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G = C823C2order 128 = 27

3rd semidirect product of C82 and C2 acting faithfully

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

Aliases: C823C2, C42.658C23, C4.6(C4○D8), (C2×C8).225D4, C4.SD168C2, C4.4D8.4C2, C4⋊Q8.83C22, (C4×C8).371C22, C41D4.44C22, C2.11(C8.12D4), C22.59(C41D4), (C2×C4).715(C2×D4), SmallGroup(128,443)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C823C2
C1C2C22C2×C4C42C4×C8C82 — C823C2
C1C22C42 — C823C2
C1C22C42 — C823C2
C1C22C22C42 — C823C2

Generators and relations for C823C2
 G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a-1b4, cbc=a4b3 >

Subgroups: 224 in 89 conjugacy classes, 36 normal (6 characteristic)
C1, C2 [×3], C2, C4 [×6], C4 [×3], C22, C22 [×3], C8 [×6], C2×C4 [×3], C2×C4 [×3], D4 [×6], Q8 [×6], C23, C42, C4⋊C4 [×6], C2×C8 [×6], C2×D4 [×3], C2×Q8 [×3], C4×C8 [×3], D4⋊C4 [×6], Q8⋊C4 [×6], C41D4, C4⋊Q8 [×3], C82, C4.4D8 [×3], C4.SD16 [×3], C823C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C41D4, C4○D8 [×6], C8.12D4 [×3], C823C2

Smallest permutation representation of C823C2
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 11 57 29 18 41 34 55)(2 12 58 30 19 42 35 56)(3 13 59 31 20 43 36 49)(4 14 60 32 21 44 37 50)(5 15 61 25 22 45 38 51)(6 16 62 26 23 46 39 52)(7 9 63 27 24 47 40 53)(8 10 64 28 17 48 33 54)
(2 17)(3 7)(4 23)(6 21)(8 19)(9 27)(10 52)(11 25)(12 50)(13 31)(14 56)(15 29)(16 54)(20 24)(26 48)(28 46)(30 44)(32 42)(33 35)(34 57)(36 63)(37 39)(38 61)(40 59)(41 51)(43 49)(45 55)(47 53)(58 64)(60 62)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111122
type+++++
imageC1C2C2C2D4C4○D8
kernelC823C2C82C4.4D8C4.SD16C2×C8C4
# reps1133624

Matrix representation of C823C2 in GL4(𝔽17) generated by

01100
31100
00130
00013
,
13000
01300
00010
001210
,
1000
11600
0010
00116
G:=sub<GL(4,GF(17))| [0,3,0,0,11,11,0,0,0,0,13,0,0,0,0,13],[13,0,0,0,0,13,0,0,0,0,0,12,0,0,10,10],[1,1,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;

C823C2 in GAP, Magma, Sage, TeX

C_8^2\rtimes_3C_2
% in TeX

G:=Group("C8^2:3C2");
// GroupNames label

G:=SmallGroup(128,443);
// by ID

G=gap.SmallGroup(128,443);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,736,422,268,1123,136,2804,172]);
// Polycyclic

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

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