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G = C42.505C23order 128 = 27

366th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.505C23, C4.262- 1+4, (C8×Q8)⋊12C2, C4⋊C4.278D4, Q83Q83C2, (C4×Q16)⋊16C2, Q8.Q810C2, C42Q1616C2, C4.76(C4○D8), (C4×C8).94C22, (C2×Q8).184D4, C2.62(Q8○D8), C4⋊C8.325C22, C4⋊C4.432C23, (C2×C8).207C23, (C2×C4).556C24, Q8.34(C4○D4), D4⋊Q8.10C2, C4.SD1632C2, (C4×SD16).13C2, C4⋊Q8.185C22, Q8.D4.1C2, C2.64(Q85D4), (C2×D4).268C23, (C4×D4).195C22, (C2×Q8).254C23, (C4×Q8).187C22, C4.Q8.176C22, C2.D8.202C22, Q8⋊C4.19C22, (C2×Q16).142C22, C4.4D4.76C22, C22.816(C22×D4), C42.C2.61C22, D4⋊C4.152C22, (C2×SD16).171C22, C22.50C24.8C2, C42.78C22.2C2, C2.74(C2×C4○D8), C4.257(C2×C4○D4), (C2×C4).176(C2×D4), SmallGroup(128,2096)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.505C23
C1C2C4C2×C4C42C4×Q8Q83Q8 — C42.505C23
C1C2C2×C4 — C42.505C23
C1C22C4×Q8 — C42.505C23
C1C2C2C2×C4 — C42.505C23

Generators and relations for C42.505C23
 G = < a,b,c,d,e | a4=b4=c2=1, d2=a2b2, e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=a2c, ece-1=bc, de=ed >

Subgroups: 280 in 168 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×2], C4 [×13], C22, C22 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], D4 [×2], Q8 [×2], Q8 [×9], C23, C42, C42 [×2], C42 [×5], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×12], C2×C8 [×2], C2×C8 [×2], SD16 [×2], Q16 [×4], C22×C4, C2×D4, C2×Q8 [×2], C2×Q8 [×2], C2×Q8, C4×C8, C4×C8 [×2], D4⋊C4, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4 [×6], C4⋊C8, C4⋊C8 [×2], C4.Q8, C2.D8 [×2], C42⋊C2, C4×D4, C4×Q8 [×2], C4×Q8 [×4], C4×Q8, C22⋊Q8, C4.4D4 [×2], C42.C2 [×2], C42.C2 [×2], C422C2 [×2], C4⋊Q8 [×2], C4⋊Q8, C2×SD16, C2×Q16 [×2], C4×SD16, C4×Q16 [×2], C8×Q8, C42Q16, Q8.D4 [×2], D4⋊Q8, Q8.Q8 [×2], C4.SD16, C42.78C22 [×2], C22.50C24, Q83Q8, C42.505C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C4○D8 [×2], C22×D4, C2×C4○D4, 2- 1+4, Q85D4, C2×C4○D8, Q8○D8, C42.505C23

Smallest permutation representation of C42.505C23
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 23 25 17)(2 24 26 18)(3 21 27 19)(4 22 28 20)(5 15 9 64)(6 16 10 61)(7 13 11 62)(8 14 12 63)(29 37 41 35)(30 38 42 36)(31 39 43 33)(32 40 44 34)(45 53 51 57)(46 54 52 58)(47 55 49 59)(48 56 50 60)
(5 13)(6 14)(7 15)(8 16)(9 62)(10 63)(11 64)(12 61)(17 23)(18 24)(19 21)(20 22)(29 37)(30 38)(31 39)(32 40)(33 43)(34 44)(35 41)(36 42)(45 49)(46 50)(47 51)(48 52)(53 55)(54 56)(57 59)(58 60)
(1 57 27 55)(2 60 28 54)(3 59 25 53)(4 58 26 56)(5 44 11 30)(6 43 12 29)(7 42 9 32)(8 41 10 31)(13 36 64 40)(14 35 61 39)(15 34 62 38)(16 33 63 37)(17 51 21 47)(18 50 22 46)(19 49 23 45)(20 52 24 48)
(1 41 25 29)(2 42 26 30)(3 43 27 31)(4 44 28 32)(5 60 9 56)(6 57 10 53)(7 58 11 54)(8 59 12 55)(13 52 62 46)(14 49 63 47)(15 50 64 48)(16 51 61 45)(17 35 23 37)(18 36 24 38)(19 33 21 39)(20 34 22 40)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D4A···4H4I···4O4P···4T8A8B8C8D8E···8J
order122224···44···44···488888···8
size111182···24···48···822224···4

35 irreducible representations

dim111111111111222244
type++++++++++++++--
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D82- 1+4Q8○D8
kernelC42.505C23C4×SD16C4×Q16C8×Q8C42Q16Q8.D4D4⋊Q8Q8.Q8C4.SD16C42.78C22C22.50C24Q83Q8C4⋊C4C2×Q8Q8C4C4C2
# reps112112121211314812

Matrix representation of C42.505C23 in GL4(𝔽17) generated by

16000
01600
00130
0004
,
0100
16000
0010
0001
,
1000
01600
0010
00016
,
13000
01300
0001
00160
,
5500
51200
0010
0001
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,4],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[13,0,0,0,0,13,0,0,0,0,0,16,0,0,1,0],[5,5,0,0,5,12,0,0,0,0,1,0,0,0,0,1] >;

C42.505C23 in GAP, Magma, Sage, TeX

C_4^2._{505}C_2^3
% in TeX

G:=Group("C4^2.505C2^3");
// GroupNames label

G:=SmallGroup(128,2096);
// by ID

G=gap.SmallGroup(128,2096);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,352,346,80,4037,1027,124]);
// Polycyclic

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

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