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G = C42.270D4order 128 = 27

252nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.270D4, C42.731C23, C4.1042+ 1+4, C4⋊D88C2, D4⋊D44C2, C42Q169C2, C4⋊SD1637C2, D4.7D44C2, C8.12D44C2, D4.D438C2, C4.114(C4○D8), C4⋊C8.287C22, C4⋊C4.151C23, (C2×C8).328C23, (C4×C8).113C22, (C2×C4).410C24, (C2×D8).25C22, (C22×C4).499D4, C23.286(C2×D4), C4⋊Q8.303C22, (C2×D4).159C23, (C4×D4).105C22, (C2×Q8).147C23, (C4×Q8).102C22, (C2×Q16).27C22, C42.12C436C2, C4⋊D4.190C22, C41D4.164C22, C22⋊C8.194C22, (C2×C42).877C22, (C2×SD16).86C22, C22.670(C22×D4), C22⋊Q8.195C22, D4⋊C4.108C22, C2.55(D8⋊C22), C42.78C228C2, C22.26C2418C2, (C22×C4).1081C23, Q8⋊C4.101C22, C4.4D4.151C22, C42.C2.126C22, C23.36C2310C2, C2.81(C22.29C24), C2.44(C2×C4○D8), (C2×C4).707(C2×D4), (C2×C4○D4).173C22, SmallGroup(128,1944)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.270D4
C1C2C4C2×C4C22×C4C2×C4○D4C22.26C24 — C42.270D4
C1C2C2×C4 — C42.270D4
C1C22C2×C42 — C42.270D4
C1C2C2C2×C4 — C42.270D4

Generators and relations for C42.270D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1b2, cbc-1=dbd=a2b-1, dcd=b2c3 >

Subgroups: 420 in 200 conjugacy classes, 86 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×12], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×16], Q8 [×6], C23, C23 [×3], C42 [×4], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C2×Q8 [×2], C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2×C42, C42⋊C2, C4×D4, C4×D4 [×3], C4×Q8, C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4, C4.4D4, C4.4D4 [×2], C42.C2, C422C2, C41D4, C4⋊Q8, C2×D8 [×2], C2×SD16 [×4], C2×Q16 [×2], C2×C4○D4 [×2], C42.12C4, D4⋊D4 [×2], D4.7D4 [×2], C4⋊D8, C4⋊SD16, D4.D4, C42Q16, C42.78C22 [×2], C8.12D4 [×2], C23.36C23, C22.26C24, C42.270D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C22×D4, 2+ 1+4 [×2], C22.29C24, C2×C4○D8, D8⋊C22, C42.270D4

Smallest permutation representation of C42.270D4
On 64 points
Generators in S64
(1 30 63 11)(2 31 64 12)(3 32 57 13)(4 25 58 14)(5 26 59 15)(6 27 60 16)(7 28 61 9)(8 29 62 10)(17 38 55 44)(18 39 56 45)(19 40 49 46)(20 33 50 47)(21 34 51 48)(22 35 52 41)(23 36 53 42)(24 37 54 43)
(1 24 5 20)(2 51 6 55)(3 18 7 22)(4 53 8 49)(9 41 13 45)(10 40 14 36)(11 43 15 47)(12 34 16 38)(17 64 21 60)(19 58 23 62)(25 42 29 46)(26 33 30 37)(27 44 31 48)(28 35 32 39)(50 63 54 59)(52 57 56 61)
(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 36)(2 35)(3 34)(4 33)(5 40)(6 39)(7 38)(8 37)(9 51)(10 50)(11 49)(12 56)(13 55)(14 54)(15 53)(16 52)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4J4K4L···4P8A···8H
order122222224···444···48···8
size111148882···248···84···4

32 irreducible representations

dim11111111111122244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D82+ 1+4D8⋊C22
kernelC42.270D4C42.12C4D4⋊D4D4.7D4C4⋊D8C4⋊SD16D4.D4C42Q16C42.78C22C8.12D4C23.36C23C22.26C24C42C22×C4C4C4C2
# reps11221111221122822

Matrix representation of C42.270D4 in GL6(𝔽17)

400000
040000
000400
0013000
000004
0000130
,
400000
040000
0040150
0004015
0000130
0000013
,
330000
1430000
00881111
0098611
00151599
0021589
,
14140000
1430000
0089215
00991515
0015298
002288

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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,15,0,13,0,0,0,0,15,0,13],[3,14,0,0,0,0,3,3,0,0,0,0,0,0,8,9,15,2,0,0,8,8,15,15,0,0,11,6,9,8,0,0,11,11,9,9],[14,14,0,0,0,0,14,3,0,0,0,0,0,0,8,9,15,2,0,0,9,9,2,2,0,0,2,15,9,8,0,0,15,15,8,8] >;

C42.270D4 in GAP, Magma, Sage, TeX

C_4^2._{270}D_4
% in TeX

G:=Group("C4^2.270D4");
// GroupNames label

G:=SmallGroup(128,1944);
// by ID

G=gap.SmallGroup(128,1944);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,675,248,4037,1027,124]);
// 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=b^2*c^3>;
// generators/relations

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