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G = C4⋊C47D4order 128 = 27

4th semidirect product of C4⋊C4 and D4 acting via D4/C2=C22

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

Aliases: C4⋊C47D4, (C2×C4).21D8, (C2×D4).105D4, (C2×C4).34SD16, C22.84(C2×D8), C2.13(C4⋊D8), C4.22(C4⋊D4), C2.6(C4.4D8), C23.913(C2×D4), (C22×C4).313D4, C2.22(C22⋊D8), C2.13(C4⋊SD16), (C22×C8).73C22, C22.97(C2×SD16), C22.223C22≀C2, C2.22(C22⋊SD16), (C2×C42).364C22, (C22×D4).79C22, C22.230(C4⋊D4), C22.138(C8⋊C22), C22.7C4211C2, (C22×C4).1447C23, C23.65C235C2, C22.90(C4.4D4), C4.71(C22.D4), C2.14(C23.10D4), C2.6(C42.29C22), (C2×D4⋊C4)⋊14C2, (C2×C41D4).10C2, (C2×C4).1039(C2×D4), (C2×C4).878(C4○D4), (C2×C4⋊C4).126C22, SmallGroup(128,773)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C4⋊C47D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×D4⋊C4 — C4⋊C47D4
C1C2C22×C4 — C4⋊C47D4
C1C23C2×C42 — C4⋊C47D4
C1C2C2C22×C4 — C4⋊C47D4

Generators and relations for C4⋊C47D4
 G = < a,b,c,d | a4=b4=c4=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=b-1, dbd=ab, dcd=c-1 >

Subgroups: 552 in 201 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×7], C22 [×7], C22 [×20], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×11], D4 [×28], C23, C23 [×16], C42 [×2], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×6], C22×C4 [×3], C22×C4 [×2], C2×D4 [×4], C2×D4 [×26], C24 [×2], C2.C42, D4⋊C4 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C41D4 [×4], C22×C8 [×2], C22×D4 [×2], C22×D4 [×2], C22.7C42, C23.65C23, C2×D4⋊C4 [×4], C2×C41D4, C4⋊C47D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, D8 [×2], SD16 [×2], C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C2×D8, C2×SD16, C8⋊C22 [×2], C23.10D4, C22⋊D8, C22⋊SD16, C4⋊D8, C4⋊SD16, C4.4D8, C42.29C22, C4⋊C47D4

Smallest permutation representation of C4⋊C47D4
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 63 11 58)(2 62 12 57)(3 61 9 60)(4 64 10 59)(5 13 21 52)(6 16 22 51)(7 15 23 50)(8 14 24 49)(17 36 39 53)(18 35 40 56)(19 34 37 55)(20 33 38 54)(25 44 30 45)(26 43 31 48)(27 42 32 47)(28 41 29 46)
(1 34 8 44)(2 35 5 41)(3 36 6 42)(4 33 7 43)(9 53 22 47)(10 54 23 48)(11 55 24 45)(12 56 21 46)(13 28 62 18)(14 25 63 19)(15 26 64 20)(16 27 61 17)(29 57 40 52)(30 58 37 49)(31 59 38 50)(32 60 39 51)
(1 44)(2 43)(3 42)(4 41)(5 33)(6 36)(7 35)(8 34)(9 47)(10 46)(11 45)(12 48)(13 37)(14 40)(15 39)(16 38)(17 50)(18 49)(19 52)(20 51)(21 54)(22 53)(23 56)(24 55)(25 57)(26 60)(27 59)(28 58)(29 63)(30 62)(31 61)(32 64)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L8A···8H
order12···222224444444444448···8
size11···188882222444488884···4

32 irreducible representations

dim111112222224
type++++++++++
imageC1C2C2C2C2D4D4D4D8SD16C4○D4C8⋊C22
kernelC4⋊C47D4C22.7C42C23.65C23C2×D4⋊C4C2×C41D4C4⋊C4C22×C4C2×D4C2×C4C2×C4C2×C4C22
# reps111412244462

Matrix representation of C4⋊C47D4 in GL6(𝔽17)

010000
1600000
000100
0016000
0000160
0000016
,
3140000
14140000
00121200
0012500
000014
0000816
,
1600000
0160000
0001600
001000
00001613
000001
,
1600000
010000
000100
001000
00001613
000001

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,0,0,0,0,1,8,0,0,0,0,4,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,13,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,13,1] >;

C4⋊C47D4 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes_7D_4
% in TeX

G:=Group("C4:C4:7D4");
// GroupNames label

G:=SmallGroup(128,773);
// by ID

G=gap.SmallGroup(128,773);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2804,1411,718,172,4037,2028,1027,124]);
// Polycyclic

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

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