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G = C4217D4order 128 = 27

11st semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C4217D4, C24.318C23, C23.435C24, C22.2252+ 1+4, (C2×D4)⋊33D4, C23.46(C2×D4), C428C440C2, C232D416C2, C2.66(D45D4), C4.165(C4⋊D4), C23.49(C4○D4), (C22×C4).92C23, C23.7Q863C2, C23.23D453C2, C23.11D440C2, (C23×C4).111C22, (C2×C42).541C22, C22.286(C22×D4), (C22×D4).526C22, C2.20(C22.29C24), C2.58(C22.19C24), C2.C42.178C22, C2.52(C22.47C24), (C2×C4×D4)⋊42C2, (C2×C4⋊D4)⋊14C2, C2.30(C2×C4⋊D4), (C2×C4).1193(C2×D4), (C2×C4).817(C4○D4), (C2×C4⋊C4).865C22, C22.312(C2×C4○D4), (C2×C22⋊C4).171C22, SmallGroup(128,1267)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4217D4
C1C2C22C23C22×C4C23×C4C2×C4×D4 — C4217D4
C1C23 — C4217D4
C1C23 — C4217D4
C1C23 — C4217D4

Generators and relations for C4217D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 772 in 366 conjugacy classes, 112 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×12], C22 [×3], C22 [×4], C22 [×40], C2×C4 [×10], C2×C4 [×44], D4 [×32], C23, C23 [×8], C23 [×24], C42 [×4], C22⋊C4 [×20], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×8], C22×C4 [×20], C2×D4 [×4], C2×D4 [×30], C24 [×4], C2.C42 [×8], C2×C42, C2×C22⋊C4 [×12], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4×D4 [×8], C4⋊D4 [×8], C23×C4 [×4], C22×D4 [×2], C22×D4 [×4], C23.7Q8 [×2], C428C4, C23.23D4 [×4], C232D4 [×2], C23.11D4 [×2], C2×C4×D4 [×2], C2×C4⋊D4 [×2], C4217D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ 1+4 [×2], C2×C4⋊D4, C22.19C24, C22.29C24, D45D4 [×2], C22.47C24 [×2], C4217D4

Smallest permutation representation of C4217D4
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 15 21 11)(2 16 22 12)(3 13 23 9)(4 14 24 10)(5 56 38 26)(6 53 39 27)(7 54 40 28)(8 55 37 25)(17 30 50 60)(18 31 51 57)(19 32 52 58)(20 29 49 59)(33 47 61 44)(34 48 62 41)(35 45 63 42)(36 46 64 43)
(1 63 37 49)(2 34 38 19)(3 61 39 51)(4 36 40 17)(5 52 22 62)(6 18 23 33)(7 50 24 64)(8 20 21 35)(9 44 53 57)(10 46 54 30)(11 42 55 59)(12 48 56 32)(13 47 27 31)(14 43 28 60)(15 45 25 29)(16 41 26 58)
(1 55)(2 26)(3 53)(4 28)(5 12)(6 13)(7 10)(8 15)(9 39)(11 37)(14 40)(16 38)(17 60)(18 31)(19 58)(20 29)(21 25)(22 56)(23 27)(24 54)(30 50)(32 52)(33 47)(34 41)(35 45)(36 43)(42 63)(44 61)(46 64)(48 62)(49 59)(51 57)

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2O4A···4H4I···4R4S4T4U4V
order12···22···24···44···44444
size11···14···42···24···48888

38 irreducible representations

dim1111111122224
type+++++++++++
imageC1C2C2C2C2C2C2C2D4D4C4○D4C4○D42+ 1+4
kernelC4217D4C23.7Q8C428C4C23.23D4C232D4C23.11D4C2×C4×D4C2×C4⋊D4C42C2×D4C2×C4C23C22
# reps1214222244482

Matrix representation of C4217D4 in GL6(𝔽5)

310000
220000
002000
000300
000010
000001
,
120000
440000
001000
000100
000010
000001
,
200000
330000
000100
004000
000032
000002
,
430000
010000
004000
000100
000040
000031

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

C4217D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{17}D_4
% in TeX

G:=Group("C4^2:17D4");
// GroupNames label

G:=SmallGroup(128,1267);
// by ID

G=gap.SmallGroup(128,1267);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,456,758,723,675,80]);
// 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*b^2,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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