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G = C4⋊C4.106D4order 128 = 27

61st non-split extension by C4⋊C4 of D4 acting via D4/C2=C22

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

Aliases: C4⋊C4.106D4, (C2×C8).162D4, (C2×D4).12Q8, (C2×C4).37SD16, C2.20(C88D4), C2.15(C82D4), C2.8(D42Q8), (C22×C4).154D4, C23.922(C2×D4), C2.10(D4.Q8), C4.35(C22⋊Q8), C4.153(C4⋊D4), C22.4Q1625C2, C2.15(C4⋊SD16), (C22×C8).80C22, C4.33(C422C2), C2.20(D4.2D4), C22.116(C4○D8), (C2×C42).376C22, C22.101(C2×SD16), (C22×D4).89C22, C2.6(C23.Q8), C22.243(C4⋊D4), C22.145(C8⋊C22), (C22×C4).1456C23, C23.65C237C2, C22.109(C22⋊Q8), C24.3C22.17C2, (C2×C4⋊C8)⋊34C2, (C2×C4.Q8)⋊22C2, (C2×C4).283(C2×Q8), (C2×C4).1055(C2×D4), (C2×D4⋊C4).15C2, (C2×C4).620(C4○D4), (C2×C4⋊C4).141C22, SmallGroup(128,797)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C4⋊C4.106D4
C1C2C4C2×C4C22×C4C22×D4C2×D4⋊C4 — C4⋊C4.106D4
C1C2C22×C4 — C4⋊C4.106D4
C1C23C2×C42 — C4⋊C4.106D4
C1C2C2C22×C4 — C4⋊C4.106D4

Generators and relations for C4⋊C4.106D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a, bab-1=cac-1=a-1, ad=da, cbc-1=a2b-1, dbd-1=a-1b, dcd-1=ac-1 >

Subgroups: 344 in 142 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×7], C22 [×7], C22 [×10], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×15], D4 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, C2.C42, D4⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C2×C4⋊C4, C22×C8 [×2], C22×D4, C22.4Q16, C23.65C23, C24.3C22, C2×D4⋊C4 [×2], C2×C4⋊C8, C2×C4.Q8, C4⋊C4.106D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, SD16 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×3], C4⋊D4 [×3], C22⋊Q8 [×3], C422C2, C2×SD16, C4○D8, C8⋊C22 [×2], C23.Q8, C4⋊SD16, D4.2D4, C88D4, C82D4, D42Q8, D4.Q8, C4⋊C4.106D4

Smallest permutation representation of C4⋊C4.106D4
On 64 points
Generators in S64
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(1 11 47 62)(2 14 48 57)(3 9 41 60)(4 12 42 63)(5 15 43 58)(6 10 44 61)(7 13 45 64)(8 16 46 59)(17 52 29 33)(18 55 30 36)(19 50 31 39)(20 53 32 34)(21 56 25 37)(22 51 26 40)(23 54 27 35)(24 49 28 38)
(1 49 50 8)(2 7 51 56)(3 55 52 6)(4 5 53 54)(9 22 29 57)(10 64 30 21)(11 20 31 63)(12 62 32 19)(13 18 25 61)(14 60 26 17)(15 24 27 59)(16 58 28 23)(33 44 41 36)(34 35 42 43)(37 48 45 40)(38 39 46 47)
(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)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G4H4I···4N8A···8H
order12···222444444444···48···8
size11···188222244448···84···4

32 irreducible representations

dim111111122222224
type++++++++++-+
imageC1C2C2C2C2C2C2D4D4D4Q8SD16C4○D4C4○D8C8⋊C22
kernelC4⋊C4.106D4C22.4Q16C23.65C23C24.3C22C2×D4⋊C4C2×C4⋊C8C2×C4.Q8C4⋊C4C2×C8C22×C4C2×D4C2×C4C2×C4C22C22
# reps111121122224642

Matrix representation of C4⋊C4.106D4 in GL6(𝔽17)

0160000
100000
001000
000100
000010
000001
,
3140000
14140000
0091500
007800
0000013
0000130
,
12120000
1250000
001000
0091600
000001
0000160
,
12120000
5120000
001000
000100
000001
000010

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],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,9,7,0,0,0,0,15,8,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[12,12,0,0,0,0,12,5,0,0,0,0,0,0,1,9,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[12,5,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C4⋊C4.106D4 in GAP, Magma, Sage, TeX

C_4\rtimes C_4._{106}D_4
% in TeX

G:=Group("C4:C4.106D4");
// GroupNames label

G:=SmallGroup(128,797);
// by ID

G=gap.SmallGroup(128,797);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,64,422,387,2019,521,248,2804,718,172,4037,1027,124]);
// Polycyclic

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

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