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G = C42.528C23order 128 = 27

389th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.528C23, C4.1452+ 1+4, (C8×Q8)⋊17C2, C85D429C2, C4⋊C4.286D4, C42Q1617C2, (C4×SD16)⋊50C2, C4.78(C4○D8), C8.97(C4○D4), C4⋊D8.11C2, (C2×Q8).190D4, D4.2D410C2, C8.12D410C2, C4⋊C4.445C23, C4⋊C8.331C22, (C2×C4).586C24, (C2×C8).379C23, (C4×C8).127C22, Q8.D411C2, (C2×D8).41C22, C4⋊Q8.214C22, C2.40(Q86D4), (C2×D4).281C23, (C4×D4).220C22, (C2×Q16).40C22, (C4×Q8).211C22, (C2×Q8).265C23, C4.Q8.187C22, C2.112(D4○SD16), C41D4.106C22, C4.4D4.87C22, C22.846(C22×D4), D4⋊C4.176C22, C22.53C246C2, Q8⋊C4.191C22, (C2×SD16).175C22, C22.50C2414C2, C2.80(C2×C4○D8), C4.164(C2×C4○D4), (C2×C4).182(C2×D4), SmallGroup(128,2126)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.528C23
C1C2C4C2×C4C42C4×D4C22.50C24 — C42.528C23
C1C2C2×C4 — C42.528C23
C1C22C4×Q8 — C42.528C23
C1C2C2C2×C4 — C42.528C23

Generators and relations for C42.528C23
 G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=e2=a2b2, ab=ba, cac-1=eae-1=a-1b2, ad=da, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ce=ec, de=ed >

Subgroups: 368 in 186 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×9], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], D4 [×9], Q8 [×8], C23 [×3], C42, C42 [×2], C42 [×3], C22⋊C4 [×11], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×2], D8 [×2], SD16 [×8], Q16 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8 [×2], C4×C8, C4×C8 [×2], D4⋊C4, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4.Q8, C42⋊C2, C4×D4, C4×D4 [×2], C4×D4, C4×Q8 [×2], C4×Q8 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4 [×4], C4.4D4, C422C2 [×2], C41D4, C4⋊Q8, C2×D8 [×2], C2×SD16, C2×SD16 [×4], C2×Q16 [×2], C4×SD16, C4×SD16 [×2], C8×Q8, C4⋊D8, C42Q16, D4.2D4 [×2], Q8.D4 [×2], C85D4, C8.12D4 [×2], C22.50C24, C22.53C24, C42.528C23
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, Q86D4, C2×C4○D8, D4○SD16, C42.528C23

Smallest permutation representation of C42.528C23
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 53 12 26)(2 54 9 27)(3 55 10 28)(4 56 11 25)(5 43 39 23)(6 44 40 24)(7 41 37 21)(8 42 38 22)(13 31 50 60)(14 32 51 57)(15 29 52 58)(16 30 49 59)(17 62 48 33)(18 63 45 34)(19 64 46 35)(20 61 47 36)
(1 49 12 16)(2 15 9 52)(3 51 10 14)(4 13 11 50)(5 17 39 48)(6 47 40 20)(7 19 37 46)(8 45 38 18)(21 64 41 35)(22 34 42 63)(23 62 43 33)(24 36 44 61)(25 31 56 60)(26 59 53 30)(27 29 54 58)(28 57 55 32)
(1 18 10 47)(2 19 11 48)(3 20 12 45)(4 17 9 46)(5 58 37 31)(6 59 38 32)(7 60 39 29)(8 57 40 30)(13 43 52 21)(14 44 49 22)(15 41 50 23)(16 42 51 24)(25 62 54 35)(26 63 55 36)(27 64 56 33)(28 61 53 34)
(1 15 10 50)(2 51 11 16)(3 13 12 52)(4 49 9 14)(5 63 37 36)(6 33 38 64)(7 61 39 34)(8 35 40 62)(17 22 46 44)(18 41 47 23)(19 24 48 42)(20 43 45 21)(25 30 54 57)(26 58 55 31)(27 32 56 59)(28 60 53 29)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F4A···4H4I···4O4P4Q4R8A8B8C8D8E···8J
order12222224···44···444488888···8
size11118882···24···488822224···4

35 irreducible representations

dim11111111111222244
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D82+ 1+4D4○SD16
kernelC42.528C23C4×SD16C8×Q8C4⋊D8C42Q16D4.2D4Q8.D4C85D4C8.12D4C22.50C24C22.53C24C4⋊C4C2×Q8C8C4C4C2
# reps13111221211314812

Matrix representation of C42.528C23 in GL4(𝔽17) generated by

13000
01300
00130
0094
,
0100
16000
0010
0001
,
4000
01300
00413
00813
,
3300
31400
0040
0004
,
1000
0100
00116
00216
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,13,9,0,0,0,4],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,13,0,0,0,0,4,8,0,0,13,13],[3,3,0,0,3,14,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,1,2,0,0,16,16] >;

C42.528C23 in GAP, Magma, Sage, TeX

C_4^2._{528}C_2^3
% in TeX

G:=Group("C4^2.528C2^3");
// GroupNames label

G:=SmallGroup(128,2126);
// by ID

G=gap.SmallGroup(128,2126);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,100,346,80,4037,1027,124]);
// Polycyclic

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

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