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G = C42.281D4order 128 = 27

263rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.281D4, C42.414C23, C4.532- 1+4, C83Q822C2, Q8⋊Q837C2, (C2×C4).58SD16, C4.87(C2×SD16), C4⋊C8.338C22, C4⋊C4.168C23, (C2×C4).427C24, (C4×C8).268C22, (C2×C8).332C23, C4.SD1627C2, (C22×C4).510D4, C23.699(C2×D4), C4⋊Q8.311C22, C4.Q8.85C22, C4.29(C8.C22), (C2×Q8).161C23, (C4×Q8).108C22, C22.26(C2×SD16), C2.25(C22×SD16), C22⋊C8.221C22, (C2×C42).888C22, C23.47D4.5C2, C22.687(C22×D4), C22⋊Q8.201C22, C42.12C4.40C2, (C22×C4).1092C23, Q8⋊C4.104C22, C23.37C23.38C2, C2.75(C23.38C23), (C2×C4⋊Q8).54C2, (C2×C4).870(C2×D4), C2.60(C2×C8.C22), (C2×C4⋊C4).647C22, SmallGroup(128,1961)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.281D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×C4⋊Q8 — C42.281D4
C1C2C2×C4 — C42.281D4
C1C22C2×C42 — C42.281D4
C1C2C2C2×C4 — C42.281D4

Generators and relations for C42.281D4
 G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=a2b2c3 >

Subgroups: 316 in 180 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×16], Q8 [×14], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×6], C4⋊C4 [×13], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×2], C2×Q8 [×9], C4×C8 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×Q8, C42.12C4, Q8⋊Q8 [×4], C23.47D4 [×4], C4.SD16 [×2], C83Q8 [×2], C2×C4⋊Q8, C23.37C23, C42.281D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C24, C2×SD16 [×6], C8.C22 [×2], C22×D4, 2- 1+4 [×2], C23.38C23, C22×SD16, C2×C8.C22, C42.281D4

Smallest permutation representation of C42.281D4
On 64 points
Generators in S64
(1 31 17 45)(2 32 18 46)(3 25 19 47)(4 26 20 48)(5 27 21 41)(6 28 22 42)(7 29 23 43)(8 30 24 44)(9 64 50 38)(10 57 51 39)(11 58 52 40)(12 59 53 33)(13 60 54 34)(14 61 55 35)(15 62 56 36)(16 63 49 37)
(1 23 5 19)(2 4 6 8)(3 17 7 21)(9 11 13 15)(10 49 14 53)(12 51 16 55)(18 20 22 24)(25 45 29 41)(26 28 30 32)(27 47 31 43)(33 57 37 61)(34 36 38 40)(35 59 39 63)(42 44 46 48)(50 52 54 56)(58 60 62 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 54 5 50)(2 12 6 16)(3 52 7 56)(4 10 8 14)(9 17 13 21)(11 23 15 19)(18 53 22 49)(20 51 24 55)(25 58 29 62)(26 39 30 35)(27 64 31 60)(28 37 32 33)(34 41 38 45)(36 47 40 43)(42 63 46 59)(44 61 48 57)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J4K···4R8A···8H
order1222224···4444···48···8
size1111222···2448···84···4

32 irreducible representations

dim1111111122244
type++++++++++--
imageC1C2C2C2C2C2C2C2D4D4SD16C8.C222- 1+4
kernelC42.281D4C42.12C4Q8⋊Q8C23.47D4C4.SD16C83Q8C2×C4⋊Q8C23.37C23C42C22×C4C2×C4C4C4
# reps1144221122822

Matrix representation of C42.281D4 in GL6(𝔽17)

1150000
1160000
0016000
0001600
0000160
0000016
,
1620000
1610000
0013000
000400
0009130
0011004
,
070000
570000
00101015
0016220
00651516
005617
,
1300000
1340000
00115150
00101015
00815162
00108716

G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,13,0,0,11,0,0,0,4,9,0,0,0,0,0,13,0,0,0,0,0,0,4],[0,5,0,0,0,0,7,7,0,0,0,0,0,0,10,16,6,5,0,0,1,2,5,6,0,0,0,2,15,1,0,0,15,0,16,7],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,1,10,8,10,0,0,15,1,15,8,0,0,15,0,16,7,0,0,0,15,2,16] >;

C42.281D4 in GAP, Magma, Sage, TeX

C_4^2._{281}D_4
% in TeX

G:=Group("C4^2.281D4");
// GroupNames label

G:=SmallGroup(128,1961);
// by ID

G=gap.SmallGroup(128,1961);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,100,675,4037,1027,124]);
// Polycyclic

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

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