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G = C42.95D4order 128 = 27

77th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.95D4, C24.52(C2×C4), (C22×C4).42Q8, C23.25(C4⋊C4), (C22×C4).264D4, (C22×C8).9C22, C4.180(C4⋊D4), C22.46(C8○D4), C4.109(C22⋊Q8), C23.307(C22×C4), C22.7C423C2, (C2×C42).244C22, (C23×C4).235C22, C2.7(C23.7Q8), (C22×C4).1618C23, C22.78(C42⋊C2), C2.6(C42.7C22), C2.10(C42.6C22), (C2×C4⋊C8)⋊10C2, (C2×C4⋊C4).50C4, (C2×C4).41(C4⋊C4), C22.89(C2×C4⋊C4), (C2×C4).336(C2×Q8), (C2×C4).1514(C2×D4), (C2×C22⋊C4).36C4, (C2×C22⋊C8).14C2, (C2×C4).926(C4○D4), (C22×C4).112(C2×C4), (C2×C4).122(C22⋊C4), (C2×C42⋊C2).13C2, C22.243(C2×C22⋊C4), C2.25((C22×C8)⋊C2), SmallGroup(128,530)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.95D4
C1C2C4C2×C4C22×C4C23×C4C2×C42⋊C2 — C42.95D4
C1C23 — C42.95D4
C1C22×C4 — C42.95D4
C1C2C2C22×C4 — C42.95D4

Generators and relations for C42.95D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 268 in 156 conjugacy classes, 68 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×10], C2×C4 [×20], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×12], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C22⋊C8 [×4], C4⋊C8 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C22×C8 [×4], C23×C4, C22.7C42 [×2], C2×C22⋊C8 [×2], C2×C4⋊C8 [×2], C2×C42⋊C2, C42.95D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C8○D4 [×4], C23.7Q8, (C22×C8)⋊C2 [×2], C42.6C22 [×2], C42.7C22 [×2], C42.95D4

Smallest permutation representation of C42.95D4
On 64 points
Generators in S64
(1 21 30 13)(2 10 31 18)(3 23 32 15)(4 12 25 20)(5 17 26 9)(6 14 27 22)(7 19 28 11)(8 16 29 24)(33 53 57 47)(34 44 58 50)(35 55 59 41)(36 46 60 52)(37 49 61 43)(38 48 62 54)(39 51 63 45)(40 42 64 56)
(1 43 5 47)(2 44 6 48)(3 45 7 41)(4 46 8 42)(9 57 13 61)(10 58 14 62)(11 59 15 63)(12 60 16 64)(17 33 21 37)(18 34 22 38)(19 35 23 39)(20 36 24 40)(25 52 29 56)(26 53 30 49)(27 54 31 50)(28 55 32 51)
(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 42 43 4 5 46 47 8)(2 3 44 45 6 7 48 41)(9 60 57 16 13 64 61 12)(10 15 58 63 14 11 62 59)(17 36 33 24 21 40 37 20)(18 23 34 39 22 19 38 35)(25 26 52 53 29 30 56 49)(27 28 54 55 31 32 50 51)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4R8A···8P
order12···2224···44···48···8
size11···1441···14···44···4

44 irreducible representations

dim111111122222
type+++++++-
imageC1C2C2C2C2C4C4D4D4Q8C4○D4C8○D4
kernelC42.95D4C22.7C42C2×C22⋊C8C2×C4⋊C8C2×C42⋊C2C2×C22⋊C4C2×C4⋊C4C42C22×C4C22×C4C2×C4C22
# reps1222144422416

Matrix representation of C42.95D4 in GL6(𝔽17)

13110000
040000
001000
000100
0000415
0000013
,
100000
010000
0016000
0001600
000040
000004
,
1080000
1570000
0041500
00161300
0000150
0000015
,
7100000
2100000
0041500
0001300
0000150
000092

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

C42.95D4 in GAP, Magma, Sage, TeX

C_4^2._{95}D_4
% in TeX

G:=Group("C4^2.95D4");
// GroupNames label

G:=SmallGroup(128,530);
// by ID

G=gap.SmallGroup(128,530);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,723,124]);
// Polycyclic

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

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