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G = C42.305D4order 128 = 27

1st non-split extension by C42 of D4 acting via D4/C4=C2

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

Aliases: C42.305D4, C42.615C23, D4⋊C831C2, Q8⋊C835C2, C22.3C4≀C2, C4.3(C8○D4), C4⋊D4.6C4, C22⋊Q8.6C4, C4.4D4.5C4, (C4×D4).6C22, C42.C2.7C4, (C4×Q8).6C22, C4.121(C4○D8), C4⋊C8.199C22, C42.259(C2×C4), (C4×C8).365C22, (C22×C4).540D4, C42.6C427C2, C23.99(C22⋊C4), (C2×C42).1037C22, C2.7(C23.24D4), C23.36C23.3C2, (C2×C4×C8)⋊4C2, C2.14(C2×C4≀C2), C4⋊C4.55(C2×C4), (C2×D4).53(C2×C4), (C2×Q8).48(C2×C4), (C2×C4).1143(C2×D4), (C2×C4).320(C22×C4), (C22×C4).398(C2×C4), (C2×C4).169(C22⋊C4), C22.170(C2×C22⋊C4), C2.20((C22×C8)⋊C2), SmallGroup(128,226)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.305D4
C1C2C22C2×C4C42C2×C42C23.36C23 — C42.305D4
C1C2C2×C4 — C42.305D4
C1C2×C4C2×C42 — C42.305D4
C1C22C22C42 — C42.305D4

Generators and relations for C42.305D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 212 in 115 conjugacy classes, 48 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], 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 [×8], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8 [×2], C4×C8, C8⋊C4, C22⋊C8, C4⋊C8 [×2], C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C22×C8, D4⋊C8 [×2], Q8⋊C8 [×2], C2×C4×C8, C42.6C4, C23.36C23, C42.305D4
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], C4○D8 [×2], (C22×C8)⋊C2, C23.24D4, C2×C4≀C2, C42.305D4

Smallest permutation representation of C42.305D4
On 64 points
Generators in S64
(1 43 17 35)(2 44 18 36)(3 45 19 37)(4 46 20 38)(5 47 21 39)(6 48 22 40)(7 41 23 33)(8 42 24 34)(9 28 50 63)(10 29 51 64)(11 30 52 57)(12 31 53 58)(13 32 54 59)(14 25 55 60)(15 26 56 61)(16 27 49 62)
(1 30 21 61)(2 31 22 62)(3 32 23 63)(4 25 24 64)(5 26 17 57)(6 27 18 58)(7 28 19 59)(8 29 20 60)(9 45 54 33)(10 46 55 34)(11 47 56 35)(12 48 49 36)(13 41 50 37)(14 42 51 38)(15 43 52 39)(16 44 53 40)
(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 29 30 20 21 60 61 8)(2 19 31 59 22 7 62 28)(3 58 32 6 23 27 63 18)(4 5 25 26 24 17 64 57)(9 48 45 49 54 36 33 12)(10 56 46 35 55 11 34 47)(13 44 41 53 50 40 37 16)(14 52 42 39 51 15 38 43)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4N4O4P4Q8A···8P8Q8R8S8T
order122222244444···44448···88888
size111122811112···28882···28888

44 irreducible representations

dim111111111122222
type++++++++
imageC1C2C2C2C2C2C4C4C4C4D4D4C8○D4C4○D8C4≀C2
kernelC42.305D4D4⋊C8Q8⋊C8C2×C4×C8C42.6C4C23.36C23C4⋊D4C22⋊Q8C4.4D4C42.C2C42C22×C4C4C4C22
# reps122111222222888

Matrix representation of C42.305D4 in GL4(𝔽17) generated by

161500
0100
00130
00013
,
4000
0400
0010
0001
,
2000
0200
00014
00611
,
2000
151500
00113
00116
G:=sub<GL(4,GF(17))| [16,0,0,0,15,1,0,0,0,0,13,0,0,0,0,13],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[2,0,0,0,0,2,0,0,0,0,0,6,0,0,14,11],[2,15,0,0,0,15,0,0,0,0,11,11,0,0,3,6] >;

C42.305D4 in GAP, Magma, Sage, TeX

C_4^2._{305}D_4
% in TeX

G:=Group("C4^2.305D4");
// GroupNames label

G:=SmallGroup(128,226);
// by ID

G=gap.SmallGroup(128,226);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,723,184,1123,570,136,172]);
// 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,a*c=c*a,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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