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G = C8215C2order 128 = 27

15th semidirect product of C82 and C2 acting faithfully

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C8215C2, C8.11M4(2), C23.12C42, C42.745C23, C4⋊C8.20C4, C8⋊C821C2, C8⋊C4.12C4, C4.55(C8○D4), C22⋊C8.19C4, C42.48(C2×C4), (C2×C4).16C42, C2.6(C4×M4(2)), (C4×C8).306C22, C4.59(C2×M4(2)), (C4×M4(2)).14C2, (C2×M4(2)).22C4, C22.42(C2×C42), C2.8(C82M4(2)), (C2×C42).142C22, C42.12C4.43C2, (C2×C8).119(C2×C4), (C22×C4).172(C2×C4), (C2×C4).582(C22×C4), SmallGroup(128,185)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C8215C2
C1C2C22C2×C4C42C2×C42C42.12C4 — C8215C2
C1C22 — C8215C2
C1C42 — C8215C2
C1C22C22C42 — C8215C2

Generators and relations for C8215C2
 G = < a,b,c | a8=b8=c2=1, ab=ba, cac=ab4, cbc=a4b >

Subgroups: 124 in 95 conjugacy classes, 68 normal (16 characteristic)
C1, C2, C2 [×2], C2, C4 [×6], C4 [×3], C22, C22 [×3], C8 [×4], C8 [×10], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], C23, C42 [×2], C42 [×2], C2×C8 [×12], M4(2) [×4], C22×C4, C22×C4 [×2], C4×C8 [×2], C4×C8 [×4], C8⋊C4 [×2], C22⋊C8 [×4], C4⋊C8 [×4], C2×C42, C2×M4(2) [×2], C82, C8⋊C8, C8⋊C8 [×2], C4×M4(2), C42.12C4 [×2], C8215C2
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×4], C22×C4 [×3], C2×C42, C2×M4(2) [×2], C8○D4 [×4], C4×M4(2), C82M4(2) [×2], C8215C2

Smallest permutation representation of C8215C2
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 42 10 54 31 36 57 21)(2 43 11 55 32 37 58 22)(3 44 12 56 25 38 59 23)(4 45 13 49 26 39 60 24)(5 46 14 50 27 40 61 17)(6 47 15 51 28 33 62 18)(7 48 16 52 29 34 63 19)(8 41 9 53 30 35 64 20)
(2 32)(4 26)(6 28)(8 30)(9 64)(11 58)(13 60)(15 62)(17 21)(18 55)(19 23)(20 49)(22 51)(24 53)(33 43)(34 38)(35 45)(36 40)(37 47)(39 41)(42 46)(44 48)(50 54)(52 56)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D4A···4L4M4N4O8A···8X8Y···8AJ
order122224···44448···88···8
size111141···14442···24···4

56 irreducible representations

dim11111111122
type+++++
imageC1C2C2C2C2C4C4C4C4M4(2)C8○D4
kernelC8215C2C82C8⋊C8C4×M4(2)C42.12C4C8⋊C4C22⋊C8C4⋊C8C2×M4(2)C8C4
# reps113124884816

Matrix representation of C8215C2 in GL4(𝔽17) generated by

0100
4000
00152
0062
,
0100
4000
00150
00015
,
1000
01600
0010
00216
G:=sub<GL(4,GF(17))| [0,4,0,0,1,0,0,0,0,0,15,6,0,0,2,2],[0,4,0,0,1,0,0,0,0,0,15,0,0,0,0,15],[1,0,0,0,0,16,0,0,0,0,1,2,0,0,0,16] >;

C8215C2 in GAP, Magma, Sage, TeX

C_8^2\rtimes_{15}C_2
% in TeX

G:=Group("C8^2:15C2");
// GroupNames label

G:=SmallGroup(128,185);
// by ID

G=gap.SmallGroup(128,185);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,120,1430,387,136,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,c*a*c=a*b^4,c*b*c=a^4*b>;
// generators/relations

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