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G = C42.384D4order 128 = 27

17th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.384D4, C42.700C23, (C4×D8)⋊4C2, (C4×Q16)⋊3C2, C42(Q8.Q8), C42(D4.Q8), Q8.Q854C2, D4.Q854C2, C42(D4⋊D4), (C4×SD16)⋊28C2, D4.2(C4○D4), D4⋊D4.5C2, C4⋊C4.55C23, (C4×C8).69C22, Q8.1(C4○D4), C42(D4.7D4), C42(D4.2D4), D4.2D453C2, D4.7D453C2, C4.135(C4○D8), C4⋊C8.309C22, (C2×C8).315C23, (C2×C4).300C24, C42(Q8.D4), Q8.D453C2, (C2×D4).85C23, (C4×D4).72C22, C23.249(C2×D4), (C22×C4).169D4, (C4×Q8).69C22, (C2×D8).122C22, (C2×Q8).371C23, C2.D8.169C22, C4.Q8.150C22, C42.12C428C2, C42(C23.20D4), C42(C23.19D4), C23.19D452C2, C23.20D453C2, C4⋊D4.159C22, C22⋊C8.188C22, (C2×C42).827C22, (C2×Q16).119C22, C22.560(C22×D4), C22⋊Q8.164C22, D4⋊C4.160C22, C2.27(D8⋊C22), (C22×C4).1016C23, C23.36C231C2, Q8⋊C4.173C22, (C2×SD16).137C22, C4.4D4.128C22, C42.C2.105C22, C42⋊C2.319C22, C2.101(C22.19C24), (C4×C4○D4)⋊9C2, C2.23(C2×C4○D8), C4.185(C2×C4○D4), (C2×C4).493(C2×D4), (C2×C4○D4).310C22, SmallGroup(128,1834)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.384D4
C1C2C4C2×C4C42C4×D4C4×C4○D4 — C42.384D4
C1C2C2×C4 — C42.384D4
C1C2×C4C2×C42 — C42.384D4
C1C2C2C2×C4 — C42.384D4

Generators and relations for C42.384D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=dad=ab2, bc=cb, bd=db, dcd=a2c3 >

Subgroups: 356 in 198 conjugacy classes, 90 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×10], C8 [×4], C2×C4 [×6], C2×C4 [×21], D4 [×2], D4 [×9], Q8 [×2], Q8 [×3], C23, C23 [×2], C42 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42, C42⋊C2, C42⋊C2 [×2], C4×D4 [×3], C4×D4 [×3], C4×Q8 [×3], C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C2×D8, C2×SD16 [×2], C2×Q16, C2×C4○D4, C42.12C4, C4×D8, C4×SD16 [×2], C4×Q16, D4⋊D4, D4.7D4, D4.2D4, Q8.D4, D4.Q8, Q8.Q8, C23.19D4, C23.20D4, C4×C4○D4, C23.36C23, C42.384D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4○D8 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C4○D8, D8⋊C22, C42.384D4

Smallest permutation representation of C42.384D4
On 64 points
Generators in S64
(1 57 5 61)(2 55 6 51)(3 59 7 63)(4 49 8 53)(9 24 13 20)(10 47 14 43)(11 18 15 22)(12 41 16 45)(17 34 21 38)(19 36 23 40)(25 50 29 54)(26 62 30 58)(27 52 31 56)(28 64 32 60)(33 42 37 46)(35 44 39 48)
(1 39 29 11)(2 40 30 12)(3 33 31 13)(4 34 32 14)(5 35 25 15)(6 36 26 16)(7 37 27 9)(8 38 28 10)(17 64 47 53)(18 57 48 54)(19 58 41 55)(20 59 42 56)(21 60 43 49)(22 61 44 50)(23 62 45 51)(24 63 46 52)
(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 8)(3 7)(4 6)(9 13)(10 12)(14 16)(17 41)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(26 32)(27 31)(28 30)(33 37)(34 36)(38 40)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 64)(56 63)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4L4M···4S4T4U4V8A···8H
order1222222244444···44···44448···8
size1111444811112···24···48884···4

38 irreducible representations

dim111111111111111222224
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D4C4○D8D8⋊C22
kernelC42.384D4C42.12C4C4×D8C4×SD16C4×Q16D4⋊D4D4.7D4D4.2D4Q8.D4D4.Q8Q8.Q8C23.19D4C23.20D4C4×C4○D4C23.36C23C42C22×C4D4Q8C4C2
# reps111211111111111224482

Matrix representation of C42.384D4 in GL4(𝔽17) generated by

4000
0400
00713
001210
,
16000
01600
00130
00013
,
14300
141400
00104
00137
,
1000
01600
0010
001216
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,7,12,0,0,13,10],[16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[14,14,0,0,3,14,0,0,0,0,10,13,0,0,4,7],[1,0,0,0,0,16,0,0,0,0,1,12,0,0,0,16] >;

C42.384D4 in GAP, Magma, Sage, TeX

C_4^2._{384}D_4
% in TeX

G:=Group("C4^2.384D4");
// GroupNames label

G:=SmallGroup(128,1834);
// by ID

G=gap.SmallGroup(128,1834);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,80,4037,1027,124]);
// Polycyclic

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

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