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G = C4247D4order 128 = 27

6th semidirect product of C42 and D4 acting via D4/C22=C2

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

Aliases: C4247D4, C24.122C23, C23.756C24, C43(C4⋊D4), (C22×C4)⋊48D4, C429C440C2, C221(C41D4), (C22×C42)⋊17C2, C23.374(C2×D4), (C23×C4).683C22, C22.466(C22×D4), (C22×C4).1486C23, (C2×C42).1091C22, (C22×D4).313C22, C24.3C22101C2, C2.56(C22.26C24), (C2×C4⋊D4)⋊42C2, (C2×C41D4)⋊12C2, (C2×C4).687(C2×D4), C2.49(C2×C4⋊D4), C2.16(C2×C41D4), (C2×C4).672(C4○D4), (C2×C4⋊C4).559C22, C22.597(C2×C4○D4), (C2×C22⋊C4).366C22, SmallGroup(128,1588)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4247D4
C1C2C22C23C24C23×C4C22×C42 — C4247D4
C1C23 — C4247D4
C1C23 — C4247D4
C1C23 — C4247D4

Generators and relations for C4247D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 948 in 462 conjugacy classes, 144 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×12], C4 [×10], C22, C22 [×10], C22 [×40], C2×C4 [×24], C2×C4 [×42], D4 [×40], C23, C23 [×6], C23 [×32], C42 [×4], C42 [×6], C22⋊C4 [×24], C4⋊C4 [×12], C22×C4 [×22], C22×C4 [×12], C2×D4 [×60], C24, C24 [×4], C2×C42, C2×C42 [×3], C2×C42 [×4], C2×C22⋊C4 [×12], C2×C4⋊C4 [×6], C4⋊D4 [×24], C41D4 [×4], C23×C4 [×3], C22×D4 [×10], C429C4, C24.3C22 [×6], C22×C42, C2×C4⋊D4 [×6], C2×C41D4, C4247D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×16], C23 [×15], C2×D4 [×24], C4○D4 [×6], C24, C4⋊D4 [×12], C41D4 [×4], C22×D4 [×4], C2×C4○D4 [×3], C2×C4⋊D4 [×3], C2×C41D4, C22.26C24 [×3], C4247D4

Smallest permutation representation of C4247D4
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 49 41 61)(2 50 42 62)(3 51 43 63)(4 52 44 64)(5 60 20 12)(6 57 17 9)(7 58 18 10)(8 59 19 11)(13 34 25 21)(14 35 26 22)(15 36 27 23)(16 33 28 24)(29 53 45 38)(30 54 46 39)(31 55 47 40)(32 56 48 37)
(1 34 11 55)(2 33 12 54)(3 36 9 53)(4 35 10 56)(5 30 50 16)(6 29 51 15)(7 32 52 14)(8 31 49 13)(17 45 63 27)(18 48 64 26)(19 47 61 25)(20 46 62 28)(21 59 40 41)(22 58 37 44)(23 57 38 43)(24 60 39 42)
(1 61)(2 64)(3 63)(4 62)(5 58)(6 57)(7 60)(8 59)(9 17)(10 20)(11 19)(12 18)(13 40)(14 39)(15 38)(16 37)(21 31)(22 30)(23 29)(24 32)(25 55)(26 54)(27 53)(28 56)(33 48)(34 47)(35 46)(36 45)(41 49)(42 52)(43 51)(44 50)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4X4Y4Z4AA4AB
order12···2222222224···44444
size11···1222288882···28888

44 irreducible representations

dim111111222
type++++++++
imageC1C2C2C2C2C2D4D4C4○D4
kernelC4247D4C429C4C24.3C22C22×C42C2×C4⋊D4C2×C41D4C42C22×C4C2×C4
# reps11616141212

Matrix representation of C4247D4 in GL6(𝔽5)

100000
010000
000100
004000
000024
000003
,
030000
300000
000100
004000
000031
000002
,
200000
030000
000100
001000
000020
000033
,
030000
200000
000400
004000
000024
000033

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

C4247D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{47}D_4
% in TeX

G:=Group("C4^2:47D4");
// GroupNames label

G:=SmallGroup(128,1588);
// by ID

G=gap.SmallGroup(128,1588);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,184,2019]);
// Polycyclic

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

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