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G = C4225D4order 128 = 27

19th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C4225D4, C23.509C24, C24.357C23, C22.2892+ 1+4, C22.2102- 1+4, C428C449C2, C23.8Q881C2, C23.Q835C2, C23.162(C4○D4), (C23×C4).413C22, (C22×C4).555C23, (C2×C42).596C22, C22.335(C22×D4), C23.23D4.44C2, C23.10D4.30C2, (C22×D4).186C22, C23.81C2354C2, C24.C22102C2, C2.78(C22.19C24), C23.63C23110C2, C2.72(C22.45C24), C2.C42.238C22, C2.20(C22.31C24), C2.31(C22.49C24), C2.77(C22.46C24), (C2×C4).370(C2×D4), (C2×C42⋊C2)⋊34C2, (C2×C4).411(C4○D4), (C2×C4⋊C4).348C22, C22.385(C2×C4○D4), (C2×C22⋊C4).205C22, SmallGroup(128,1341)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4225D4
C1C2C22C23C22×C4C23×C4C2×C42⋊C2 — C4225D4
C1C23 — C4225D4
C1C23 — C4225D4
C1C23 — C4225D4

Generators and relations for C4225D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=dbd=a2b-1, dcd=c-1 >

Subgroups: 500 in 270 conjugacy classes, 100 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×18], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×10], C2×C4 [×50], D4 [×4], C23, C23 [×4], C23 [×12], C42 [×4], C42 [×4], C22⋊C4 [×18], C4⋊C4 [×15], C22×C4 [×3], C22×C4 [×10], C22×C4 [×12], C2×D4 [×5], C24 [×2], C2.C42 [×2], C2.C42 [×8], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×3], C2×C4⋊C4 [×6], C42⋊C2 [×8], C23×C4 [×2], C22×D4, C428C4, C23.8Q8 [×2], C23.23D4 [×2], C23.63C23 [×2], C24.C22 [×2], C23.10D4, C23.Q8 [×2], C23.81C23, C2×C42⋊C2 [×2], C4225D4
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- 1+4, C22.19C24 [×2], C22.31C24, C22.45C24, C22.46C24 [×2], C22.49C24, C4225D4

Smallest permutation representation of C4225D4
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 50 29 16)(2 51 30 13)(3 52 31 14)(4 49 32 15)(5 45 61 44)(6 46 62 41)(7 47 63 42)(8 48 64 43)(9 36 28 37)(10 33 25 38)(11 34 26 39)(12 35 27 40)(17 53 24 58)(18 54 21 59)(19 55 22 60)(20 56 23 57)
(1 57 25 63)(2 60 26 62)(3 59 27 61)(4 58 28 64)(5 31 54 12)(6 30 55 11)(7 29 56 10)(8 32 53 9)(13 17 34 43)(14 20 35 42)(15 19 36 41)(16 18 33 44)(21 38 45 50)(22 37 46 49)(23 40 47 52)(24 39 48 51)
(1 63)(2 6)(3 61)(4 8)(5 31)(7 29)(9 58)(10 56)(11 60)(12 54)(13 48)(14 42)(15 46)(16 44)(17 39)(18 33)(19 37)(20 35)(21 38)(22 36)(23 40)(24 34)(25 57)(26 55)(27 59)(28 53)(30 62)(32 64)(41 49)(43 51)(45 50)(47 52)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4V4W4X4Y4Z
order12···222224···44···44444
size11···144442···24···48888

38 irreducible representations

dim111111111122244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC4225D4C428C4C23.8Q8C23.23D4C23.63C23C24.C22C23.10D4C23.Q8C23.81C23C2×C42⋊C2C42C2×C4C23C22C22
# reps112222121248811

Matrix representation of C4225D4 in GL6(𝔽5)

110000
340000
000100
004000
000040
000004
,
220000
130000
002000
000200
000010
000001
,
440000
010000
004000
000100
000022
000003
,
440000
010000
004000
000400
000022
000013

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

C4225D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{25}D_4
% in TeX

G:=Group("C4^2:25D4");
// GroupNames label

G:=SmallGroup(128,1341);
// by ID

G=gap.SmallGroup(128,1341);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,675,248]);
// 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,d*a*d=a^-1*b^2,c*b*c^-1=d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations

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