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G = C42.324D4order 128 = 27

20th non-split extension by C42 of D4 acting via D4/C4=C2

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

Aliases: C42.324D4, (C4×C8).27C4, C8.20(C4⋊C4), (C2×C8).49Q8, (C2×C8).267D4, C4.46(C4⋊Q8), C41(C8.C4), C22.1(C4⋊Q8), (C22×C4).92Q8, C23.83(C2×Q8), C4.46(C41D4), C42.326(C2×C4), C2.9(C429C4), C4⋊M4(2).29C2, (C22×C8).550C22, (C22×C4).1347C23, (C2×C42).1062C22, (C2×M4(2)).169C22, (C2×C4×C8).45C2, C4.37(C2×C4⋊C4), (C2×C8).235(C2×C4), (C2×C4).732(C2×D4), (C2×C4).197(C2×Q8), (C2×C4).133(C4⋊C4), C2.13(C2×C8.C4), C22.106(C2×C4⋊C4), (C2×C8.C4).11C2, (C2×C4).546(C22×C4), SmallGroup(128,580)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.324D4
C1C2C4C2×C4C22×C4C2×C42C2×C4×C8 — C42.324D4
C1C2C2×C4 — C42.324D4
C1C2×C4C2×C42 — C42.324D4
C1C2C2C22×C4 — C42.324D4

Generators and relations for C42.324D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 156 in 110 conjugacy classes, 72 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], C23, C42 [×2], C42 [×2], C2×C8 [×12], C2×C8 [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C4×C8 [×2], C4×C8 [×2], C4⋊C8 [×4], C8.C4 [×8], C2×C42, C22×C8 [×2], C2×M4(2) [×4], C2×C4×C8, C4⋊M4(2) [×2], C2×C8.C4 [×4], C42.324D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×6], C23, C4⋊C4 [×12], C22×C4, C2×D4 [×3], C2×Q8 [×3], C8.C4 [×4], C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], C429C4, C2×C8.C4 [×2], C42.324D4

Smallest permutation representation of C42.324D4
On 64 points
Generators in S64
(1 40 27 10)(2 33 28 11)(3 34 29 12)(4 35 30 13)(5 36 31 14)(6 37 32 15)(7 38 25 16)(8 39 26 9)(17 49 46 62)(18 50 47 63)(19 51 48 64)(20 52 41 57)(21 53 42 58)(22 54 43 59)(23 55 44 60)(24 56 45 61)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(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 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 49 3 55 5 53 7 51)(2 52 4 50 6 56 8 54)(9 43 11 41 13 47 15 45)(10 46 12 44 14 42 16 48)(17 34 23 36 21 38 19 40)(18 37 24 39 22 33 20 35)(25 64 27 62 29 60 31 58)(26 59 28 57 30 63 32 61)

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim1111122222
type++++++--
imageC1C2C2C2C4D4D4Q8Q8C8.C4
kernelC42.324D4C2×C4×C8C4⋊M4(2)C2×C8.C4C4×C8C42C2×C8C2×C8C22×C4C4
# reps11248244216

Matrix representation of C42.324D4 in GL4(𝔽17) generated by

4000
01300
00115
00116
,
4000
0400
0010
0001
,
9000
01500
0010
0001
,
01500
2000
0025
001315
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,1,1,0,0,15,16],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,15,0,0,0,0,1,0,0,0,0,1],[0,2,0,0,15,0,0,0,0,0,2,13,0,0,5,15] >;

C42.324D4 in GAP, Magma, Sage, TeX

C_4^2._{324}D_4
% in TeX

G:=Group("C4^2.324D4");
// GroupNames label

G:=SmallGroup(128,580);
// by ID

G=gap.SmallGroup(128,580);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,100,2019,248,124]);
// Polycyclic

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

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