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G = C42.374D4order 128 = 27

7th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.374D4, Q8.3M4(2), D4.3M4(2), C42.609C23, D4⋊C827C2, Q8⋊C831C2, (C4×D4).16C4, (C4×C8).7C22, (C4×Q8).16C4, C42.62(C2×C4), C4.120(C4○D8), C4⋊C8.194C22, (C22×C4).204D4, C4.20(C2×M4(2)), C42⋊C2.17C4, C42.6C425C2, (C4×D4).265C22, (C4×Q8).252C22, C42.12C412C2, C23.46(C22⋊C4), (C2×C42).165C22, C2.14(C24.4C4), C2.6(C23.24D4), C2.11(C42⋊C22), (C4×C4○D4).3C2, C4⋊C4.181(C2×C4), (C2×C4○D4).15C4, (C2×D4).193(C2×C4), (C2×C4).1451(C2×D4), (C2×Q8).176(C2×C4), (C2×C4).314(C22×C4), (C22×C4).187(C2×C4), (C2×C4).315(C22⋊C4), C22.164(C2×C22⋊C4), SmallGroup(128,220)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.374D4
C1C2C22C2×C4C42C2×C42C4×C4○D4 — C42.374D4
C1C2C2×C4 — C42.374D4
C1C2×C4C2×C42 — C42.374D4
C1C22C22C42 — C42.374D4

Generators and relations for C42.374D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 228 in 123 conjugacy classes, 50 normal (36 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4×C8 [×2], C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C2×C4○D4, D4⋊C8 [×2], Q8⋊C8 [×2], C42.12C4, C42.6C4, C4×C4○D4, C42.374D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C2×M4(2) [×2], C4○D8 [×2], C24.4C4, C23.24D4, C42⋊C22, C42.374D4

Smallest permutation representation of C42.374D4
On 64 points
Generators in S64
(1 42 37 49)(2 54 38 47)(3 44 39 51)(4 56 40 41)(5 46 33 53)(6 50 34 43)(7 48 35 55)(8 52 36 45)(9 24 31 62)(10 59 32 21)(11 18 25 64)(12 61 26 23)(13 20 27 58)(14 63 28 17)(15 22 29 60)(16 57 30 19)
(1 61 33 19)(2 62 34 20)(3 63 35 21)(4 64 36 22)(5 57 37 23)(6 58 38 24)(7 59 39 17)(8 60 40 18)(9 43 27 54)(10 44 28 55)(11 45 29 56)(12 46 30 49)(13 47 31 50)(14 48 32 51)(15 41 25 52)(16 42 26 53)
(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 18 61 8 33 60 19 40)(2 7 62 59 34 39 20 17)(3 58 63 38 35 24 21 6)(4 37 64 23 36 5 22 57)(9 14 43 48 27 32 54 51)(10 47 44 31 28 50 55 13)(11 30 45 49 29 12 56 46)(15 26 41 53 25 16 52 42)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4L4M···4S8A···8H8I8J8K8L
order122222244444···44···48···88888
size111144411112···24···44···48888

38 irreducible representations

dim1111111111222224
type++++++++
imageC1C2C2C2C2C2C4C4C4C4D4D4M4(2)M4(2)C4○D8C42⋊C22
kernelC42.374D4D4⋊C8Q8⋊C8C42.12C4C42.6C4C4×C4○D4C42⋊C2C4×D4C4×Q8C2×C4○D4C42C22×C4D4Q8C4C2
# reps1221112222224482

Matrix representation of C42.374D4 in GL4(𝔽17) generated by

13000
01300
00160
00161
,
1000
0100
00130
00013
,
31400
3300
00115
00716
,
31400
141400
00162
0061
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,16,16,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[3,3,0,0,14,3,0,0,0,0,1,7,0,0,15,16],[3,14,0,0,14,14,0,0,0,0,16,6,0,0,2,1] >;

C42.374D4 in GAP, Magma, Sage, TeX

C_4^2._{374}D_4
% in TeX

G:=Group("C4^2.374D4");
// GroupNames label

G:=SmallGroup(128,220);
// by ID

G=gap.SmallGroup(128,220);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,184,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,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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