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G = C42.265D4order 128 = 27

247th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.265D4, C42.728C23, C4.992+ 1+4, C84D49C2, D4⋊D43C2, C85D421C2, D4.2D41C2, C4.28(C4○D8), C4.4D826C2, (C4×C8).73C22, C4⋊C8.311C22, C4⋊C4.146C23, C4.26(C8⋊C22), (C2×C4).405C24, (C2×C8).159C23, C4.SD1611C2, (C2×D8).23C22, C23.284(C2×D4), (C22×C4).495D4, C4⋊Q8.300C22, (C2×D4).155C23, (C4×D4).104C22, (C2×Q8).142C23, Q8⋊C4.2C22, C42.12C433C2, C4⋊D4.188C22, C41D4.161C22, C22⋊C8.192C22, (C2×C42).872C22, (C2×SD16).84C22, C22.665(C22×D4), D4⋊C4.106C22, (C22×C4).1076C23, C22.26C2417C2, C4.4D4.149C22, C2.76(C22.29C24), C2.41(C2×C4○D8), (C2×C4).537(C2×D4), C2.54(C2×C8⋊C22), (C2×C4○D4).171C22, SmallGroup(128,1939)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.265D4
C1C2C4C2×C4C22×C4C2×C4○D4C22.26C24 — C42.265D4
C1C2C2×C4 — C42.265D4
C1C22C2×C42 — C42.265D4
C1C2C2C2×C4 — C42.265D4

Generators and relations for C42.265D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1, cbc-1=dbd=a2b, dcd=b2c3 >

Subgroups: 492 in 219 conjugacy classes, 88 normal (34 characteristic)
C1, C2 [×3], C2 [×5], C4 [×6], C4 [×7], C22, C22 [×15], C8 [×4], C2×C4 [×6], C2×C4 [×18], D4 [×24], Q8 [×6], C23, C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×4], D8 [×4], SD16 [×4], C22×C4 [×3], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×2], C2×Q8, C4○D4 [×8], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2×C42, C4×D4 [×2], C4×D4 [×3], C4⋊D4 [×2], C4⋊D4 [×3], C4.4D4 [×2], C4.4D4, C41D4 [×2], C4⋊Q8 [×2], C2×D8 [×4], C2×SD16 [×4], C2×C4○D4 [×2], C2×C4○D4, C42.12C4, D4⋊D4 [×4], D4.2D4 [×4], C4.4D8, C4.SD16, C85D4, C84D4, C22.26C24 [×2], C42.265D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C8⋊C22 [×2], C22×D4, 2+ 1+4 [×2], C22.29C24, C2×C4○D8, C2×C8⋊C22, C42.265D4

Smallest permutation representation of C42.265D4
On 64 points
Generators in S64
(1 61 29 53)(2 62 30 54)(3 63 31 55)(4 64 32 56)(5 57 25 49)(6 58 26 50)(7 59 27 51)(8 60 28 52)(9 19 40 41)(10 20 33 42)(11 21 34 43)(12 22 35 44)(13 23 36 45)(14 24 37 46)(15 17 38 47)(16 18 39 48)
(1 39 5 35)(2 9 6 13)(3 33 7 37)(4 11 8 15)(10 27 14 31)(12 29 16 25)(17 64 21 60)(18 49 22 53)(19 58 23 62)(20 51 24 55)(26 36 30 40)(28 38 32 34)(41 50 45 54)(42 59 46 63)(43 52 47 56)(44 61 48 57)
(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)
(2 8)(3 7)(4 6)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 40)(16 39)(17 19)(20 24)(21 23)(26 32)(27 31)(28 30)(41 47)(42 46)(43 45)(49 57)(50 64)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4J4K4L4M4N4O8A···8H
order1222222224···4444448···8
size1111488882···2488884···4

32 irreducible representations

dim11111111122244
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4C4○D8C8⋊C222+ 1+4
kernelC42.265D4C42.12C4D4⋊D4D4.2D4C4.4D8C4.SD16C85D4C84D4C22.26C24C42C22×C4C4C4C4
# reps11441111222822

Matrix representation of C42.265D4 in GL6(𝔽17)

100000
010000
00012014
00120140
000305
003050
,
1300000
0130000
000100
001000
000001
000010
,
330000
1430000
000010
0000016
001000
0001600
,
010000
100000
001000
0001600
000010
0000016

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

C42.265D4 in GAP, Magma, Sage, TeX

C_4^2._{265}D_4
% in TeX

G:=Group("C4^2.265D4");
// GroupNames label

G:=SmallGroup(128,1939);
// by ID

G=gap.SmallGroup(128,1939);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,675,248,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,d*a*d=a^-1,c*b*c^-1=d*b*d=a^2*b,d*c*d=b^2*c^3>;
// generators/relations

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