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G = C8212C2order 128 = 27

12nd semidirect product of C82 and C2 acting faithfully

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

Aliases: C8212C2, C8.21SD16, C42.655C23, C83Q817C2, C4.3(C4○D8), (C2×C8).222D4, C4.4(C2×SD16), C85D4.12C2, C2.8(C85D4), C4.SD167C2, C4.4D8.3C2, C4⋊Q8.80C22, (C4×C8).429C22, C2.8(C8.12D4), C41D4.42C22, C22.56(C41D4), (C2×C4).712(C2×D4), SmallGroup(128,440)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C8212C2
C1C2C22C2×C4C42C4×C8C82 — C8212C2
C1C22C42 — C8212C2
C1C22C42 — C8212C2
C1C22C22C42 — C8212C2

Generators and relations for C8212C2
 G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a3, cbc=a4b3 >

Subgroups: 240 in 95 conjugacy classes, 40 normal (12 characteristic)
C1, C2, C2 [×2], C2, C4 [×6], C4 [×3], C22, C22 [×3], C8 [×4], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×6], Q8 [×6], C23, C42, C4⋊C4 [×6], C2×C8 [×6], SD16 [×4], C2×D4 [×3], C2×Q8 [×3], C4×C8, C4×C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C41D4, C4⋊Q8, C4⋊Q8 [×2], C2×SD16 [×2], C82, C4.4D8 [×2], C4.SD16 [×2], C85D4, C83Q8, C8212C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, SD16 [×4], C2×D4 [×3], C41D4, C2×SD16 [×2], C4○D8 [×4], C85D4, C8.12D4 [×2], C8212C2

Smallest permutation representation of C8212C2
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 46 38 12 57 25 51 17)(2 47 39 13 58 26 52 18)(3 48 40 14 59 27 53 19)(4 41 33 15 60 28 54 20)(5 42 34 16 61 29 55 21)(6 43 35 9 62 30 56 22)(7 44 36 10 63 31 49 23)(8 45 37 11 64 32 50 24)
(2 4)(3 7)(6 8)(9 41)(10 44)(11 47)(12 42)(13 45)(14 48)(15 43)(16 46)(17 29)(18 32)(19 27)(20 30)(21 25)(22 28)(23 31)(24 26)(33 52)(34 55)(35 50)(36 53)(37 56)(38 51)(39 54)(40 49)(58 60)(59 63)(62 64)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A···4F4G4H4I8A···8X
order122224···44448···8
size1111162···21616162···2

38 irreducible representations

dim111111222
type+++++++
imageC1C2C2C2C2C2D4SD16C4○D8
kernelC8212C2C82C4.4D8C4.SD16C85D4C83Q8C2×C8C8C4
# reps1122116816

Matrix representation of C8212C2 in GL4(𝔽17) generated by

51200
5500
00162
00161
,
13000
01300
0007
0057
,
1000
01600
0010
00116
G:=sub<GL(4,GF(17))| [5,5,0,0,12,5,0,0,0,0,16,16,0,0,2,1],[13,0,0,0,0,13,0,0,0,0,0,5,0,0,7,7],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;

C8212C2 in GAP, Magma, Sage, TeX

C_8^2\rtimes_{12}C_2
% in TeX

G:=Group("C8^2:12C2");
// GroupNames label

G:=SmallGroup(128,440);
// by ID

G=gap.SmallGroup(128,440);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,64,422,268,1123,136,2804,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,c*a*c=a^3,c*b*c=a^4*b^3>;
// generators/relations

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