C12:2D8 - GroupNames

G = C₁₂⋊₂D₈ order 192 = 2⁶·3

2^nd semidirect product of C₁₂ and D₈ acting via D₈/C₄=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₂⋊₂D₈, D₁₂⋊₉D₄, C₄².₇₃D₆, C₄⋊₂(D₄⋊S₃), C₄⋊₁D₄⋊₂S₃, C₃⋊₄(C₄⋊D₈), C₄.₅₃(S₃×D₄), C₆.₅₇(C₂×D₈), (C₄×D₁₂)⋊₂₂C₂, (C₂×D₄).₅₅D₆, C₁₂⋊C₈⋊₃₁C₂, C₁₂.₃₀(C₂×D₄), (C₂×C₁₂).₁₄₇D₄, C₁₂.₇₆(C₄○D₄), C₄.₃(D₄⋊₂S₃), D₄⋊Dic₃⋊₂₂C₂, C₆.₉₄(C₈⋊C₂²), (C₆×D₄).₇₁C₂², C₂.₁₂(D₆⋊₃D₄), C₆.₁₀₃(C₄⋊D₄), (C₄×C₁₂).₁₂₀C₂², (C₂×C₁₂).₃₉₀C₂³, C₂.₁₅(D₁₂⋊₆C₂²), (C₂×D₁₂).₂₄₆C₂², C₄⋊Dic₃.₃₄₄C₂², (C₂×D₄⋊S₃)⋊₁₃C₂, (C₃×C₄⋊₁D₄)⋊₁C₂, C₂.₁₂(C₂×D₄⋊S₃), (C₂×C₆).₅₂₁(C₂×D₄), (C₂×C₃⋊C₈).₁₃₀C₂², (C₂×C₄).₁₈₅(C₃⋊D₄), (C₂×C₄).₄₈₈(C₂²×S₃), C₂².₁₉₄(C₂×C₃⋊D₄), SmallGroup(192,631)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — C₁₂⋊₂D₈

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂×C₁₂ — C₂×D₁₂ — C₄×D₁₂ — C₁₂⋊₂D₈

Lower central

C₃ — C₆ — C₂×C₁₂ — C₁₂⋊₂D₈

Upper central

C₁ — C₂² — C₄² — C₄⋊₁D₄

Generators and relations for C₁₂⋊₂D₈
G = < a,b,c | a¹²=b⁸=c²=1, bab^-1=a^-1, cac=a⁵, cbc=b^-1 >

Subgroups: 448 in 140 conjugacy classes, 45 normal (29 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, Dic₃, C₁₂, C₁₂, C₁₂, D₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, D₈, C₂²×C₄, C₂×D₄, C₂×D₄, C₃⋊C₈, C₄×S₃, D₁₂, D₁₂, C₂×Dic₃, C₂×C₁₂, C₃×D₄, C₂²×S₃, C₂²×C₆, D₄⋊C₄, C₄⋊C₈, C₄×D₄, C₄⋊₁D₄, C₂×D₈, C₂×C₃⋊C₈, C₄⋊Dic₃, D₆⋊C₄, D₄⋊S₃, C₄×C₁₂, S₃×C₂×C₄, C₂×D₁₂, C₆×D₄, C₆×D₄, C₄⋊D₈, C₁₂⋊C₈, D₄⋊Dic₃, C₄×D₁₂, C₂×D₄⋊S₃, C₃×C₄⋊₁D₄, C₁₂⋊₂D₈
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, D₈, C₂×D₄, C₄○D₄, C₃⋊D₄, C₂²×S₃, C₄⋊D₄, C₂×D₈, C₈⋊C₂², D₄⋊S₃, S₃×D₄, D₄⋊₂S₃, C₂×C₃⋊D₄, C₄⋊D₈, C₂×D₄⋊S₃, D₁₂⋊₆C₂², D₆⋊₃D₄, C₁₂⋊₂D₈

Smallest permutation representation of C₁₂⋊₂D₈
►On 96 points

Generators in S₉₆

(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 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)

(1 36 48 50 85 21 78 64)(2 35 37 49 86 20 79 63)(3 34 38 60 87 19 80 62)(4 33 39 59 88 18 81 61)(5 32 40 58 89 17 82 72)(6 31 41 57 90 16 83 71)(7 30 42 56 91 15 84 70)(8 29 43 55 92 14 73 69)(9 28 44 54 93 13 74 68)(10 27 45 53 94 24 75 67)(11 26 46 52 95 23 76 66)(12 25 47 51 96 22 77 65)

(2 6)(3 11)(5 9)(8 12)(13 58)(14 51)(15 56)(16 49)(17 54)(18 59)(19 52)(20 57)(21 50)(22 55)(23 60)(24 53)(25 69)(26 62)(27 67)(28 72)(29 65)(30 70)(31 63)(32 68)(33 61)(34 66)(35 71)(36 64)(37 83)(38 76)(39 81)(40 74)(41 79)(42 84)(43 77)(44 82)(45 75)(46 80)(47 73)(48 78)(86 90)(87 95)(89 93)(92 96)

