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G = C4227D4order 128 = 27

21st semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C4227D4, C23.519C24, C24.362C23, C22.2972+ 1+4, (C22×C4)⋊35D4, C428C450C2, C4.99(C4⋊D4), C232D425C2, C23.191(C2×D4), C23.10D456C2, (C2×C42).600C22, (C23×C4).422C22, (C22×C4).129C23, C22.344(C22×D4), C24.3C2264C2, (C22×D4).191C22, C2.34(C22.29C24), C2.C42.246C22, C2.24(C22.34C24), (C2×C41D4)⋊7C2, (C2×C4⋊D4)⋊23C2, (C2×C4).379(C2×D4), C2.43(C2×C4⋊D4), (C2×C42⋊C2)⋊36C2, (C2×C4).655(C4○D4), (C2×C4⋊C4).886C22, C22.391(C2×C4○D4), (C2×C22⋊C4).211C22, SmallGroup(128,1351)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4227D4
C1C2C22C23C22×C4C22×D4C2×C4⋊D4 — C4227D4
C1C23 — C4227D4
C1C23 — C4227D4
C1C23 — C4227D4

Generators and relations for C4227D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 852 in 364 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×4], C4 [×12], C22 [×3], C22 [×4], C22 [×38], C2×C4 [×12], C2×C4 [×32], D4 [×40], C23, C23 [×2], C23 [×34], C42 [×4], C42 [×2], C22⋊C4 [×20], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×12], C22×C4 [×4], C2×D4 [×44], C24, C24 [×4], C2.C42 [×4], C2×C42 [×2], C2×C22⋊C4 [×14], C2×C4⋊C4 [×4], C42⋊C2 [×4], C4⋊D4 [×8], C41D4 [×4], C23×C4, C22×D4 [×10], C428C4, C24.3C22 [×2], C232D4 [×4], C23.10D4 [×4], C2×C42⋊C2, C2×C4⋊D4 [×2], C2×C41D4, C4227D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, 2+ 1+4 [×4], C2×C4⋊D4, C22.29C24 [×4], C22.34C24 [×2], C4227D4

Smallest permutation representation of C4227D4
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 39 11)(2 44 40 12)(3 41 37 9)(4 42 38 10)(5 45 17 13)(6 46 18 14)(7 47 19 15)(8 48 20 16)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 36)(30 62 58 33)(31 63 59 34)(32 64 60 35)
(1 63 51 19)(2 33 52 6)(3 61 49 17)(4 35 50 8)(5 37 36 21)(7 39 34 23)(9 57 53 13)(10 32 54 48)(11 59 55 15)(12 30 56 46)(14 44 58 28)(16 42 60 26)(18 40 62 24)(20 38 64 22)(25 45 41 29)(27 47 43 31)
(1 15)(2 48)(3 13)(4 46)(5 41)(6 10)(7 43)(8 12)(9 17)(11 19)(14 38)(16 40)(18 42)(20 44)(21 29)(22 58)(23 31)(24 60)(25 36)(26 62)(27 34)(28 64)(30 50)(32 52)(33 54)(35 56)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4N4O4P4Q4R
order12···222222244444···44444
size11···144888822224···48888

32 irreducible representations

dim111111112224
type+++++++++++
imageC1C2C2C2C2C2C2C2D4D4C4○D42+ 1+4
kernelC4227D4C428C4C24.3C22C232D4C23.10D4C2×C42⋊C2C2×C4⋊D4C2×C41D4C42C22×C4C2×C4C22
# reps112441214444

Matrix representation of C4227D4 in GL8(ℤ)

10000000
01000000
00010000
00-100000
00000010
000011-1-1
00001000
0000000-1
,
10000000
01000000
00100000
00010000
00000100
0000-1000
000011-1-1
0000-2021
,
1-2000000
1-1000000
00-100000
00010000
00000100
00001000
0000-1-111
0000220-1
,
1-2000000
0-1000000
00-100000
000-10000
00001000
00000100
000000-10
0000220-1

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,-2,0,0,0,0,1,0,1,0,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,-1,1],[1,1,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,-1,2,0,0,0,0,1,0,-1,2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,2,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;

C4227D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{27}D_4
% in TeX

G:=Group("C4^2:27D4");
// GroupNames label

G:=SmallGroup(128,1351);
// by ID

G=gap.SmallGroup(128,1351);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,185,80]);
// 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=b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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