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G = C42.114D4order 128 = 27

96th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.114D4, M4(2).28D4, C22.59(C4×D4), C4.18(C41D4), C41(C4.10D4), C4.79(C4⋊D4), C4.64(C4.4D4), (C4×M4(2)).25C2, C4⋊M4(2).32C2, (C2×C42).320C22, (C22×C4).700C23, C23.198(C22×C4), (C22×Q8).36C22, (C2×M4(2)).209C22, C2.18(C24.3C22), (C2×C4⋊C4).24C4, (C2×C4⋊Q8).11C2, (C2×C4).65(C4○D4), (C2×C4).1352(C2×D4), (C22×C4).24(C2×C4), (C2×C4.10D4).9C2, C2.26(C2×C4.10D4), (C2×C4).258(C22⋊C4), C22.288(C2×C22⋊C4), SmallGroup(128,698)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.114D4
C1C2C4C2×C4C22×C4C2×C42C4×M4(2) — C42.114D4
C1C2C23 — C42.114D4
C1C22C2×C42 — C42.114D4
C1C2C2C22×C4 — C42.114D4

Generators and relations for C42.114D4
 G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b-1c3 >

Subgroups: 260 in 146 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×6], C4 [×6], C22, C22 [×2], C22 [×2], C8 [×6], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], Q8 [×8], C23, C42 [×2], C42 [×2], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×4], M4(2) [×6], C22×C4, C22×C4 [×6], C2×Q8 [×12], C4×C8, C8⋊C4, C4.10D4 [×8], C4⋊C8 [×2], C2×C42, C2×C4⋊C4 [×4], C4⋊Q8 [×4], C2×M4(2) [×4], C22×Q8 [×2], C4×M4(2), C2×C4.10D4 [×4], C4⋊M4(2), C2×C4⋊Q8, C42.114D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C4.10D4 [×4], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C24.3C22, C2×C4.10D4 [×2], C42.114D4

Smallest permutation representation of C42.114D4
On 64 points
Generators in S64
(1 11 27 39)(2 12 28 40)(3 13 29 33)(4 14 30 34)(5 15 31 35)(6 16 32 36)(7 9 25 37)(8 10 26 38)(17 49 45 58)(18 50 46 59)(19 51 47 60)(20 52 48 61)(21 53 41 62)(22 54 42 63)(23 55 43 64)(24 56 44 57)
(1 25 5 29)(2 30 6 26)(3 27 7 31)(4 32 8 28)(9 35 13 39)(10 40 14 36)(11 37 15 33)(12 34 16 38)(17 43 21 47)(18 48 22 44)(19 45 23 41)(20 42 24 46)(49 64 53 60)(50 61 54 57)(51 58 55 62)(52 63 56 59)
(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 49 5 53)(2 63 6 59)(3 55 7 51)(4 61 8 57)(9 19 13 23)(10 44 14 48)(11 17 15 21)(12 42 16 46)(18 40 22 36)(20 38 24 34)(25 60 29 64)(26 56 30 52)(27 58 31 62)(28 54 32 50)(33 43 37 47)(35 41 39 45)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J4K4L4M4N8A···8H8I8J8K8L
order1222224···44444448···88888
size1111222···24488884···48888

32 irreducible representations

dim1111112224
type+++++++-
imageC1C2C2C2C2C4D4D4C4○D4C4.10D4
kernelC42.114D4C4×M4(2)C2×C4.10D4C4⋊M4(2)C2×C4⋊Q8C2×C4⋊C4C42M4(2)C2×C4C4
# reps1141184444

Matrix representation of C42.114D4 in GL6(𝔽17)

0160000
100000
000100
0016000
000001
0000160
,
100000
010000
000100
0016000
0000016
000010
,
100000
010000
000010
000001
000100
0016000
,
100000
0160000
0011000
00101600
0000167
000071

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,10,0,0,0,0,10,16,0,0,0,0,0,0,16,7,0,0,0,0,7,1] >;

C42.114D4 in GAP, Magma, Sage, TeX

C_4^2._{114}D_4
% in TeX

G:=Group("C4^2.114D4");
// GroupNames label

G:=SmallGroup(128,698);
// by ID

G=gap.SmallGroup(128,698);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,100,2019,1018,2028,124]);
// 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=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c^3>;
// generators/relations

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