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G = C42.278D4order 128 = 27

260th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.278D4, C42.412C23, C4.502- 1+4, (C2×C4).53D8, C4.73(C2×D8), D4⋊Q88C2, C82Q812C2, C4.4D813C2, (C4×C8).76C22, C2.16(C22×D8), C22.25(C2×D8), C4⋊C4.165C23, C4⋊C8.288C22, (C2×C4).424C24, (C2×C8).165C23, C22.D86C2, C23.697(C2×D4), (C22×C4).507D4, C4⋊Q8.308C22, C2.D8.35C22, D4⋊C4.3C22, (C2×D4).173C23, (C4×D4).111C22, C4.28(C8.C22), C42.12C427C2, C41D4.169C22, C4⋊D4.196C22, C22⋊C8.180C22, (C2×C42).885C22, C22.684(C22×D4), (C22×C4).1089C23, C22.26C24.43C2, C2.72(C23.38C23), (C2×C4⋊Q8)⋊42C2, (C2×C4).868(C2×D4), C2.59(C2×C8.C22), (C2×C4⋊C4).645C22, SmallGroup(128,1958)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.278D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×C4⋊Q8 — C42.278D4
C1C2C2×C4 — C42.278D4
C1C22C2×C42 — C42.278D4
C1C2C2C2×C4 — C42.278D4

Generators and relations for C42.278D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1, cbc-1=a2b-1, dbd=a2b, dcd=a2c3 >

Subgroups: 412 in 202 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×18], D4 [×12], Q8 [×10], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×9], C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C2.D8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C4⋊Q8 [×4], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, C42.12C4, D4⋊Q8 [×4], C22.D8 [×4], C4.4D8 [×2], C82Q8 [×2], C2×C4⋊Q8, C22.26C24, C42.278D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C8.C22 [×2], C22×D4, 2- 1+4 [×2], C23.38C23, C22×D8, C2×C8.C22, C42.278D4

Smallest permutation representation of C42.278D4
On 64 points
Generators in S64
(1 56 25 63)(2 49 26 64)(3 50 27 57)(4 51 28 58)(5 52 29 59)(6 53 30 60)(7 54 31 61)(8 55 32 62)(9 47 38 19)(10 48 39 20)(11 41 40 21)(12 42 33 22)(13 43 34 23)(14 44 35 24)(15 45 36 17)(16 46 37 18)
(1 16 5 12)(2 34 6 38)(3 10 7 14)(4 36 8 40)(9 26 13 30)(11 28 15 32)(17 55 21 51)(18 59 22 63)(19 49 23 53)(20 61 24 57)(25 37 29 33)(27 39 31 35)(41 58 45 62)(42 56 46 52)(43 60 47 64)(44 50 48 54)
(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 28)(3 7)(4 26)(6 32)(8 30)(9 11)(10 35)(12 33)(13 15)(14 39)(16 37)(17 43)(19 41)(20 24)(21 47)(23 45)(27 31)(34 36)(38 40)(44 48)(49 51)(50 61)(52 59)(53 55)(54 57)(56 63)(58 64)(60 62)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I4J4K···4P8A···8H
order122222224···4444···48···8
size111122882···2448···84···4

32 irreducible representations

dim1111111122244
type+++++++++++--
imageC1C2C2C2C2C2C2C2D4D4D8C8.C222- 1+4
kernelC42.278D4C42.12C4D4⋊Q8C22.D8C4.4D8C82Q8C2×C4⋊Q8C22.26C24C42C22×C4C2×C4C4C4
# reps1144221122822

Matrix representation of C42.278D4 in GL6(𝔽17)

0160000
100000
001000
000100
000010
000001
,
010000
1600000
000001
0000160
000100
0016000
,
3140000
330000
001521515
001515215
0022215
0015222
,
100000
0160000
001000
0001600
0000160
000001

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1,0,0,0],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,15,15,2,15,0,0,2,15,2,2,0,0,15,2,2,2,0,0,15,15,15,2],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;

C42.278D4 in GAP, Magma, Sage, TeX

C_4^2._{278}D_4
% in TeX

G:=Group("C4^2.278D4");
// GroupNames label

G:=SmallGroup(128,1958);
// by ID

G=gap.SmallGroup(128,1958);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,100,675,4037,1027,124]);
// 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=a^2*b^-1,d*b*d=a^2*b,d*c*d=a^2*c^3>;
// generators/relations

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