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G = C4226D4order 128 = 27

20th semidirect product of C42 and D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C4226D4, C24.38C23, C23.510C24, C22.2902+ 1+4, C424C428C2, C23⋊Q825C2, C232D4.16C2, C23.163(C4○D4), C23.10D451C2, C23.23D465C2, (C2×C42).597C22, (C23×C4).414C22, C22.336(C22×D4), (C22×C4).1262C23, C24.3C2263C2, (C22×D4).187C22, (C22×Q8).149C22, C23.78C2323C2, C24.C22103C2, C2.79(C22.19C24), C2.30(C22.29C24), C2.73(C22.45C24), C2.C42.239C22, C2.48(C22.26C24), C2.32(C22.49C24), (C2×C4).371(C2×D4), (C2×C4.4D4)⋊19C2, (C2×C42⋊C2)⋊35C2, (C2×C4).412(C4○D4), (C2×C4⋊C4).349C22, C22.386(C2×C4○D4), (C2×C22⋊C4).206C22, SmallGroup(128,1342)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4226D4
C1C2C22C23C22×C4C23×C4C2×C42⋊C2 — C4226D4
C1C23 — C4226D4
C1C23 — C4226D4
C1C23 — C4226D4

Generators and relations for C4226D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=a-1, cbc-1=dbd=a2b-1, dcd=c-1 >

Subgroups: 596 in 288 conjugacy classes, 100 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×18], C22 [×3], C22 [×4], C22 [×24], C2×C4 [×12], C2×C4 [×38], D4 [×12], Q8 [×4], C23, C23 [×2], C23 [×20], C42 [×4], C42 [×6], C22⋊C4 [×24], C4⋊C4 [×6], C22×C4 [×6], C22×C4 [×6], C22×C4 [×6], C2×D4 [×14], C2×Q8 [×4], C24, C24 [×2], C2.C42 [×2], C2.C42 [×6], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C4.4D4 [×4], C23×C4, C22×D4, C22×D4 [×2], C22×Q8, C424C4, C23.23D4 [×2], C24.C22 [×4], C24.3C22 [×2], C232D4, C23⋊Q8, C23.10D4, C23.78C23, C2×C42⋊C2, C2×C4.4D4, C4226D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22×D4, C2×C4○D4 [×4], 2+ 1+4 [×2], C22.19C24, C22.26C24, C22.29C24, C22.45C24 [×2], C22.49C24 [×2], C4226D4

Smallest permutation representation of C4226D4
On 64 points
Generators in S64
(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 43 53 16)(2 44 54 13)(3 41 55 14)(4 42 56 15)(5 17 37 26)(6 18 38 27)(7 19 39 28)(8 20 40 25)(9 31 45 58)(10 32 46 59)(11 29 47 60)(12 30 48 57)(21 35 52 62)(22 36 49 63)(23 33 50 64)(24 34 51 61)
(1 34 47 28)(2 64 48 18)(3 36 45 26)(4 62 46 20)(5 16 49 29)(6 42 50 59)(7 14 51 31)(8 44 52 57)(9 17 55 63)(10 25 56 35)(11 19 53 61)(12 27 54 33)(13 21 30 40)(15 23 32 38)(22 60 37 43)(24 58 39 41)
(2 4)(5 24)(6 23)(7 22)(8 21)(10 12)(13 44)(14 43)(15 42)(16 41)(17 63)(18 62)(19 61)(20 64)(25 33)(26 36)(27 35)(28 34)(29 58)(30 57)(31 60)(32 59)(37 51)(38 50)(39 49)(40 52)(46 48)(54 56)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4X4Y4Z
order12···222224···44···444
size11···144882···24···488

38 irreducible representations

dim111111111112224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC4226D4C424C4C23.23D4C24.C22C24.3C22C232D4C23⋊Q8C23.10D4C23.78C23C2×C42⋊C2C2×C4.4D4C42C2×C4C23C22
# reps1124211111141242

Matrix representation of C4226D4 in GL6(𝔽5)

210000
030000
000100
004000
000040
000004
,
420000
010000
002000
000200
000010
000001
,
420000
410000
000400
001000
000004
000010
,
400000
410000
001000
000400
000010
000004

G:=sub<GL(6,GF(5))| [2,0,0,0,0,0,1,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,2,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,4,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[4,4,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4] >;

C4226D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{26}D_4
% in TeX

G:=Group("C4^2:26D4");
// GroupNames label

G:=SmallGroup(128,1342);
// by ID

G=gap.SmallGroup(128,1342);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,568,758,723,675,248]);
// Polycyclic

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

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