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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, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, D4⋊C4, C4⋊C8, C4.Q8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C22.4Q16, C23.65C23, C24.3C22, C2×D4⋊C4, C2×C4⋊C8, C2×C4.Q8, C4⋊C4.106D4
Quotients: C1, C2, C22, D4, Q8, C23, SD16, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C422C2, C2×SD16, C4○D8, C8⋊C22, 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 61 46 16)(2 64 47 11)(3 59 48 14)(4 62 41 9)(5 57 42 12)(6 60 43 15)(7 63 44 10)(8 58 45 13)(17 55 26 40)(18 50 27 35)(19 53 28 38)(20 56 29 33)(21 51 30 36)(22 54 31 39)(23 49 32 34)(24 52 25 37)
(1 53 54 8)(2 7 55 52)(3 51 56 6)(4 5 49 50)(9 61 23 31)(10 30 24 60)(11 59 17 29)(12 28 18 58)(13 57 19 27)(14 26 20 64)(15 63 21 25)(16 32 22 62)(33 43 48 36)(34 35 41 42)(37 47 44 40)(38 39 45 46)
(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), (1,61,46,16)(2,64,47,11)(3,59,48,14)(4,62,41,9)(5,57,42,12)(6,60,43,15)(7,63,44,10)(8,58,45,13)(17,55,26,40)(18,50,27,35)(19,53,28,38)(20,56,29,33)(21,51,30,36)(22,54,31,39)(23,49,32,34)(24,52,25,37), (1,53,54,8)(2,7,55,52)(3,51,56,6)(4,5,49,50)(9,61,23,31)(10,30,24,60)(11,59,17,29)(12,28,18,58)(13,57,19,27)(14,26,20,64)(15,63,21,25)(16,32,22,62)(33,43,48,36)(34,35,41,42)(37,47,44,40)(38,39,45,46), (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), (1,61,46,16)(2,64,47,11)(3,59,48,14)(4,62,41,9)(5,57,42,12)(6,60,43,15)(7,63,44,10)(8,58,45,13)(17,55,26,40)(18,50,27,35)(19,53,28,38)(20,56,29,33)(21,51,30,36)(22,54,31,39)(23,49,32,34)(24,52,25,37), (1,53,54,8)(2,7,55,52)(3,51,56,6)(4,5,49,50)(9,61,23,31)(10,30,24,60)(11,59,17,29)(12,28,18,58)(13,57,19,27)(14,26,20,64)(15,63,21,25)(16,32,22,62)(33,43,48,36)(34,35,41,42)(37,47,44,40)(38,39,45,46), (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)], [(1,61,46,16),(2,64,47,11),(3,59,48,14),(4,62,41,9),(5,57,42,12),(6,60,43,15),(7,63,44,10),(8,58,45,13),(17,55,26,40),(18,50,27,35),(19,53,28,38),(20,56,29,33),(21,51,30,36),(22,54,31,39),(23,49,32,34),(24,52,25,37)], [(1,53,54,8),(2,7,55,52),(3,51,56,6),(4,5,49,50),(9,61,23,31),(10,30,24,60),(11,59,17,29),(12,28,18,58),(13,57,19,27),(14,26,20,64),(15,63,21,25),(16,32,22,62),(33,43,48,36),(34,35,41,42),(37,47,44,40),(38,39,45,46)], [(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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