2, C43(D42Q8), C43(C4.Q16), C43(C4⋊SD16), C4⋊SD1649C2, C4.Q1650C2, D42Q848C2, C4⋊C4.59C23, C4.112(C4○D8), C4⋊C8.285C22, (C2×C4).304C24, (C4×C8).107C22, (C2×C8).149C23, (C4×D4).74C22, (C2×D4).88C23, (C22×C4).445D4, C23.250(C2×D4), C4⋊Q8.265C22, C4.149(C8⋊C22), (C2×D8).124C22, (C2×Q8).374C23, (C4×Q8).301C22, C2.D8.171C22, C42.12C430C2, C4.Q8.152C22, C43(C23.19D4), C41D4.140C22, C23.19D453C2, C4⋊D4.162C22, C22⋊C8.189C22, C22.26C245C2, (C2×C42).831C22, C22.564(C22×D4), D4⋊C4.184C22, (C22×C4).1020C23, Q8⋊C4.153C22, (C2×SD16).140C22, C42⋊C2.320C22, C2.105(C22.19C24), (C4×C4○D4)⋊10C2, C2.25(C2×C4○D8), C4.189(C2×C4○D4), C2.31(C2×C8⋊C22), (C2×C4).1583(C2×D4), (C2×C4○D4).311C22, SmallGroup(128,1838)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.450D4
C1C2C4C2×C4C42C4×D4C4×C4○D4 — C42.450D4
C1C2C2×C4 — C42.450D4
C1C2×C4C2×C42 — C42.450D4
C1C2C2C2×C4 — C42.450D4

Generators and relations for C42.450D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=dad=a-1b2, bc=cb, bd=db, dcd=a2c3 >

Subgroups: 428 in 217 conjugacy classes, 92 normal (44 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×13], C8 [×4], C2×C4 [×6], C2×C4 [×23], D4 [×2], D4 [×17], Q8 [×2], Q8 [×3], C23, C23 [×3], C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×4], D8 [×4], SD16 [×4], C22×C4 [×3], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C2×Q8, C4○D4 [×8], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×D4 [×3], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×D8 [×2], C2×SD16 [×2], C2×C4○D4, C2×C4○D4, C42.12C4, C4×D8 [×2], C4×SD16 [×2], D4⋊D4 [×2], C4⋊D8, C4⋊SD16, D42Q8, C4.Q16, C23.19D4 [×2], C4×C4○D4, C22.26C24, C42.450D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4○D8 [×2], C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C4○D8, C2×C8⋊C22, C42.450D4

Smallest permutation representation of C42.450D4
On 64 points
Generators in S64
(1 57 5 61)(2 55 6 51)(3 59 7 63)(4 49 8 53)(9 20 13 24)(10 43 14 47)(11 22 15 18)(12 45 16 41)(17 34 21 38)(19 36 23 40)(25 50 29 54)(26 62 30 58)(27 52 31 56)(28 64 32 60)(33 42 37 46)(35 44 39 48)
(1 35 25 15)(2 36 26 16)(3 37 27 9)(4 38 28 10)(5 39 29 11)(6 40 30 12)(7 33 31 13)(8 34 32 14)(17 64 43 49)(18 57 44 50)(19 58 45 51)(20 59 46 52)(21 60 47 53)(22 61 48 54)(23 62 41 55)(24 63 42 56)
(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)
(2 8)(3 7)(4 6)(9 13)(10 12)(14 16)(17 41)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(26 32)(27 31)(28 30)(33 37)(34 36)(38 40)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 64)(56 63)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E···4L4M···4S4T4U8A···8H
order12222222244444···44···4448···8
size11114448811112···24···4884···4

38 irreducible representations

dim111111111111222224
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D4C4○D8C8⋊C22
kernelC42.450D4C42.12C4C4×D8C4×SD16D4⋊D4C4⋊D8C4⋊SD16D42Q8C4.Q16C23.19D4C4×C4○D4C22.26C24C42C22×C4D4Q8C4C4
# reps112221111211224482

Matrix representation of C42.450D4 in GL4(𝔽17) generated by

16200
0100
0040
0004
,
13000
01300
0040
0004
,
16200
16100
00314
0033
,
1000
11600
0010
00016
G:=sub<GL(4,GF(17))| [16,0,0,0,2,1,0,0,0,0,4,0,0,0,0,4],[13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[16,16,0,0,2,1,0,0,0,0,3,3,0,0,14,3],[1,1,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C42.450D4 in GAP, Magma, Sage, TeX

C_4^2._{450}D_4
% in TeX

G:=Group("C4^2.450D4");
// GroupNames label

G:=SmallGroup(128,1838);
// by ID

G=gap.SmallGroup(128,1838);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,304,4037,1027,124]);
// Polycyclic

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

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