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G = C42.378D4order 128 = 27

11st non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.378D4, C23.34C42, C23.29M4(2), C22⋊C810C4, C4.156(C4×D4), (C2×C4).44C42, (C23×C4).28C4, (C2×C42).33C4, C221(C8⋊C4), C24.110(C2×C4), C2.10(C4×M4(2)), (C2×C4).59M4(2), (C22×C42).6C2, C22.49(C2×C42), C2.2(C24.4C4), C2.2(C42.6C4), (C22×C8).377C22, (C2×C42).988C22, C23.251(C22×C4), (C23×C4).629C22, C22.37(C2×M4(2)), C22.7C4235C2, (C22×C4).1604C23, C22.49(C42⋊C2), (C2×C8)⋊23(C2×C4), C2.7(C2×C8⋊C4), (C2×C8⋊C4)⋊17C2, C2.8(C4×C22⋊C4), (C2×C4).1494(C2×D4), (C2×C22⋊C8).40C2, (C2×C4).914(C4○D4), (C2×C4).594(C22×C4), (C22×C4).434(C2×C4), (C2×C4).325(C22⋊C4), C22.112(C2×C22⋊C4), SmallGroup(128,481)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.378D4
C1C2C4C2×C4C22×C4C2×C42C2×C8⋊C4 — C42.378D4
C1C22 — C42.378D4
C1C22×C4 — C42.378D4
C1C2C2C22×C4 — C42.378D4

Generators and relations for C42.378D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b-1, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 316 in 206 conjugacy classes, 96 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×8], C22 [×12], C8 [×8], C2×C4 [×16], C2×C4 [×34], C23, C23 [×6], C23 [×4], C42 [×4], C42 [×6], C2×C8 [×8], C2×C8 [×8], C22×C4 [×2], C22×C4 [×8], C22×C4 [×16], C24, C8⋊C4 [×4], C22⋊C8 [×8], C2×C42 [×2], C2×C42 [×2], C2×C42 [×4], C22×C8 [×4], C23×C4, C23×C4 [×2], C22.7C42 [×2], C2×C8⋊C4 [×2], C2×C22⋊C8 [×2], C22×C42, C42.378D4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], M4(2) [×8], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C8⋊C4 [×4], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C2×M4(2) [×4], C4×C22⋊C4, C2×C8⋊C4, C4×M4(2), C24.4C4 [×2], C42.6C4 [×2], C42.378D4

Smallest permutation representation of C42.378D4
On 64 points
Generators in S64
(1 57 51 16)(2 62 52 13)(3 59 53 10)(4 64 54 15)(5 61 55 12)(6 58 56 9)(7 63 49 14)(8 60 50 11)(17 37 27 44)(18 34 28 41)(19 39 29 46)(20 36 30 43)(21 33 31 48)(22 38 32 45)(23 35 25 42)(24 40 26 47)
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 47 13 43)(10 48 14 44)(11 41 15 45)(12 42 16 46)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(33 63 37 59)(34 64 38 60)(35 57 39 61)(36 58 40 62)
(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 36 25 13 5 40 29 9)(2 57 26 42 6 61 30 46)(3 38 27 15 7 34 31 11)(4 59 28 44 8 63 32 48)(10 18 37 50 14 22 33 54)(12 20 39 52 16 24 35 56)(17 64 49 41 21 60 53 45)(19 58 51 43 23 62 55 47)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AB8A···8P
order12···222224···44···48···8
size11···122221···12···24···4

56 irreducible representations

dim111111112222
type++++++
imageC1C2C2C2C2C4C4C4D4M4(2)C4○D4M4(2)
kernelC42.378D4C22.7C42C2×C8⋊C4C2×C22⋊C8C22×C42C22⋊C8C2×C42C23×C4C42C2×C4C2×C4C23
# reps1222116444848

Matrix representation of C42.378D4 in GL5(𝔽17)

40000
013000
00400
000130
000013
,
10000
04000
00400
00010
00001
,
10000
00100
04000
0001516
00052
,
130000
001600
013000
000913
000128

G:=sub<GL(5,GF(17))| [4,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,15,5,0,0,0,16,2],[13,0,0,0,0,0,0,13,0,0,0,16,0,0,0,0,0,0,9,12,0,0,0,13,8] >;

C42.378D4 in GAP, Magma, Sage, TeX

C_4^2._{378}D_4
% in TeX

G:=Group("C4^2.378D4");
// GroupNames label

G:=SmallGroup(128,481);
// by ID

G=gap.SmallGroup(128,481);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,100,124]);
// Polycyclic

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

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