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G = C42.21C23order 128 = 27

21st non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.21C23, (C2×Q8)⋊7Q8, Q8.5(C2×Q8), C4⋊C4.342D4, Q8⋊Q83C2, Q8.Q814C2, C4.Q1620C2, C4⋊C8.45C22, C4⋊C4.45C23, (C2×C8).29C23, C2.13(Q8○D8), C4.33(C22×Q8), (C2×C4).280C24, C22⋊C4.143D4, C23.449(C2×D4), C4⋊Q8.102C22, C4.68(C22⋊Q8), (C4×Q8).66C22, C2.21(D4○SD16), (C2×Q8).368C23, C4.Q8.148C22, C2.D8.167C22, C42.C2.9C22, (C22×C8).182C22, (C22×C4).999C23, Q8⋊C4.27C22, C23.25D4.6C2, C23.38D4.3C2, C22.540(C22×D4), C22.11(C22⋊Q8), M4(2)⋊C4.11C2, (C2×M4(2)).69C22, C42.6C22.3C2, (C22×Q8).288C22, C42⋊C2.119C22, C23.32C23.5C2, C23.41C23.7C2, C4.90(C2×C4○D4), (C2×C4).482(C2×D4), (C2×C4).104(C2×Q8), C2.61(C2×C22⋊Q8), (C2×C4).482(C4○D4), (C2×C4⋊C4).606C22, (C2×Q8⋊C4).26C2, SmallGroup(128,1814)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.21C23
C1C2C4C2×C4C22×C4C22×Q8C23.32C23 — C42.21C23
C1C2C2×C4 — C42.21C23
C1C22C42⋊C2 — C42.21C23
C1C2C2C2×C4 — C42.21C23

Generators and relations for C42.21C23
 G = < a,b,c,d,e | a4=b4=1, c2=e2=a2b2, d2=b2, ab=ba, cac-1=a-1b2, ad=da, eae-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, de=ed >

Subgroups: 292 in 181 conjugacy classes, 100 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×14], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×13], Q8 [×4], Q8 [×8], C23, C42 [×2], C42 [×6], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×11], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×5], Q8⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×2], C4.Q8 [×2], C2.D8 [×2], C2.D8 [×2], C2×C4⋊C4, C42⋊C2 [×3], C42⋊C2 [×2], C4×Q8 [×4], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C42.C2, C4⋊Q8 [×2], C4⋊Q8, C22×C8, C2×M4(2), C22×Q8, C2×Q8⋊C4, C23.38D4, C42.6C22, C23.25D4, M4(2)⋊C4, Q8⋊Q8 [×2], C4.Q16 [×2], Q8.Q8 [×4], C23.32C23, C23.41C23, C42.21C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, D4○SD16, Q8○D8, C42.21C23

Smallest permutation representation of C42.21C23
On 64 points
Generators in S64
(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 26 50)(2 30 27 51)(3 31 28 52)(4 32 25 49)(5 60 61 53)(6 57 62 54)(7 58 63 55)(8 59 64 56)(9 40 13 35)(10 37 14 36)(11 38 15 33)(12 39 16 34)(17 48 24 43)(18 45 21 44)(19 46 22 41)(20 47 23 42)
(1 53 28 58)(2 59 25 54)(3 55 26 60)(4 57 27 56)(5 31 63 50)(6 51 64 32)(7 29 61 52)(8 49 62 30)(9 45 15 42)(10 43 16 46)(11 47 13 44)(12 41 14 48)(17 39 22 36)(18 33 23 40)(19 37 24 34)(20 35 21 38)
(1 9 26 13)(2 10 27 14)(3 11 28 15)(4 12 25 16)(5 44 61 45)(6 41 62 46)(7 42 63 47)(8 43 64 48)(17 59 24 56)(18 60 21 53)(19 57 22 54)(20 58 23 55)(29 35 50 40)(30 36 51 37)(31 33 52 38)(32 34 49 39)
(1 27 28 4)(2 3 25 26)(5 62 63 8)(6 7 64 61)(9 14 15 12)(10 11 16 13)(17 18 22 23)(19 20 24 21)(29 51 52 32)(30 31 49 50)(33 39 40 36)(34 35 37 38)(41 42 48 45)(43 44 46 47)(53 57 58 56)(54 55 59 60)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4P4Q4R4S4T8A8B8C8D8E8F
order12222244444···44444888888
size11112222224···48888444488

32 irreducible representations

dim11111111111222244
type+++++++++++++--
imageC1C2C2C2C2C2C2C2C2C2C2D4D4Q8C4○D4D4○SD16Q8○D8
kernelC42.21C23C2×Q8⋊C4C23.38D4C42.6C22C23.25D4M4(2)⋊C4Q8⋊Q8C4.Q16Q8.Q8C23.32C23C23.41C23C22⋊C4C4⋊C4C2×Q8C2×C4C2C2
# reps11111122411224422

Matrix representation of C42.21C23 in GL6(𝔽17)

010000
1600000
00160161
000001
00216116
0001600
,
100000
010000
00116016
00216116
000001
0000160
,
1100000
10160000
007404
00510116
0000136
000064
,
1600000
0160000
009101
0038107
0000167
000071
,
0160000
100000
001000
000001
00151161
000100

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,2,0,0,0,0,0,16,16,0,0,16,0,1,0,0,0,1,1,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,16,16,0,0,0,0,0,1,0,16,0,0,16,16,1,0],[1,10,0,0,0,0,10,16,0,0,0,0,0,0,7,5,0,0,0,0,4,10,0,0,0,0,0,11,13,6,0,0,4,6,6,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,9,3,0,0,0,0,1,8,0,0,0,0,0,10,16,7,0,0,1,7,7,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,15,0,0,0,0,0,1,1,0,0,0,0,16,0,0,0,0,1,1,0] >;

C42.21C23 in GAP, Magma, Sage, TeX

C_4^2._{21}C_2^3
% in TeX

G:=Group("C4^2.21C2^3");
// GroupNames label

G:=SmallGroup(128,1814);
// by ID

G=gap.SmallGroup(128,1814);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,352,1018,4037,1027,124]);
// Polycyclic

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

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