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G = C42.47D4order 128 = 27

29th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.47D4, C42.604C23, D4⋊C824C2, Q8⋊C828C2, C22.6C4≀C2, C4⋊D4.3C4, C22⋊Q8.3C4, C4.27(C8○D4), C42.57(C2×C4), C4.4D4.4C4, (C4×D4).3C22, C42.C2.6C4, (C4×Q8).3C22, C4⋊C8.250C22, (C4×M4(2))⋊12C2, (C4×C8).309C22, (C22×C4).657D4, C4.131(C8⋊C22), C4.125(C8.C22), C23.96(C22⋊C4), (C2×C42).160C22, C2.5(C23.36D4), C23.36C23.2C2, (C2×C4⋊C8)⋊4C2, C2.6(C2×C4≀C2), C4⋊C4.50(C2×C4), (C2×D4).50(C2×C4), (C2×Q8).45(C2×C4), (C2×C4).1137(C2×D4), (C2×C4).77(C22⋊C4), (C2×C4).309(C22×C4), (C22×C4).182(C2×C4), C22.159(C2×C22⋊C4), C2.15((C22×C8)⋊C2), SmallGroup(128,215)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.47D4
C1C2C22C2×C4C42C2×C42C23.36C23 — C42.47D4
C1C2C2×C4 — C42.47D4
C1C2×C4C2×C42 — C42.47D4
C1C22C22C42 — C42.47D4

Generators and relations for C42.47D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 220 in 117 conjugacy classes, 48 normal (36 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×5], C8 [×6], C2×C4 [×6], C2×C4 [×11], D4 [×4], Q8 [×2], C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×6], M4(2) [×4], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8 [×2], C8⋊C4, C4⋊C8 [×2], C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C22×C8, C2×M4(2), D4⋊C8 [×2], Q8⋊C8 [×2], C4×M4(2), C2×C4⋊C8, C23.36C23, C42.47D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C8○D4 [×2], C8⋊C22, C8.C22, (C22×C8)⋊C2, C23.36D4, C2×C4≀C2, C42.47D4

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4J4K4L4M4N4O8A···8P
order122222244444···4444448···8
size111122811112···2448884···4

38 irreducible representations

dim1111111111222244
type+++++++++-
imageC1C2C2C2C2C2C4C4C4C4D4D4C8○D4C4≀C2C8⋊C22C8.C22
kernelC42.47D4D4⋊C8Q8⋊C8C4×M4(2)C2×C4⋊C8C23.36C23C4⋊D4C22⋊Q8C4.4D4C42.C2C42C22×C4C4C22C4C4
# reps1221112222228811

Matrix representation of C42.47D4 in GL4(𝔽17) generated by

13200
1400
00116
00216
,
4000
0400
0040
0004
,
81300
0900
001211
00120
,
81300
15900
001211
00125
G:=sub<GL(4,GF(17))| [13,1,0,0,2,4,0,0,0,0,1,2,0,0,16,16],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[8,0,0,0,13,9,0,0,0,0,12,12,0,0,11,0],[8,15,0,0,13,9,0,0,0,0,12,12,0,0,11,5] >;

C42.47D4 in GAP, Magma, Sage, TeX

C_4^2._{47}D_4
% in TeX

G:=Group("C4^2.47D4");
// GroupNames label

G:=SmallGroup(128,215);
// by ID

G=gap.SmallGroup(128,215);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1059,1123,570,136,172]);
// 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=a^-1*b^2,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations

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