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G = C42.379D4order 128 = 27

12nd non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.379D4, C23.16C42, C22⋊C811C4, C4.157(C4×D4), C24.49(C2×C4), (C2×C4).23C42, C22.21(C8○D4), C22.50(C2×C42), (C23×C4).224C22, (C22×C8).470C22, (C2×C42).989C22, C23.252(C22×C4), C2.12(C82M4(2)), C22.7C4236C2, (C22×C4).1605C23, C22.50(C42⋊C2), C2.2(C42.7C22), (C2×C4×C8)⋊6C2, (C2×C4⋊C4).46C4, (C2×C8⋊C4)⋊18C2, C2.9(C4×C22⋊C4), (C2×C8).130(C2×C4), (C2×C4).1495(C2×D4), (C2×C22⋊C4).22C4, (C2×C22⋊C8).41C2, (C2×C4).915(C4○D4), (C22×C4).106(C2×C4), (C2×C4).595(C22×C4), (C2×C4).188(C22⋊C4), (C2×C42⋊C2).12C2, C2.2((C22×C8)⋊C2), C22.113(C2×C22⋊C4), SmallGroup(128,482)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.379D4
C1C2C4C2×C4C22×C4C2×C42C2×C4×C8 — C42.379D4
C1C22 — C42.379D4
C1C22×C4 — C42.379D4
C1C2C2C22×C4 — C42.379D4

Generators and relations for C42.379D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 268 in 168 conjugacy classes, 84 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C22 [×10], C8 [×8], 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 [×8], C2×C8 [×8], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C22×C8 [×4], C23×C4, C22.7C42 [×2], C2×C4×C8, C2×C8⋊C4, C2×C22⋊C8 [×2], C2×C42⋊C2, C42.379D4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C8○D4 [×4], C4×C22⋊C4, C82M4(2) [×2], (C22×C8)⋊C2 [×2], C42.7C22 [×2], C42.379D4

Smallest permutation representation of C42.379D4
On 64 points
Generators in S64
(1 64 10 19)(2 61 11 24)(3 58 12 21)(4 63 13 18)(5 60 14 23)(6 57 15 20)(7 62 16 17)(8 59 9 22)(25 34 51 44)(26 39 52 41)(27 36 53 46)(28 33 54 43)(29 38 55 48)(30 35 56 45)(31 40 49 42)(32 37 50 47)
(1 28 5 32)(2 29 6 25)(3 30 7 26)(4 31 8 27)(9 53 13 49)(10 54 14 50)(11 55 15 51)(12 56 16 52)(17 41 21 45)(18 42 22 46)(19 43 23 47)(20 44 24 48)(33 60 37 64)(34 61 38 57)(35 62 39 58)(36 63 40 59)
(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 47 10 37)(2 63 11 18)(3 41 12 39)(4 57 13 20)(5 43 14 33)(6 59 15 22)(7 45 16 35)(8 61 9 24)(17 52 62 26)(19 54 64 28)(21 56 58 30)(23 50 60 32)(25 36 51 46)(27 38 53 48)(29 40 55 42)(31 34 49 44)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4P4Q···4V8A···8P8Q···8X
order12···2224···44···44···48···88···8
size11···1441···12···24···42···24···4

56 irreducible representations

dim111111111222
type+++++++
imageC1C2C2C2C2C2C4C4C4D4C4○D4C8○D4
kernelC42.379D4C22.7C42C2×C4×C8C2×C8⋊C4C2×C22⋊C8C2×C42⋊C2C22⋊C8C2×C22⋊C4C2×C4⋊C4C42C2×C4C22
# reps12112116444416

Matrix representation of C42.379D4 in GL5(𝔽17)

40000
016000
015100
00001
00010
,
10000
013000
001300
000130
000013
,
130000
021500
001500
00002
000150
,
40000
013400
09400
000013
00040

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

C42.379D4 in GAP, Magma, Sage, TeX

C_4^2._{379}D_4
% in TeX

G:=Group("C4^2.379D4");
// GroupNames label

G:=SmallGroup(128,482);
// by ID

G=gap.SmallGroup(128,482);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,100,124]);
// Polycyclic

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

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