G:=sub<Sym(96)| (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)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,36,48,50,85,21,78,64)(2,35,37,49,86,20,79,63)(3,34,38,60,87,19,80,62)(4,33,39,59,88,18,81,61)(5,32,40,58,89,17,82,72)(6,31,41,57,90,16,83,71)(7,30,42,56,91,15,84,70)(8,29,43,55,92,14,73,69)(9,28,44,54,93,13,74,68)(10,27,45,53,94,24,75,67)(11,26,46,52,95,23,76,66)(12,25,47,51,96,22,77,65), (2,6)(3,11)(5,9)(8,12)(13,58)(14,51)(15,56)(16,49)(17,54)(18,59)(19,52)(20,57)(21,50)(22,55)(23,60)(24,53)(25,69)(26,62)(27,67)(28,72)(29,65)(30,70)(31,63)(32,68)(33,61)(34,66)(35,71)(36,64)(37,83)(38,76)(39,81)(40,74)(41,79)(42,84)(43,77)(44,82)(45,75)(46,80)(47,73)(48,78)(86,90)(87,95)(89,93)(92,96)>;

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)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,36,48,50,85,21,78,64)(2,35,37,49,86,20,79,63)(3,34,38,60,87,19,80,62)(4,33,39,59,88,18,81,61)(5,32,40,58,89,17,82,72)(6,31,41,57,90,16,83,71)(7,30,42,56,91,15,84,70)(8,29,43,55,92,14,73,69)(9,28,44,54,93,13,74,68)(10,27,45,53,94,24,75,67)(11,26,46,52,95,23,76,66)(12,25,47,51,96,22,77,65), (2,6)(3,11)(5,9)(8,12)(13,58)(14,51)(15,56)(16,49)(17,54)(18,59)(19,52)(20,57)(21,50)(22,55)(23,60)(24,53)(25,69)(26,62)(27,67)(28,72)(29,65)(30,70)(31,63)(32,68)(33,61)(34,66)(35,71)(36,64)(37,83)(38,76)(39,81)(40,74)(41,79)(42,84)(43,77)(44,82)(45,75)(46,80)(47,73)(48,78)(86,90)(87,95)(89,93)(92,96) );

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),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,36,48,50,85,21,78,64),(2,35,37,49,86,20,79,63),(3,34,38,60,87,19,80,62),(4,33,39,59,88,18,81,61),(5,32,40,58,89,17,82,72),(6,31,41,57,90,16,83,71),(7,30,42,56,91,15,84,70),(8,29,43,55,92,14,73,69),(9,28,44,54,93,13,74,68),(10,27,45,53,94,24,75,67),(11,26,46,52,95,23,76,66),(12,25,47,51,96,22,77,65)], [(2,6),(3,11),(5,9),(8,12),(13,58),(14,51),(15,56),(16,49),(17,54),(18,59),(19,52),(20,57),(21,50),(22,55),(23,60),(24,53),(25,69),(26,62),(27,67),(28,72),(29,65),(30,70),(31,63),(32,68),(33,61),(34,66),(35,71),(36,64),(37,83),(38,76),(39,81),(40,74),(41,79),(42,84),(43,77),(44,82),(45,75),(46,80),(47,73),(48,78),(86,90),(87,95),(89,93),(92,96)]])

33 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3	4A	4B	4C	4D	4E	4F	4G	6A	6B	6C	6D	6E	6F	6G	8A	8B	8C	8D	12A	···	12F
order	1	2	2	2	2	2	2	2	3	4	4	4	4	4	4	4	6	6	6	6	6	6	6	8	8	8	8	12	···	12
size	1	1	1	1	8	8	12	12	2	2	2	2	2	4	12	12	2	2	2	8	8	8	8	12	12	12	12	4	···	4

33 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+			+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₆	D₆	D₈	C₄○D₄	C₃⋊D₄	C₈⋊C₂²	D₄⋊S₃	S₃×D₄	D₄⋊₂S₃	D₁₂⋊₆C₂²
kernel	C₁₂⋊₂D₈	C₁₂⋊C₈	D₄⋊Dic₃	C₄×D₁₂	C₂×D₄⋊S₃	C₃×C₄⋊₁D₄	C₄⋊₁D₄	D₁₂	C₂×C₁₂	C₄²	C₂×D₄	C₁₂	C₁₂	C₂×C₄	C₆	C₄	C₄	C₄	C₂
# reps	1	1	2	1	2	1	1	2	2	1	2	4	2	4	1	2	1	1	2

Matrix representation of C₁₂⋊₂D₈ ►in GL₆(𝔽₇₃)

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	1	0	0
0	0	72	72	0	0
0	0	0	0	27	0
0	0	0	0	0	46

41	41	0	0	0	0
16	0	0	0	0	0
0	0	72	0	0	0
0	0	1	1	0	0
0	0	0	0	0	72
0	0	0	0	1	0

1	0	0	0	0	0
72	72	0	0	0	0
0	0	1	0	0	0
0	0	72	72	0	0
0	0	0	0	1	0
0	0	0	0	0	72

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,27,0,0,0,0,0,0,46],[41,16,0,0,0,0,41,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;

C₁₂⋊₂D₈ in GAP, Magma, Sage, TeX

C_{12}\rtimes_2D_8

% in TeX

G:=Group("C12:2D8");

// GroupNames label

G:=SmallGroup(192,631);

// by ID

G=gap.SmallGroup(192,631);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,1123,297,136,6278]);

// Polycyclic

G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^5,c*b*c=b^-1>;

// generators/relations

G = C12⋊2D8 order 192 = 26·3

2nd semidirect product of C12 and D8 acting via D8/C4=C22

G = C₁₂⋊₂D₈ order 192 = 2⁶·3

2^nd semidirect product of C₁₂ and D₈ acting via D₈/C₄=C₂²