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G = C42.284D4order 128 = 27

266th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.284D4, C42.734C23, C4.562- 1+4, Q8.Q82C2, D4.Q82C2, C8.5Q83C2, C4⋊C8.315C22, C4⋊C4.171C23, (C2×C4).430C24, (C4×C8).115C22, (C2×C8).334C23, (C22×C4).173D4, C23.701(C2×D4), C2.D8.39C22, C4.Q8.87C22, (C2×D4).176C23, D4⋊C4.4C22, (C4×D4).114C22, C22.34(C4○D8), C23.48D46C2, (C4×Q8).111C22, (C2×Q8).164C23, C22.D8.2C2, C42.12C438C2, C4⋊D4.199C22, C23.47D430C2, C22⋊C8.182C22, (C2×C42).891C22, C23.46D4.5C2, C22.690(C22×D4), C22⋊Q8.204C22, C2.61(D8⋊C22), (C22×C4).1095C23, C42.78C229C2, Q8⋊C4.105C22, C4.4D4.158C22, C42.C2.131C22, C23.36C23.25C2, C2.78(C23.38C23), C2.47(C2×C4○D8), (C2×C4).555(C2×D4), (C2×C42.C2)⋊36C2, (C2×C4⋊C4).649C22, SmallGroup(128,1964)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.284D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×C42.C2 — C42.284D4
C1C2C2×C4 — C42.284D4
C1C22C2×C42 — C42.284D4
C1C2C2C2×C4 — C42.284D4

Generators and relations for C42.284D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=a2b2c3 >

Subgroups: 292 in 167 conjugacy classes, 86 normal (42 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×6], C2×C4 [×18], D4 [×4], Q8 [×2], C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2 [×4], C42.C2 [×2], C422C2, C42.12C4, D4.Q8 [×2], Q8.Q8 [×2], C22.D8, C23.46D4, C23.47D4, C23.48D4, C42.78C22 [×2], C8.5Q8 [×2], C2×C42.C2, C23.36C23, C42.284D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C22×D4, 2- 1+4 [×2], C23.38C23, C2×C4○D8, D8⋊C22, C42.284D4

Smallest permutation representation of C42.284D4
On 64 points
Generators in S64
(1 45 29 63)(2 46 30 64)(3 47 31 57)(4 48 32 58)(5 41 25 59)(6 42 26 60)(7 43 27 61)(8 44 28 62)(9 52 37 22)(10 53 38 23)(11 54 39 24)(12 55 40 17)(13 56 33 18)(14 49 34 19)(15 50 35 20)(16 51 36 21)
(1 13 5 9)(2 34 6 38)(3 15 7 11)(4 36 8 40)(10 30 14 26)(12 32 16 28)(17 48 21 44)(18 59 22 63)(19 42 23 46)(20 61 24 57)(25 37 29 33)(27 39 31 35)(41 52 45 56)(43 54 47 50)(49 60 53 64)(51 62 55 58)
(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)
(2 28)(3 7)(4 26)(6 32)(8 30)(9 33)(10 12)(11 39)(13 37)(14 16)(15 35)(17 49)(19 55)(20 24)(21 53)(23 51)(27 31)(34 36)(38 40)(41 63)(42 44)(43 61)(45 59)(46 48)(47 57)(50 54)(58 64)(60 62)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A···4H4I4J4K···4Q8A···8H
order12222224···4444···48···8
size11112282···2448···84···4

32 irreducible representations

dim11111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D82- 1+4D8⋊C22
kernelC42.284D4C42.12C4D4.Q8Q8.Q8C22.D8C23.46D4C23.47D4C23.48D4C42.78C22C8.5Q8C2×C42.C2C23.36C23C42C22×C4C22C4C2
# reps11221111221122822

Matrix representation of C42.284D4 in GL6(𝔽17)

040000
1300000
004000
000400
000040
000004
,
010000
1600000
00016015
001020
000001
0000160
,
1430000
14140000
00101699
0011089
007171
00167167
,
100000
0160000
001000
0001600
00160160
000101

G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,2,0,16,0,0,15,0,1,0],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,10,1,7,16,0,0,16,10,1,7,0,0,9,8,7,16,0,0,9,9,1,7],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,16,0,0,0,0,16,0,1,0,0,0,0,16,0,0,0,0,0,0,1] >;

C42.284D4 in GAP, Magma, Sage, TeX

C_4^2._{284}D_4
% in TeX

G:=Group("C4^2.284D4");
// GroupNames label

G:=SmallGroup(128,1964);
// by ID

G=gap.SmallGroup(128,1964);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,100,675,4037,1027,124]);
// Polycyclic

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

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