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G = C42.322D4order 128 = 27

18th non-split extension by C42 of D4 acting via D4/C4=C2

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

Aliases: C42.322D4, (C4×C8).21C4, C4.7(C8.C4), (C22×C4).91Q8, C23.81(C2×Q8), C42.321(C2×C4), C4.82(C4.4D4), C4.C42.9C2, C4.60(C42⋊C2), C2.10(C428C4), C4⋊M4(2).27C2, (C22×C8).480C22, C22.1(C42.C2), (C22×C4).1341C23, (C2×C42).1057C22, (C2×M4(2)).167C22, (C2×C4×C8).19C2, (C2×C4).87(C4⋊C4), (C2×C8).213(C2×C4), C22.99(C2×C4⋊C4), C2.12(C2×C8.C4), (C2×C4).1521(C2×D4), (C2×C4).559(C4○D4), (C2×C4).539(C22×C4), SmallGroup(128,569)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.322D4
C1C2C4C2×C4C22×C4C22×C8C2×C4×C8 — C42.322D4
C1C2C2×C4 — C42.322D4
C1C2×C4C2×C42 — C42.322D4
C1C2C2C22×C4 — C42.322D4

Generators and relations for C42.322D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=a2b2c3 >

Subgroups: 140 in 94 conjugacy classes, 56 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], C23, C42 [×2], C42 [×2], C2×C8 [×4], C2×C8 [×8], M4(2) [×8], C22×C4, C22×C4 [×2], C4×C8 [×4], C4⋊C8 [×4], C2×C42, C22×C8 [×2], C2×M4(2) [×4], C4.C42 [×4], C2×C4×C8, C4⋊M4(2) [×2], C42.322D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C8.C4 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], C428C4, C2×C8.C4 [×2], C42.322D4

Smallest permutation representation of C42.322D4
On 64 points
Generators in S64
(1 34 27 15)(2 35 28 16)(3 36 29 9)(4 37 30 10)(5 38 31 11)(6 39 32 12)(7 40 25 13)(8 33 26 14)(17 60 44 55)(18 61 45 56)(19 62 46 49)(20 63 47 50)(21 64 48 51)(22 57 41 52)(23 58 42 53)(24 59 43 54)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 34 13 38)(10 35 14 39)(11 36 15 40)(12 37 16 33)(17 46 21 42)(18 47 22 43)(19 48 23 44)(20 41 24 45)(49 64 53 60)(50 57 54 61)(51 58 55 62)(52 59 56 63)
(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 58 7 60 5 62 3 64)(2 52 8 54 6 56 4 50)(9 48 15 42 13 44 11 46)(10 20 16 22 14 24 12 18)(17 38 19 36 21 34 23 40)(25 55 31 49 29 51 27 53)(26 59 32 61 30 63 28 57)(33 43 39 45 37 47 35 41)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N8A···8P8Q···8X
order12222244444···48···88···8
size11112211112···22···28···8

44 irreducible representations

dim111112222
type+++++-
imageC1C2C2C2C4D4Q8C4○D4C8.C4
kernelC42.322D4C4.C42C2×C4×C8C4⋊M4(2)C4×C8C42C22×C4C2×C4C4
# reps1412822816

Matrix representation of C42.322D4 in GL4(𝔽17) generated by

4000
01300
0014
00816
,
13000
01300
00160
00016
,
8000
0200
00131
0024
,
01500
15000
001510
00152
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,1,8,0,0,4,16],[13,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[8,0,0,0,0,2,0,0,0,0,13,2,0,0,1,4],[0,15,0,0,15,0,0,0,0,0,15,15,0,0,10,2] >;

C42.322D4 in GAP, Magma, Sage, TeX

C_4^2._{322}D_4
% in TeX

G:=Group("C4^2.322D4");
// GroupNames label

G:=SmallGroup(128,569);
// by ID

G=gap.SmallGroup(128,569);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,58,2019,248,2804,172,124]);
// Polycyclic

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

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