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G = C42.90D4order 128 = 27

72nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.90D4, C42.17Q8, C42.627C23, C4⋊C8.17C4, C82C826C2, C4.5(C4.Q8), C4.43(C8○D4), C22⋊C8.15C4, C4.99(C2×SD16), (C22×C4).29Q8, C4⋊C8.267C22, C23.52(C4⋊C4), C42.123(C2×C4), (C4×C8).236C22, (C2×C4).118SD16, (C22×C4).747D4, C22.5(C4.Q8), C4⋊M4(2).23C2, (C2×C42).226C22, C2.5(M4(2).C4), C42.12C4.39C2, C2.5(C42.6C22), (C2×C4⋊C8).19C2, C2.5(C2×C4.Q8), (C2×C4).34(C4⋊C4), (C2×C8).100(C2×C4), C22.84(C2×C4⋊C4), (C2×C4).154(C2×Q8), (C2×C4).1463(C2×D4), (C2×C4).509(C22×C4), (C22×C4).248(C2×C4), SmallGroup(128,302)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.90D4
C1C2C22C2×C4C42C2×C42C42.12C4 — C42.90D4
C1C2C2×C4 — C42.90D4
C1C2×C4C2×C42 — C42.90D4
C1C22C22C42 — C42.90D4

Generators and relations for C42.90D4
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=a2b, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 132 in 84 conjugacy classes, 54 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C4 [×3], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×6], C2×C4 [×4], C2×C4 [×3], C23, C42 [×4], C2×C8 [×4], C2×C8 [×6], M4(2) [×2], C22×C4 [×3], C4×C8 [×2], C22⋊C8 [×2], C4⋊C8 [×6], C4⋊C8 [×2], C2×C42, C22×C8, C2×M4(2), C82C8 [×4], C2×C4⋊C8, C4⋊M4(2), C42.12C4, C42.90D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], SD16 [×4], C22×C4, C2×D4, C2×Q8, C4.Q8 [×4], C2×C4⋊C4, C8○D4 [×2], C2×SD16 [×2], C42.6C22, C2×C4.Q8, M4(2).C4, C42.90D4

Smallest permutation representation of C42.90D4
On 64 points
Generators in S64
(1 3 5 7)(2 45 6 41)(4 47 8 43)(9 19 13 23)(10 12 14 16)(11 21 15 17)(18 20 22 24)(25 58 29 62)(26 28 30 32)(27 60 31 64)(33 35 37 39)(34 52 38 56)(36 54 40 50)(42 44 46 48)(49 51 53 55)(57 59 61 63)
(1 39 42 55)(2 40 43 56)(3 33 44 49)(4 34 45 50)(5 35 46 51)(6 36 47 52)(7 37 48 53)(8 38 41 54)(9 62 17 31)(10 63 18 32)(11 64 19 25)(12 57 20 26)(13 58 21 27)(14 59 22 28)(15 60 23 29)(16 61 24 30)
(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 27 35 9 42 58 51 17)(2 30 36 12 43 61 52 20)(3 25 37 15 44 64 53 23)(4 28 38 10 45 59 54 18)(5 31 39 13 46 62 55 21)(6 26 40 16 47 57 56 24)(7 29 33 11 48 60 49 19)(8 32 34 14 41 63 50 22)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4J4K4L8A···8P8Q8R8S8T
order12222244444···4448···88888
size11112211112···2444···48888

38 irreducible representations

dim11111112222224
type++++++-+-
imageC1C2C2C2C2C4C4D4Q8D4Q8SD16C8○D4M4(2).C4
kernelC42.90D4C82C8C2×C4⋊C8C4⋊M4(2)C42.12C4C22⋊C8C4⋊C8C42C42C22×C4C22×C4C2×C4C4C2
# reps14111441111882

Matrix representation of C42.90D4 in GL4(𝔽17) generated by

0100
16000
0010
00016
,
1000
0100
0040
0004
,
51200
5500
00015
0080
,
01300
13000
0004
0010
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[5,5,0,0,12,5,0,0,0,0,0,8,0,0,15,0],[0,13,0,0,13,0,0,0,0,0,0,1,0,0,4,0] >;

C42.90D4 in GAP, Magma, Sage, TeX

C_4^2._{90}D_4
% in TeX

G:=Group("C4^2.90D4");
// GroupNames label

G:=SmallGroup(128,302);
// by ID

G=gap.SmallGroup(128,302);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,723,1123,136,172]);
// Polycyclic

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

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