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G = C42.282D4order 128 = 27

264th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.282D4, C42.415C23, C4.542- 1+4, C4.Q168C2, C82Q814C2, (C2×C4).26Q16, C4.48(C2×Q16), (C4×C8).78C22, C4⋊C4.169C23, C4⋊C8.289C22, C4.30(C8⋊C22), (C2×C4).428C24, (C2×C8).167C23, C4.SD1613C2, (C22×C4).511D4, C23.700(C2×D4), C4⋊Q8.312C22, C2.16(C22×Q16), C22.19(C2×Q16), C2.D8.37C22, (C4×Q8).109C22, (C2×Q8).162C23, Q8⋊C4.5C22, C22⋊C8.181C22, (C2×C42).889C22, C23.48D4.2C2, C22.688(C22×D4), C22⋊Q8.202C22, C42.12C4.36C2, (C22×C4).1093C23, C23.37C23.39C2, C2.76(C23.38C23), (C2×C4⋊Q8).55C2, (C2×C4).871(C2×D4), C2.61(C2×C8⋊C22), (C2×C4⋊C4).648C22, SmallGroup(128,1962)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.282D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×C4⋊Q8 — C42.282D4
C1C2C2×C4 — C42.282D4
C1C22C2×C42 — C42.282D4
C1C2C2C2×C4 — C42.282D4

Generators and relations for C42.282D4
 G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=a2c3 >

Subgroups: 316 in 180 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×16], Q8 [×14], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×6], C4⋊C4 [×13], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×2], C2×Q8 [×9], C4×C8 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C2.D8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×Q8, C42.12C4, C4.Q16 [×4], C23.48D4 [×4], C4.SD16 [×2], C82Q8 [×2], C2×C4⋊Q8, C23.37C23, C42.282D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C24, C2×Q16 [×6], C8⋊C22 [×2], C22×D4, 2- 1+4 [×2], C23.38C23, C22×Q16, C2×C8⋊C22, C42.282D4

Smallest permutation representation of C42.282D4
On 64 points
Generators in S64
(1 21 30 44)(2 22 31 45)(3 23 32 46)(4 24 25 47)(5 17 26 48)(6 18 27 41)(7 19 28 42)(8 20 29 43)(9 57 55 38)(10 58 56 39)(11 59 49 40)(12 60 50 33)(13 61 51 34)(14 62 52 35)(15 63 53 36)(16 64 54 37)
(1 32 5 28)(2 8 6 4)(3 26 7 30)(9 49 13 53)(10 16 14 12)(11 51 15 55)(17 42 21 46)(18 24 22 20)(19 44 23 48)(25 31 29 27)(33 39 37 35)(34 63 38 59)(36 57 40 61)(41 47 45 43)(50 56 54 52)(58 64 62 60)
(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 50 5 54)(2 15 6 11)(3 56 7 52)(4 13 8 9)(10 28 14 32)(12 26 16 30)(17 64 21 60)(18 40 22 36)(19 62 23 58)(20 38 24 34)(25 51 29 55)(27 49 31 53)(33 48 37 44)(35 46 39 42)(41 59 45 63)(43 57 47 61)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J4K···4R8A···8H
order1222224···4444···48···8
size1111222···2448···84···4

32 irreducible representations

dim1111111122244
type++++++++++-+-
imageC1C2C2C2C2C2C2C2D4D4Q16C8⋊C222- 1+4
kernelC42.282D4C42.12C4C4.Q16C23.48D4C4.SD16C82Q8C2×C4⋊Q8C23.37C23C42C22×C4C2×C4C4C4
# reps1144221122822

Matrix representation of C42.282D4 in GL6(𝔽17)

010000
1600000
001000
000100
000010
000001
,
0160000
100000
0011500
0011600
000101
00161160
,
3140000
330000
001020
001011
00016160
0000160
,
1250000
550000
00160150
0000161
001010
0011610

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,1,0,16,0,0,15,16,1,1,0,0,0,0,0,16,0,0,0,0,1,0],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,1,1,0,0,0,0,0,0,16,0,0,0,2,1,16,16,0,0,0,1,0,0],[12,5,0,0,0,0,5,5,0,0,0,0,0,0,16,0,1,1,0,0,0,0,0,16,0,0,15,16,1,1,0,0,0,1,0,0] >;

C42.282D4 in GAP, Magma, Sage, TeX

C_4^2._{282}D_4
% in TeX

G:=Group("C4^2.282D4");
// GroupNames label

G:=SmallGroup(128,1962);
// by ID

G=gap.SmallGroup(128,1962);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,436,675,4037,1027,124]);
// Polycyclic

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

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