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G = C42.285D4order 128 = 27

267th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.285D4, C42.735C23, C4.572- 1+4, C4.Q169C2, D4⋊Q89C2, C8.5Q84C2, Q8⋊Q838C2, D42Q836C2, C4.115(C4○D8), C4⋊C8.290C22, C4⋊C4.172C23, (C4×C8).116C22, (C2×C4).431C24, (C2×C8).335C23, C23.298(C2×D4), (C22×C4).513D4, C4⋊Q8.314C22, C4.Q8.88C22, C2.D8.40C22, (C2×D4).177C23, (C4×D4).115C22, C23.20D43C2, (C2×Q8).165C23, (C4×Q8).112C22, C42.12C439C2, C4⋊D4.200C22, C22⋊C8.197C22, (C2×C42).892C22, C23.19D4.1C2, C22.691(C22×D4), C22⋊Q8.205C22, D4⋊C4.112C22, C2.62(D8⋊C22), (C22×C4).1096C23, Q8⋊C4.106C22, C4.4D4.159C22, C42.C2.132C22, C23.37C2321C2, C42.78C2210C2, C42⋊C2.165C22, C23.36C23.26C2, C2.79(C23.38C23), C2.48(C2×C4○D8), (C2×C4).708(C2×D4), SmallGroup(128,1965)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.285D4
C1C2C4C2×C4C22×C4C42⋊C2C23.37C23 — C42.285D4
C1C2C2×C4 — C42.285D4
C1C22C2×C42 — C42.285D4
C1C2C2C2×C4 — C42.285D4

Generators and relations for C42.285D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1b2, cbc-1=dbd=a2b-1, dcd=a2c3 >

Subgroups: 292 in 166 conjugacy classes, 86 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×14], D4 [×4], Q8 [×6], C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C42⋊C2 [×2], C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8 [×2], C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4, C4.4D4, C42.C2, C42.C2 [×2], C422C2, C4⋊Q8 [×2], C42.12C4, D4⋊Q8, Q8⋊Q8, D42Q8, C4.Q16, C23.19D4 [×2], C23.20D4 [×2], C42.78C22 [×2], C8.5Q8 [×2], C23.36C23, C23.37C23, C42.285D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C22×D4, 2- 1+4 [×2], C23.38C23, C2×C4○D8, D8⋊C22, C42.285D4

Smallest permutation representation of C42.285D4
On 64 points
Generators in S64
(1 57 25 41)(2 58 26 42)(3 59 27 43)(4 60 28 44)(5 61 29 45)(6 62 30 46)(7 63 31 47)(8 64 32 48)(9 24 33 56)(10 17 34 49)(11 18 35 50)(12 19 36 51)(13 20 37 52)(14 21 38 53)(15 22 39 54)(16 23 40 55)
(1 39 5 35)(2 12 6 16)(3 33 7 37)(4 14 8 10)(9 31 13 27)(11 25 15 29)(17 60 21 64)(18 41 22 45)(19 62 23 58)(20 43 24 47)(26 36 30 40)(28 38 32 34)(42 51 46 55)(44 53 48 49)(50 57 54 61)(52 59 56 63)
(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 28)(3 7)(4 26)(6 32)(8 30)(9 33)(10 12)(11 39)(13 37)(14 16)(15 35)(17 55)(19 53)(20 24)(21 51)(23 49)(27 31)(34 36)(38 40)(41 61)(42 48)(43 59)(44 46)(45 57)(47 63)(52 56)(58 64)(60 62)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4J4K4L···4R8A···8H
order1222224···444···48···8
size1111482···248···84···4

32 irreducible representations

dim11111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D82- 1+4D8⋊C22
kernelC42.285D4C42.12C4D4⋊Q8Q8⋊Q8D42Q8C4.Q16C23.19D4C23.20D4C42.78C22C8.5Q8C23.36C23C23.37C23C42C22×C4C4C4C2
# reps11111122221122822

Matrix representation of C42.285D4 in GL6(𝔽17)

400000
040000
000400
0013000
000004
0000130
,
1300000
0130000
000010
000001
0016000
0001600
,
660000
1400000
0013455
001313125
0055413
0012544
,
100000
16160000
001000
0001600
0000160
000001

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[6,14,0,0,0,0,6,0,0,0,0,0,0,0,13,13,5,12,0,0,4,13,5,5,0,0,5,12,4,4,0,0,5,5,13,4],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;

C42.285D4 in GAP, Magma, Sage, TeX

C_4^2._{285}D_4
% in TeX

G:=Group("C4^2.285D4");
// GroupNames label

G:=SmallGroup(128,1965);
// by ID

G=gap.SmallGroup(128,1965);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,100,675,248,4037,1027,124]);
// Polycyclic

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

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